Karapetyants Drakin The Structure of Matter Mir 1978 PDF

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THE STRUCTURE
K UMPHTMTS. S.IHKIN
TRANSLATED FROM THE RUSSIAN

by Y. NADLER and G. KITTELL
MIR PUBLISHERS *MOSCOW

First published 1974
Second printing 1978
Revised from the 1970 second Russian edition
Treated in this book arc present-day ideas on the structure

of atoms, molecules and crystals, as well as the nature of che
mical bonds. It is supposed that the reader is acquainted with
chemistry within the scope of the secondary-school course and
that he has some notion of differential and integral calculus.
The study of this book can precede the learning of a higher-
school course in chemistry; it will be conducive in widening the
student’s knowledge of inorganic chemistry as well as in study
ing more deeply organic and analytical chemistry.
The book is designed for students in chemical engineering,
polytechnical and other institutes of learning in which inorganic
and organic chemistry are studied. It can be found useful for
engineers, technicians and scientific workers whose industrial
or research work is associated with chemical technology, metal
lurgy, biochemistry, or geochemistry.
Ha amjiuucKOM statute
© English translation, Mir Publishers, 1978

CONTENTS
P r e f a c e ................................................................................................................. 9
Part I. ATOMIC STRUCTURE
Chapter One. In trod u ction .............................................................................. 11
1.1. A t o m s ..................................................................................... 11
1.2. The Avogadro Number ................................................................. 12
1.3. Mass and Size of A t o m s ..................................................... 15
1.4. The Constituents of an Atom: Electrons and the Nucleus 17
Chapter Two. Atomic S p e c tr a .......................................................................... 19
2.1. Principle of Operation of Spectrographs; Kinds of Spectra 19
2.2. The Atomic Spectrum of H yd rogen................................ 20
2.3. The Spectra of Other E le m e n ts......................................... 21
2.4. The Concept of Light Q u an tu m .................................... 22
2.5. History of the Development of the Concepts of Atomic Struc
ture ........................................................................................................ 25
Chapter Three. The Wave Properties of Material P a rticles........................ 28
3.1. Dual Nature of L i g h t ..................................................................... 28
3.2. The Law of Interdependence of Mass and E n e r g y .................... 30
3.3. Compton Effect . . . ......... ........................................................... 31
3.4. De Broglie W a v e s ......................................................................... 33
3.5. Quantum Mechanics;the Schrodinger E q u a tio n ........................ 35
Chapter Four. Quantum-Mechanical Explanation of Atomic Structure • 39
4.1. Solution of the Schrodinger Equation for the One-Dimensional
Square-Well M o d el............................................................................. 39
4.2. Three-Dimensional Square-WellM o d e l ........................................ 43
4.3. Quantum-Mechanical Explanation of Structure of Hydrogen
A t o m ......................................................................................... 46
4.4. Quantum Numbers of Electrons inA t o m s ................................. 50
4.5. Many-Electron Atoms ...................................................................... 54
4.6. Origination of Spectra..................................................................... 57
4.7. Energy Characteristics of Atoms: Ionization Energy and Ele
ctron A f f in it y ..................................................................................... 60
Part II.'MENDELEEV’S PERIODIC LAW AND THE STRUCTURE
OF ATOMS OF ELEMENTS
Chapter Five. Introduction .............................................................................. 04
5.1. The Modern Formulation of the Periodic L a w ........................ 64
5.2. The Structure of the Periodic S y s te m ......................................... 69
5.3. Predicting the Properties of Substances with the Aid of the
Periodic L a w ...................................................................................... 74
6 CONTENTS
Chapter Six. The Periodic System of the Elements and Their Atomic Stru
cture ............................................................................................................. 76
6.1.. Filling of Electron Shells and S u b sh ells..................................... 76
6.2. Variation of Ionization E n e r g ie s ..................................................... 85
6.3. Secondary P e r io d ic it y ......................................................................... 87
Chapter Seven. Elementary Principles of Forms and Properties of Chemical
Compounds..................................................................................... 89
7.1. Oxidation S t a t e ................................................................................. 89
7.2. Atomic and Ionic R a d i i ................................................................. 91
7.3. Coordination Num ber......................................................................... 97
7.4. Compounds Containing R —H and R —0 — B o n d s ................ 98
7.5. Acids, Bases andAmphoteric C om pounds................................... 99
7.6. Dependence of the Strength of Acids and Bases on the Charge
and Radius of the Ion ofthe Element Forming Them . . . . 100
Chapter Eight. Electronic Structure and Properties of Elements and Their
Compounds................................................................................. 102
8.1. First G roup......................................................................................... 102
8.2. Second G r o u p ..................................................................................... 103
8.3. Third Group ..................................................................................... 104
8.4. Fourth G ro u p ..................................................................................... 106
8.5. Fifth G roup........................................................................................ 107
8.6. Sixth G roup......................................................................................... 108
8.7. Seventh G roup..................................................................................... 108
8.8. Eighth G roup..................................................................................... 110
8.9. Zero G r o u p ......................................................................................... Ill
8.10. Some Conclusions............................................................................. 112
Chapter Nine. Significance of the Periodic L a w ............................................ 113
Part III. THE STRUCTURE OF MOLECULES AND THE CHEMICAL

BOND
Chapter Ten. In tro d u ctio n ..................... 115
10.1. Molecules, Ions and Free R a d ic a ls............................................ 115
10.2. Development of Conceptions of the Chemical Bond and
V a l e n c e ............................................................................................. 116
10.3. A. Butlerov’s Theory of Chemical Structure........................ 118
10.4. Structural Iso m e r ism ..................................................................... 120
10.5. Spatial Isom erism ............................................................................. 122
Chapter Eleven. Basic Characteristics of the Chemical Bond—Length,
Direction, Strength............................................................... 127
11.1. Length of B o n d s ............................................................................. 128
11.2. Valence Angles . .............................................................................. 129
11.3. Strength of the B o n d ..................................................................... 133
Chapter Twelve. Physical Methods of Determining Molecular Structure 137
12.1. Electron-Diffraction E x a m in a tio n ............................................. 138
12.2. Molecular Spectra............................................................................ 145
Chapter Thirteen. Basic Types of the Chemical Bond—Ionic and Covalent
B o n d ..................................................................................... 149
13.1. Electronegativity of the E le m e n ts ............................................. 149
13.2. Ionic and Covalent B o n d ............................................................ 151
13.3. The Dipole Moment and Molecular Structure........................ 153
13.4. Effective Charges............................................................................ 158
CONTENTS 7
Chapter Fourteen. Quantum-Mechanical Explanation of the Covalent Bond 159

14.4. Solution of the Schrodinger Equation Using Approximate
F u n c t io n s ..................................................................... .... 160
. 14.2. Potential Energy Curves for Molecules . . ............................. 166
14.3. Results of Quantum-Mechanical Treatment of the Hydrogen
Molecule by Heitler and L o n d o n ............................................. 168
14.4. Valence of the Elements on the Basis of the Heitler and
London T h eory................................................................................. 174
14.5. Explanation of the Orientation of V a le n c e ............................. 179
14.6. Single, Double and Triple .Bonds . ......................................... 187
14.7. The Donor-Acceptor B o n d ..................................... 196
14.8. The Bond in Electron-Deficient M olecu les............................. 201
14.9. Molecular Orbital M ethod............................................................. 203
14.10. Molecular Orbitals in Diatomic M olecu les............................. 206
14.11. Hiickel M e t h o d ..................................................... ........................ 211
Chapter Fifteen. The Ionic B o n d ................................................................. 221
15.1. Energy of the Ionic B o n d ........................................................ 221
15.2. Ionic Polarization............................................................................. 225
15.3. Effect of Polarization on Properties of Substances . . . . 229
15.4. The Polar Bond and Electronegativity . . . ................ 231
Chapter Sixteen. The Chemical Bond in Complex Compounds.................... 233
16.1. Complex Compounds......................................................................... 233
16.2. Isomerism of Complex C om pounds.............................................. 235
16.3. Explanation of the Chemical Bond in Complexes on the
Basis of Electrostatic C on cep tion s............................................. 237
16.4. Quantum-Mechanical Interpretation of the Chemical Bond
in Complex Com pounds................................................................. 238
16.5. Valence Bond M ethod..................................................................... 239
16.6. Crystal Field T h e o r y ...................................................................... 242
16.7. Molecular Orbitals in ComplexCom pounds............................... 247
Chapter Seventeen. The Hydrogen B o n d ..................................................... 254
Part IV. THE STRUCTURE OF MATTER IN THE CONDENSED STATE 260
Chapter Eighteen. In tr o d u c tio n ...................................................................... 260
18.1. Aggregate States . . ..................................................................... 260
18.2. Molecular In tera ctio n ..................................................................... 263
Chapter Nineteen. The Crystalline S t a t e ......................................................... 267
19.1. Characteristics of the Crystalline State ..................................... 267
19.2. Study of Crystal S tr u c tu r e .......................................................... 273
19.3. Types of Crystal L a t t ic e s .............................................................. 277
19.4. Some Crystal Structures . .......................................................... 279
19.5. Energetics of Ionic C r y s t a l s ...................................................... 287
Chapter Twenty. Liquid and Amorphous S t a t e s ......................................... 293
20.1. Structure of L iq u id s.......................................................................... 293
20.2. The Structure of W a t e r ................................................................. 290
20.3. ' Solutions of E le c tr o ly te s .......................................................... 297
20.4. The Amorphous S t a t e ...................................................................... 303
Appendix. I. Determination of the Ratio e/m for an Electron . . . . 306
Appendix II. Characteristics of Wave M otion............................................ 307
Appendix III. Construction of the Schrodinger E q u a tio n ....................... 311
8 CON T EN T S
Appendix IV. Polarization of L ig h t ........................................................... ?,H

Appendix V. Derivation of Relationship Describing Electron Dif
fraction by M olecules............................................... 018
Appendix VI. Moment of I n e r t i a .......................................... 017
Appendix VII. Expressions for Wave Functions of Hybrid Orbitals . . 010
Appendix VIII. Electron Spin and Magnetic Properties of Matter . . .'#20
Appendix IX. Calculation of the Absorption Spectra of Poly methy
lene D y e s ....................... 021
Appendix X. Solution of Homogeneous Sets of Linear Equations . . 020
Values of Units of Measure and Physical C o n sta n ts.................... 020
NAME INDEX .............................................................................. 02f>
SUBJECTINDEX ............................................................................................ 028
PREFACE
In our days, often called a period of scientific and technological

revolution, reorientation in the various branches of science is con
stantly taking place. In particular, the study of the structure of
matter, which lies in the borderland between physics and chemistry
and was formerly a part of these sciences, is gradually being separated
into an independent science with its own mathematical apparatus
and methodology. This is so first of all because quantum mechanics
has become the principal means of studying and solving problems
in this field. On the other hand, the theory of the structure of matter
is becoming essential for working in the most diverse branches of
science and technology — from astrophysics to agriculture.
Because of the aforesaid it is now necessary that the chemical
student in the first year of study in higher institutes of learning
(colleges and universities) acquire a deeper knowledge of the structure
of matter. This book was written to satisfy this requirement. It
presents the modern concept of the structure of atoms, molecules
and crystals, as well as the nature of chemical bonds; it describes
some methods used in structure determination. The book is based
on the course of lectures regularly delivered by the authors since 1964
in the Mendeleev Chemical Engineering Institute in Moscow. This
course precedes that of general and inorganic chemistry. The authors
believe that inorganic chemistry cannot be satisfactorily studied
without the knowledge of the structure of matter and the energetics
of chemical reactions, i.e., chemical thermodynamics1.
The theory of the structure of matter is probably the most compli
cated branch of modern natural science; it utilizes practically all
the achievements of physics and an enormous mathematical appara
tus. It is obvious that in a book intended for students with only
a rudimentary knowledge of mathematical analysis, many important
1 Information about the energetics of chemical reactions, necessary for

studying inorganic chemistry, is given in the book by M. Karapetyants: Intro
duction to the Theory of Chemical Processes, ‘Vysshaya Shkola* Publishing
House, Moscow, 1970.
10 PREFACE
problems cannot be treated quantitatively. Accordingly, the purpose

of this book is to acquaint the reader with the fundamental laws
and concepts involved, discussing the quantitative aspect of the
problems only where this can be done in such a way that it can be
easily understood. When using the mathematical apparatus we tried
where possible to give a ‘key’ to a deeper understanding while avoid
ing unwieldy calculations. As might well be expected, the discussion
of a number of problems had to be simplified.
Inasmuch as in colleges this book may be studied prior to the course
in chemistry, it was necessary to touch on certain things usually
discussed in such a course, such as oxidation state, acid and basic
properties of compounds, formation of complexes. The information
given is, however, very concise. It was also necessary to present cer
tain facts in the field of physics; these are given in the appendices
at the end of the book. In the appendices are also given the deduction
of a number of relationships and other information that can be omitt
ed in an initial discussion of the problems considered.
The authors welcome apy comments and suggestions that would
improve the contents of the book.
PART I
ATOMIC STRUCTURE
CHAPTER ONE
INTRODUCTION
1.1. Atoms
According to the latest investigations matter consists of protons,
neutrons and other elementary particles that are now considered
indivisible. Thanks to the powerful accelerators now available and
the intensive study of cosmic rays, about 200 elementary particles
have come to be known in the last decades. Sometimes instead of
‘elementary particles’, they are called ‘fundamental units of m atter’.
Atoms are the simplest electrically neutral systems made up of ele
mentary particles. More complex systems, molecules, consist of
several atoms. Chemists deal with atoms which form substances,
i.e., with atoms of the chemical elements; they are the smallest parti
cles. of an element which can take part in a chemical reaction. An
atom of an element consists of a positively charged nucleus containing
protons and neutrons and of electrons1 moving around the nucleus.
Many atoms are stable, i.e., they can exist indefinitely. There are
also a great many radioactive atoms that after the elapse of some time,
are converted to other atoms as a result of changes that take place
in the nucleus.
The number bf electrons in neutral atoms of elements is equal to the
positive charge of the nucleus (expressed in units of elementary charge).
The charge carried by the nucleus is equal to the sum of the charges
of the electrons and is opposite in sign to that of the latter. A positive
ion is formed when one or more electrons are removed from an atom,
while a negative ion is formed when an electron is added to it.
The number of electrons in an atom, and consequently the positive
charge of its nucleus, determines the behaviour of atoms in chemical
reactions. A chemical element is a combination of atoms with identically
charged nuclei. The charge of the nucleus determines the position of
an element' in the Mendeleev Periodic System: the atomic number of
an element in the periodic system is equal to the positive charges on the
nucleus of an atom of the element (expressed in units of elementary
electric charge).1
1 There exist atoms that are positron-and-eleetron systems (positronium),
meson-and-proton systems (mesoatoms), and others. The iifetime of such atoms
is less than a millionth fraction of a second.
12 P A R T I. A T O M I C S T R U C T U R E
The ancient Greek philosophers already supposed that matter consists

ultimately of indivisible discrete particles, atoms. The present concept of
atoms as minute particles of chemical elements capable of combining to form
larger particles, molecules, of which matter consists was first stated by M. Lo
monosov in 1741 in his work “Elements of Mathematical Chemistry” and sub
sequently in other works. Although these works were published by the St.
Petersburg Academy of Sciences whose publications were available to all the
large libraries of that time, they were passed unnoticed by Lomonosov’s con
temporaries at the time.
At the beginning of the XIX century Dalton (England) stated that matter
consists of atoms and that chemical action takes place between the atoms
which combine in simple proportions. Dalton advanced the concept of relative
masses of atoms (atomic weights) and pointed out the necessity of precisely
determining these quantities. A few years after his works were published they
attracted the attention of many researchers. From this time the atom-molecule
concept has been widely used in chemistry and physics.
1.2. The Avogadro Number

The number of atoms in a gram-atom of any element is called the
Avogadro number; this constant is denoted as N 0. The most exact
determinations show that
N0= 6.02296 .K F g -a r1
This is also the number of molecules in a gram-molecule of any sub
stance. This quantity equal to the ratio of the mass of a gram to 1/12
of the mass of the C12 atom is one of the most important universal
constants in chemistry and physics; it does not depend on the nature
of the substance or its state of aggregation (physical state).
The Avogadro number can be found by different absolutely inde
pendent methods; about 60 methods are known at present. We shall
consider two of them: one of the most easy-to-understand and one
of the most exact ones.
1. When certain radioactive elements decay (disintegrate), a-
particles (nuclei of helium atoms) are emitted. These particles are
detained by any material that lies in their path and, on adding two
electrons, are converted to helium atoms. The amount of helium form
ed is not very great but it can be determined by micromethods.
In this way it was found that 1 g of radium (containing decay pro
ducts) yields 159 mm3 of helium per annum or 5.03 •10“9 cm3 He per
second.
The a-particle from which the atom is formed possesses sufficient
energy to produce a visible effect. For example, when a-particles
impinge on a screen coated with zinc sulphide, scintillation is observ
ed, which can be seen through a magnifying glass; this is made use
of to count the number of a-particles. There are other methods of
detecting and counting a-particles. Thus, the number of a-particles
emitted by a very small but definite amount of radioactive substance
can be counted. Since each a-particle is converted to a helium atom.
Ch. 1. I N T R O D U C T I O N 13
it is possible to calculate the number of atoms of helium obtained

from 1 g of the radioactive substance and if the volume of the helium
produced is known, it is easy to find the number of atoms contained
in 22.4 litres of helium measured under normal, conditions. The
helium molecule is monoatomic and therefore the number of atoms
in a gram-molecular volume (also called molar volume) is equal to
Fig. 1.1. Schematic drawing of layout used by Millikan to measure the charge
on the electron
1 , 2 —capacitor plates; 3 —metal chamber; 4 —thermostat; 5—oil atomizer; 6 —aperture
in plate; 7—eyepiece; 8 —storage batteries; 9-pressure gauge; 10 —X-ray tube; 11 —thermostat
liquid (kerosene)
V
the number of molecules, i.e., to the Avogadro number. Such measu
rements were first made by Rutherford and his co-workers in 1911.
They found that N 0 = 6.1 *1023 which is very close to the value
established at the present time.
2. The Avogadro number can also be found with the aid of the
charge of an electron, which is a fundamental constant that is essen
tial for the theory of the structure of matter.
The exact determination of the charge of the electron was first
effected in the years 1909-1914 by Millikan (USA).
A schematic diagram of the layout used by Millikan is shown in
Fig. 1.1. The main part of the layout was an electric capacitor
consisting of brass plates 1 and 2 contained in a metal chamber 3
placed in a thermostat 4. A fog composed of minute oil droplets was
created in the chamber by means of atomizer 5. The droplets entered
the capacitor through aperture 6 in the upper plate. The motion of
the droplets between the plates could be viewed through eyepiece 7.
Ionization was produced by exposing the air in the Apparatus to
X-rays emitted from tube 10. The free electrons (or positive ions)
obtained in this way impinged on the oil droplets which received
an electric charge ed. By varying the voltage across the plates of the
capacitor it was possible to attain such a voltage at which the electric
field strength was balanced by the force of gravity of the charged
droplet and the latter remained motionless in. the field of view. Then
mg = edE ( 1 . 1)
where m = mass of the droplet
g = acceleration due to gravity
E = strength of the electric field
For a plane capacitor
( 1 . 2)
where V = voltage applied to the plates

d = distance between the plates
From equations (1.1) and (1.2) it follows that
( 1 . 3)
wherefrom ed can be found if the mass of the droplet is known (the

mass of the droplet can be calculated from its velocity when falling
in air in the absence of an electric field).
Millikan found that the charges on the droplets were always
multiples of a certain value e, the smallest charge observed. This
could be explained by the fact that a droplet can capture one, two,
and so on electrons (or ions) but never a fraction of an electron
because an electron is indivisible. Hence, the smallest charge of
a droplet is the charge of an electron.
On taking a great number of measurements of the charge of an
electron, Millikan obtained the following value:
e = 4.77 HO-10 esu
This constant was subsequently determined more precisely. Its
value is taken at the present time to be
e = 4.80286-10"10 esu = 1.60206 -10"19 coulomb
This constant is of major importance to the chemist and physicist
and can be used, in particular, to determine the Avogadro number.
Indeed, according to Faraday’s law of electrolysis, it is necessary
to pass a quantity of electricity equal to 1 faraday, or 96,491 cou
lombs, to liberate or deposit one gram-equivalent of a substance.
This quantity of electricity is needed to produce from hydrochloric
acid 1.008 g of hydrogen and 35.453 g of chlorine. Since in the electro
lysis of IICl solution H + and Cl" ions are discharged, each of which
carries a charge equal to the charge of an electron, it is obvious
that the quotient obtained on dividing a faraday by the value of

the charge of an electron shows how many atoms are contained in
35.453 g of chlorine or in 1.008 g of hydrogen, or in general, in a gram-
atom of any element, i.e., gives the Avogadro number; hence,
tfo = - (!-4>
where F = Faraday’s constant (1 faraday). Thus,1
,.602-^10—* " 6 .0 2 3 - W
The Avogadro number is enormous. If we were to gather peas
in a quantity equal to the Avogadro number, and if we assumed
that the volume of a pea is equal to 0.3 cm3, such a number of peas
would occupy a volume equal to 0.3-6.023-1023 = 1.8-1023 cm3 =
= 1.8-108 km3. Each side of a cubic container of such a capacity
would be equal to 565 km.
Two conclusions that are very important to the chemist can be
drawn from the fact that the Avogadro number is so great.
1. Even very small quantities of a substance that are hardly
discernible in an optical microscope contain an enormous number of
atoms. That is why, macroscopically, substances seem to be conti
nuous.
2. Any substance, even the purest one, always contains some
impurities, i.e., alien atoms (atoms of different elements). At present
it is impossible to obtain an appreciable amount of a pure substance
that is absolutely free of impurities. Certain substances (silicon,
germanium, and others) can now be obtained that contain impurities
in an amount equal to 10~6 per cent and even from 10"7 to 10"8 per
cent. But even such extra-pure substances contain milliards of alien
atoms per gram.
1.3. Mass and Size of Atoms
The mass in grams and dimensions of any atom can be found from
the Avogadro number. The mass of the atom m is found by dividing
the gram-atomic weight12 A by the Avogadro number:
1 The descriptions of the methods for determining the Avogadro number are
somewhat simplified here. Those who want to acquaint themselves with all the
details of the experiments and calculations involved can do so in the following
hooks: R . M illikan. Electrons ( + and —), Protons, Photons, Neutrons, and
Cosmic Rays; E. Guggenheim and G. Prue. Physico-Chemical Calculations,
Interscience publishers inc. New York.
2 The terms ‘atomic weight’, ‘molecular weight’ are not absolutely correct;
it would be more exact to say ‘relative atomic mass’, ‘relative molecular mass’.
However, the terms ‘atomic weight’, ‘molecular weight’ are consistently used
in the chemical literature and shall be used in this book.
Hence, the mass of the hydrogen atom, for instance, is

1.008 .
6.023-1023 - 1-673*10 g
The mass of the uranium atom (mass number 238) is
238
mV— 6.023-1023 3.95-10“22 g
If we divide the volume of a gram-atom of an elementary substance
in the solid state by the Avogadro number, we will obtain the volu
me v per 1 atom. The atom can be assumed to be a sphere enclosed
in a cube of a volume equal to v. Since the atoms in solid bodies are
very close to each other, such an assumption will not lead to consi
derable error. Then the diameter of the atom can be found by taking
the cubic root of the volume occupied by 1 atom. Let us calculate
the diameter of the copper atom.
The density of copper is equal to 8.93 g/cm3; therefore, a gram-
atom of copper is equal to 63.46/8.93 = 7.10 cm3 wherefrom
7.10
Vcu'~ 6.023-1023 1.178-10-23 cm3
and
dcu « v&i = y '/ "l .178-10"23 —2.28-10~8 cm
Hence, the radius of the copper atom is approximately 1.14 .10~8 cm.
To find the exact dimensions of atoms it is necessary to know how
they are arranged in the crystals of the solid substance; this is deter
mined by X-ray diffraction analysis of crystal structure (see pp. 273-
277). Diffraction data have shown that for most metals, including
copper, the atoms are arranged the same as the most closely packed
balls (see pp. 281-282); in this case the volume of the balls consti
tutes 74 per cent of the total volume of the space occupied by them.
The exact value of the radius of the copper atom in a crystal can be
found as follows.
The volume of the copper atom is yCu — 1.178 *10"23 *0.74 =
.-= 0.870-lO"23 cm3 and hence
As can be seen, the exact value of the radius of the copper atom
in a crystal does not differ appreciably from the approximate value
determined above.
It should be noted that there is no clear demarcation between the
atoms and the space surrounding them (this will be discussed at
length later on). Therefore, the dimensions of atoms are given con
ventionally. Here, by the dimensions of the atoms are implied their
radii in the crystals of the respective simple substances. The radius of
the atom is taken to be half the distance between the nuclei of neigh
bouring atoms.
The radii of all atoms have a value of the same order, 10~8 cm.
In the theory of atomic structure .it is convenient therefore to use
a unit equal to 10“8 cm which is called the Angstrom unit (or simply
angstrom) and denoted as A. Thus, rGu = 1-28 A .
As can be seen, atoms are extremely small. Millions of copper
atoms arranged in a row would form a chain that is only 0.26 mm
long.
1.4. The Constituents of an Atom: Electrons
and the Nucleus
As has already been stated, the atoms of chemical elements consist
of a nucleus and of electrons moving around it. The properties of
electrons were studied after these particles were isolated from matter
Scale
as cathode rays in the second half of the last century. At first the
ratio of the charge of the electron to its mass, e!me, was measured.
This ratio is determined by the deflection of a beam of electrons
in electric and magnetic fields. Such measurements were first made
in 1897 by J. Thomson (England). A schematic drawing of the appa
ratus used is shown in Fig. 1.2. Cathode ray tubes of a similar design
are now being widely used (for example, in television sets). The
theory on which this method is based is briefly given in Appendix I.
With the aid of these experiments it was found that
—me = 5.273-1017 esu/g
The electron charge is determined by the method described above.
The mass of the electron can be calculated if e/me and e are known.2
2 3aK . 15648
18 PART I . ATOMIC STRUCTURE
It proved to be
me = 0.9109-10-27 g
Let us compare this value with the mass of the hydrogen atom,
already calculated (see p. 16):
nip _ 0.9109.10-27 1
roH ~ 1.673-10-24 “ 1837
We see that the mass of an electron is extremely small as compared

to the mass of the lightest atom. Consequently, almost all the mass
of the atom resides in its nucleus. The nucleus is very small; whereas
the radius of an atom is of the order of 10~8 cm, the radius of the
nucleus of the atom is within the limits of 10“lai to 10~12 cm. The
nucleus and the electrons, being charged particles, create an electric
field that ’fills’ the space ‘inside* the atom and surrounds it beyond
its ‘limiting boundary*. The electric field produced is material
like the nucleus, electrons, and other particles are.
The presence of the nucleus in the atom was experimentally proved
in 1909-1911 by Rutherford and his co-workers (England). When
investigating the passage of a-particles through very thin metal
films, they unexpectedly discovered that a certain part of the a-
particles (approximately 1 out of 10,000) did not pass through the
foil but were thrown back. This could be explained only by the
collision of the a-particles with a massive positively charged particle,
the nucleus of the atom.
The nucleus of the atom consists of two kinds of elementary par
ticles, protons and neutrons; this concept of the structure of the nucle
us was first stated and proved in 1932 by D. Ivanenko and E. Gapon
(USSR), and Heisenberg (Germany). The proton is the nucleus of
the hydrogen isotope JH; it carries a positive charge equal in magni
tude and opposite in sign to that of the electron. The neutron is an
uncharged particle. The proton and neutron are approximately equal
in mass, being respectively 1836.12 and 1838.65 times as massive as
the electron. The charge on the nucleus is determined by the number
of protons in it; the sum of the number of protons Z and the number
of neutrons N gives the mass number A :
A - Z+ N
Atoms of the same element having different nuclear masses but
identical nuclear charges are called isotopes1. The isotopes of a given
element differ from one another by the number of neutrons in the
nuclei of their atoms.
1 The Greek word ‘isotopes’ means equally positioned (i.e., they occupy the
same position in the periodic system).
Ch. 2. A T O M I C SPECTRA 19
CHAPTER TWO
ATOMIC SPECTRA
The study of the spectra of chemical elements constitutes the expe

rimental basis of the theory of atomic structure. At present, the
wavelengths of spectral lines can be measured with an accuracy
of 0.001 per cent; in many investigations the accuracy is considerably
higher. The spectral-line intensities can also be determined very
precisely. Thus, our knowledge of the structure of atoms is based on
very reliable experimental data.
2.1. Principle of Operation of Spectrographs;

Kinds of Spectra
In the spectrograph the beam of light passing through a narrow
slit is rendered parallel by a collimating lens and falls on a prism
or diffraction grating, where it is dispersed into its constituent rays
which are then focused on different parts of the photographic plate,
Fig. 2.1. Schematic drawing of spectrograph
according to their wavelength. Visible and ultraviolet radiation

is usually investigated by means of optical spectrographs in which
the radiation is dispersed by passing it through a glass prism (for
visible light) or quartz prism (for ultraviolet radiation). A schematic
drawing of the spectrograph is shown in Fig. 2.1. Light is dispersed
due to the fact that the refractive index varies with the wavelength
of light; for most media the indeix of refraction decreases with
an increase in the wavelength.
The spectra obtained by dispersing the radiation emitted by bodies
are called emission spectra. These may be continuous, line, and band
spectra. Continuous spectra are obtained from glowing solids and
liquids. The radiation of gases (which can be produced by heating
them or with the aid of an electric discharge) gives a line spectrum
consisting of separate lines or a band spectrum consisting of bands1.
1 Many gases (H2, Cl2, and others) through the agency of heat or an electric
discharge give continuous spectra in addition to line (or band) spectra.
The use of high-resolution spectrographs has shown that these bands

consist of a great number of lines that are very close to each other.
At the present time it has been established that line spectra are
obtained from the radiation emitted by atoms; while band spectra,
from radiation emitted by molecules. The atom of every element and
the molecule of every substance, each has its characteristic spectrum
which consists of a definite set of lines or bands that correspond
to strictly definite wavelengths. In this Chapter we will consider
pcm- atomic spectra. Examples of
1k300
such spectra are shown in
16700 9 Fig. 2.2 in which wavele
x:
ngth (X) and wave number
(v)1 scales are plotted.
The spectra of many ele
i i i ments are very complex.
For example, there are over
He | | five thousand lines in the
r ri : i spectrum of iron. Work
Hg
: i i : i Fig. 2.2. Schematic representa
! i
tion of spectra of certain ele
ments
The lines are of different thickness
i i i
m
j
i l : to show their relative intensity;
actually the lines are of the same
thickness, corresponding to the width
of the spectrograph slit. Only the
most intense lines are shown so as
7000 6000 5000 WOO not to make the drawing too comp
X, A licated
carried out with sensitive apparatus has shown that many lines in
atomic spectra actually consist of several lines that are very close
to each other, i.e., they are multiplets: double lines are called doub
lets; triple lines, triplets (single lines are called singlets). If the source
of radiation is placed in a magnetic field, single lines are split and,
instead of one line, a few adjacent lines will appear in the spectrum
{Zeeman effect). A similar effect is observed when the radiation source
is placed in an electric field {Stark effect).
2.2. The Atomic Spectrum of Hydrogen

The simplest spectrum is that of hydrogen. In the visible region
there are only 4 lines (see Fig. 2.2); they are denoted as Ha , H 3,
IT and / / 6. In the near ultraviolet region there are a number of
1 The wave number v is the reciprocal of the wavelength; v = 1/A,, i.e. the
mimher of wave's in a centimetre; it characterizes the frequency of vibrations
(Mm Appemlix I I).
Ch. 2. ATOMIC SPECTRA 21
other lines that together with the aforementioned four lines form
a series (Fig. 2.3) that is called the Balmer series after the Swiss
scientist who in 1885 discovered that the wave numbers v of the lines
in this series are expressed exactly by the following formula:
—_ _R____ R_
v ~ 22 rfi
where R = 109,678 cm-1 and n = 3, 4, 5, . . . The constant R in
this formula is called the Rydberg constant.
Investigation of the hydrogen spectrum in the far ultraviolet
and infrared regions revealed the presence of a few other series of
o<
Fig. 2.3. Atomic spectrumof hydrogen in §|

the visible and near ultraviolet regions rr
(Balmer series) I
Ha
lines called by the names of the men who investigated them: the
Lyman (ultraviolet region), Paschen, Brackett, and Pfund (infrared
region) series. It was found that the wave numbers of the lines in
these spectra are given by formulae similar to the Balmer formula,
but which instead of 22 contained l 2, 32, 42, and 52, respectively.
Therefore, the universal formula for the atomic spectrum of hydro
gen is
R R
( 2 . 1)
V ~ n\ n\
where nx and n2 are whole numbers and n2 > As seen from equat
ion (2.1), the number of lines in the hydrogen spectrum is infinitely
large (at the edges of the series which correspond to large values
of n2, the lines are very close to each other and cannot be distinguish
ed from one other). Thus, a very simple formula describes the large
number of lines observed in the spectrum of hydrogen.
2.3. The Spectra of Other Elements

Series of lines were found in the spectra of all the other elements.
The spectra, however, were more complex because, unlike the hydro
gen spectrum, the series are not arranged separately in different parts
of the spectrum but are superimposed on one another. Nevertheless,
they can be discerned by spectroscopists according to certain cri
teria such as the appearance of the lines (sharp or diffuse), the mode
of excitation (arc or spark), multiplicity, the way they are split
in magnetic and electric fields, etc.
In 1889 Rydberg (Sweden) found that the wave numbers of a parti

cular spectral series can always be expressed as the difference between
two functions of the integers ni and n2:
v = T (Hi) — T (n2) ( 2. 2)
where n2 > n x. The numerical values of these functions are called

spectral terms.
For the hydrogen atom, the single-charged helium ion He+, the
double-charged Li2+ ion, and other particles containing only one
electron the spectral term is given by the formula
(2.3)
For the hydrogen atom, Z = 1; for the single-charged H e+ ion,

Z = 2; for the double-charged Li2+ ion, Z = 3; etc. For the atoms
of other elements, the terms can be expressed by the formula
RZ*
(2.4)
(n + a)2
where a is a fraction smaller than unity; it is a constant for the lines
of a particular series1. The correction a for different series is denoted
as 5, p, d, /, i. e., the first letters in the names of the series:
‘sharp’, ‘principal’, ‘diffuse’, ‘fundamental’, respectively. As in
equation (2.3), Z = 1 for neutral atoms; Z = 2 for single-charged
ions, etc.
Thus, it was discovered that the vast number of spectral lines
is described by relatively simple relations that contain integer para
meters.
2.4. The Concept of Light Quantum
In 1900 Planck (Germany), in order to explain the distribution of
energy in the spectra of heated bodies, developed a theory based
on the assumption that energy is not emitted by atoms continuously
but only in minimum indivisible quantities called quanta, the
value of which depends on the frequency of the light emitted.
According to Planck's formula
E = hv (2.5)
where E = energy of a quantum
v = frequency of vibrations
h = Planck's constant
h = 6.62517-IQ"27 erg-s
1 Equation (2.4) is most precise for the spectra of alkali metals; for the
other elements there is a slight difference between the calculated and experi
mental values, but this discrepancy can be eliminated by introducing, in addi
tion to a, another small correction to the integer n.
Ch. 2. ATOMIC S PECTRA 23
Thus, the energy of a body can change by values that are multiples
of hv, just as the electric charge can change by a value that is a •
a multiple of the charge of an electron.
Experimental data splendidly confirmed Planck’s theory. Planck’s
formula expresses one of the most important laws of nature. Planck’s
constant, like the velocity of light and the charge of the electron,
is a fundamental constant which cannot be expressed by any other
simpler parameter.
Since every spectral line is characterized by a strictly definite
wavelength, and consequently by a strictly definite frequency, this
signifies that atoms can radiate only a strictly definite amount of
energy which can be calculated for a given spectral line from Planck’s
formula. For example, for the Ha line (see Fig. 2.3)
%= 6562.8 A -= 0.656 • 10'4 cm
c 2.9979-10io
v s '1
X 0.656-10-4 -4 .5 7 -1 0 14
where c = velocity of light; according to equation (2.5)
E = 6.625-10-27 -4.57.1014 = 3.03-10"12 erg
which shows that the energy of a quantum of visible light is a very
small quantity.
When an atom radiates a light quantum, it passes from one energy
state to another. Hence, the physical meaning of spectral terms lies
in the fact that the terms characterize the energy levels of the electrons
in atoms1. Thus, we come to the conclusion that the electrons in atoms
can have only strictly definite values of energy which are characterized
by a series of integers.
It follows from equations (2.2) and (2.5) that the energy of electrons
in atoms is related to the values of the respective terms according
to the equation
E = -h c T (2.6)
The energy is taken with a minus sign because it is assumed that when
the electron is at an infinitely great distance from the atom it is in
a state of zero energy; therefore, the energy of the electron in the
atom is always less than zero.
It follows from equation (2.3) that equation (2.6) for the hydrogen
atom will be
hcR
n2 (2.7)
1 The nucleus, being considerably heavier than the electrons, mky be deemed
to be at rest and therefore all the energy changes in the atom that do not affect
the structure of the nucleus can be regarded as energy changes of the electrons
(kinetic and potential energy).
The energy levels of the electron in the hydrogen atom is schema

tically represented in Fig. 2.4. The energy is expressed in electron-
volts; this is a very convenient unit to use for atoms. A n electron-volt
Et eV
Fig. 2.4. Energy levels of the electron in the hydrogen atom
(eV) is the energy gained by an electron which is accelerated by an

electric field in the region with a potential difference of one volt:
1 eV - 1.6021 -lO"12 erg
Besides the spectral data obtained, there are many other facts that afford
evidence that electrons in atoms possess strictly definite energies. This is con
firmed, in particular, by the results of experiments first carried out in 1912 by
Franck and Hertz to study electron bombardment.
In these experiments Franck and Hertz bombarded the atoms of gases (vapo
urs of morcury, krypton, and others) with electrons and measured the loss in
Ch. 2 . A T O M I C S P E C T R A 25
energy of the electrons on colliding with the atoms of the gas. The experiments
showed that if the energies of the electrons are less than a certain value, they
rebound from the atoms of the gas, practically without transferring energy to
them. This, for example, occurs when mercury vapours are bombarded by elect
rons with an energy less than 4.9 eV. When the energy of the bombarding elect
rons is greater than 4.9 eV, their energy is transferred to the mercury atoms. From
this it can be concluded that 4.9 eV is the energy needed to transfer an electron
of the mercury atom from the lowest energy level to the next energy level.
When mercury atoms are bombarded with electrons with an energy of 4.9 eV
or greater, radiation of a wavelength of 2537 A is observed, which corresponds to
the energy of a quantum equal to 4.87 eV; this value practically coincides with
the energy of electrons required to induce radiation (4.9 eV). This can be expla
ined by the fact that the electrons of atoms which have been transferred to
a higher energy level by electron bombardment, are transferred back to the*
lower energy level and emit a quantum of radiant energy.
With the aid of more powerful electron bombardment the electrons of the
atoms can be transferred to the third, fourth, etc. levels; this is indicated by
absorption of the energy of the bombarding electrons. When a certain amount
of energy is transmitted, the electrons break away from the atoms and ionization
of the gas takes place. By this method it was established that an energy of 10.4 eV
is necessary for the ionization of the mercury atom.
Thus, experimental data show that a definite set of energy levels exist in
the atoms of elements.
2.5. History of the Development of the Concepts

of Atomic Structure
Rutherford, after establishing the presence of a nucleus in the
atom in 1911, proposed the planetary model of the atom in which the
electrons revolve around the nucleus like planets revolve around
the Sun. From electrodynamics it is known howrever that a charge
revolving around a certain centre is a source of electromagnetic
waves and therefore a radiating electron would continuously lose
energy and eventually fall onto the nucleus. In 1913 Bohr (Denmark)
suggested that there are stationary orbits in atoms, in which the
electron moves without radiating energy; according to Bohr, for
such orbits the following equation must be satisfied:
mevr = n — (2.8)
where m e = mass of electron
v = .velocity of electron
r — radius of the orbit
m evr = angular momentum of the electron1
n = 1, 2, 3, . . .
h = Planck’s constant
1 The angular momentum of a particle* rotating in a circle with a radius r

is equal to mvr where m and v are the mass and velocity of the particle, respec-
tively. In the general case, the angular momentum of a material point relative
to any centre 0 is equal to the product of the mass and the distance r betweeik
The value h!2jx is designated as h (crossed h), then

mevr = nh (2.9)
Proceeding from this hypothesis, Bohr was able to develop his
theory of the structure of the hydrogen atom. Indeed, by equating
the centripetal force and attractive force of the electron to the
nucleus, we cpn write
mev2 e2
( 2 . 10)
r ~~ r2
By solving the system of equations (2.9) and (2.10), we can find the
velocity of the electron in a stationary orbit
*2
( 2 . 11)
nh
-and the radius
( 2 . 12)
mee*
By substituting the known values into equation (2.12), we find that

r = 0.529ra2 A (2.13)
Consequently, the radius of the first Bohr orbit is 0.529 A; the order
•of this value conforms with that of the known values of the sizes of
-atoms (see p. 17).
The energy of an electron is the sum of its kinetic and potential
energies. The potential energy1of a system of two bodies with charges
e is equal to —e2/r. Hence,
£ = iT + (-7 ) (2-14)
the particle and the centre O and the projection of its velocity on the line lying
in the plane of motion perpendicular to r. The concept of angular momentum is
extensively used in the theory of atomic structure. The angular momentum
(also called the moment of rotation) is a vector quantity; it is directed perpendi
cular to the plane in which rotation occurs.
1 The potential energy of two differently charged bodies with charges ei
and e2l which are at a distance r from each other, is determined by the work
jiecessary to move these bodies from the position at which the potential energy
is zero (corresponding to r = oo) to the given distance r; this work, in accor
dance with Coulomb’s law, is expressed by the integral
eie2 dr— eie2

r2 r
oo
When the charges of the bodies are of the same magnitude, as in our case, the
potential energy is equal to —ea/r.
Ch. 2. A T O M I C S P E C T R A 27
Substituting the values of v and r from equations (2.11) and (2.12),

respectively, we obtain
1 mee4 _ const
2 ft2/?2 M2 (2.15)
i.e., the equation obtained resembles the one for the energy levels
in the hydrogen atom [see equation (2.7)1. By equating these expres
sions, we can find the theoretical equation for the Rydberg constant:
(2.16)
The value of R calculated from this equation coincides with that

found experimentally.
In this way Bohr calculated theoretically the spectrum of hyd
rogen.
Later on (1916-1925) Sommerfeld (Germany) and other scientists
developed a theory of the structure of many-electron atoms, based
on the Bohr theory. It was assumed that the stationary orbits in
atoms could be not only circular but also elliptical and be positioned
differently in space; the size of the orbits and their position in space
were determined by quantization rules, a generalization of equation
(2.9). Many characteristics of the atomic spectra could be explained
with the aid of this theory.
The Bohr — Sommerfeld theory, however, does not satisfy the
present state of the art. Although it does explain many chara
cteristics of the spectra, it has many faults because of which it
must be replaced by modern concepts. The main fauLts of the
Bohr—Sommerfeld theory are as follows.
1. It is based on quantization rules which do not stem from the
laws of mechanics and electrodynamics.
2. The use of this theory in calculating a number of spectral
characteristics, in particular, the intensity of spectral lines and their
multiplet structure, yields results that do not agree with those
obtained experimentally.
3. The Bohr — Sommerfeld theory when used for the calculation
of the energy of electrons in many-electron atoms, also fails to give
results that coincide with those obtained in experiments (even for
the simplest case, the He atom).
4. It was found that this theory could not be used for the quanti
tative explanation of chemical bonding. Thus, the calculation of
the bond strength (bond-breaking energy) in the simplest system,
the ionized molecule HJ, gave a negative value, i.e., showed that
such an ion cannot exist. Actually, however, it does exist and
the bond strength in this case is equal to +61 kcal/mole (see p. 203).
CHAPTER THREE
THE WAVE PROPERTIES OF MATERIAL PARTICLES
The modern theory of the structure of atoms and molecules is based
on laws describing the motion of electrons and other particles of
very small mass, i.e., of microobjects. These laws were definitely
formulated in 1925-1926; they differ greatly from the laws governing
the motion of macroobjects which include all objects that are visible
in an optical microscope or to the naked eye.
The basis of the modern theory lies in the concept of the dual
nature of microobjects: they can be regarded as particles and as
waves, i.e., microobjects simultaneously have the properties of particles
and waves.
3.1. Dual Nature of Light
The dual wave-particle picture was first established for light.
In the first half of the last century, as a result of interference and
diffraction studies, it was experimentally proved that light consists
of transverse electromagnetic vibrations. The occurrence of inter
ference and diffraction under certain conditions is characteristic
of any wave process (see Appendix II).
However, in the XX century a great number of phenomena became
known, that showed that light consists of a stream of material
particles which were named light quanta or photons.
The concept of the quantum, as has already been stated, was
first introduced in 1900 by Planck. The corpuscular properties of
light are most clearly manifested in two phenomena: the photo
electric effect and the Compton effect.
The photoelectric effect, which was discovered in 1887 by Hertz
and developed by A. Stoletov, consists in the fact that metals (and
semiconductors) emit electrons when exposed to light. The photo
electric effect cannot be explained by the wave theory of light.
Calculations show that due to the extremely small size of the electron,
the quantity of energy transmitted to it by the electromagnetic waves
that fall on it is so small that in order that the electrons accumulate
sufficient energy to leave the metal the latter would have to be ex
posed to sunlight for no less than several hours (and that only in the
absence of transfer of the energy absorbed by the electrons to the
atoms). Emission of electrons, however, is observed immediately
after the metal is illuminated. Besides that, according to the wave
theory, the energy E e of the electrons emitted by the metal should
be proportional to the intensity of the incident light. It was, however,
established that E e does not depend on the intensity of the light,
but depends on its frequency; E e increases with v but an increase in
the intensity of light only causes a greater number of electrons to be
emitted from the metal.
Ch. 3. W A V E P R O P E R T I E S OF PARTI CLES 29
In 1905 Einstein showed that the photoelectric effect could be

explained very simply if light were regarded as a stream of particles,
photons. The photons on colliding with the electrons transmit to
them their energy equal to hv in accordance with Planck’s formula;
this also explained why radiation of long wavelengths does not
produce the photoelectric effect; the reason for this is that the energy
of the photon in this case is not sufficient to tear the electron away
from the metal.
Those electrons that escaped without giving up to the atoms of
the metal any of the energy they received from the photons will
Fig. 3.1. Dependence of the

voltage at which the photo
electric current is disconti
nued on the frequency of
the incident light (for cae
sium)
have the maximum energy. Obviously, the energy of such electrons

is equal to the difference between the energy of a photon hv and the
work required to overcome the force retaining the electron in the
metal, i.e., the work function of the electron W; hence,
(Ee)max = hv - W (3.1)
This equation, called Einstein's law for the photoelectric effect, is
fully consistent with experimental data. It was very thoroughly
checked experimentally in 1916 by Millikan. The maximum energy
of the emitted electrons was measured by applying an external
electric field at which the photoelectric current is discontinued; the
applied voltage does not allow the electrons to reach the electrode,
and in this case
Ve = -i- mev2= (£e)max (3.2)
where me, e, and v are respectively the mass, charge and velocity
of the electron; V is the potential difference of the electric field.
Planck’s constant can be found on the basis of Einstein’s law;
for this purpose it is necessary to determine the dependence of
(Ee)max on the frequency of the incident light. This dependence for
caesium, found experimentally, is represented in Fig. 3.1. As can be
seen from equations (3.1) and (3.2), the tangent of the slope in the
V—v coordinates is equal to hie. This method is one of the most exact
ones for determining Planck’s constant.
Prior to passing on to the discussion of the other phenomenon
indicative of the corpuscular nature of light, the Compton effect, it
is necessary to say something about the law of interdependence of
mass and energy.
3.2. The Law of Interdependence of Mass and Energy

In 1903, with the aid of his relativity theory, Einstein proved
that the mass of a body in motion exceeds its mass at rest, according
to the equation
rn0
m (3.3)
/~ i/2
V ‘- 7
where m = mass of the body in motion
m0 = mass of the body at rest
v = velocity of the body
c = velocity of light in vacuum
Hence, an increase in the velocity of the body, and consequently
an increase in its energy, results in an increase in its mass.
Einstein also showed that the mass of a body is related to its
energy according to the equation
E = me2 (3.4)
The above equation expresses the law of interdependence of mass
and energy, i.e., it shows how mass and energy are related to each
other; prior to the appearance of the relativity theory, they were
deemed to be independent quantities. Equation (3.4) shows the
interdependence of changes in mass Am and energy AE in any pro
cess; thus, the equation can also be written as AE — Amc2.
It cannot be concluded from equation (3.4) that mass can be con
verted to energy or, all the more, that matter can be converted to
energy. Mass and energy are merely different properties of matter.
The first is a measure of its inertness; the second, a measure of motion.
Therefore, they cannot be reduced to one another, nor can they be
converted to each other. The conversion of matter to motion would
imply the possibility of motion without matter, which is of course
absurd. Equation (3.4) only shows that one of the characteristics of
material bodies, their mass, depends on their motion.
The relationship between the wavelength of light and the mass
of a photon can be determined by Planck’s and Einstein’s formulae.
The photon has no rest mass; this is due to the fact that it moves
with the velocity of light (if the mass were at rest, then in accor
dance with equation (3.3) the mass and energy of a photon would
Ch. 3 . W A V E P R O P E R T I E S OF PARTICLES 31
be infinitely large). Therefore, all the mass of a photon is dynamic,

i.e., due to motion1, and can be calculated from the energy of a photon
by means of equation (3.4).
On the other hand, according to Planck’s formula
= (3.5)
Combining this equation with equation (3.4), we obtain
me1
2= h —
wherefrom
This equation expresses the interdependence of the momentum2*

of a photon me and the wavelength of light. It can be written as
K= — (3.7)
P ' 7
where p = momentum (impulse) of a photon.
3.3. Compton Effect

In this phenomenon, when a photon, collides with an electron it
gives up part of its energy to it; as a result, the radiation is scattered
and its wavelength is increased. This effect was discovered by Comp
ton (USA) in 1923. He found that when various substances are ex
posed to X-rays, the wavelength of the scattered radiation is greater
than the original wavelength. The change in the wavelength AX does
not depend on the nature of tjie substance or the wavelength of the
original radiation; it is always a definite value that is determined
by the scattering angle cp (the angle between the directions of the
scattered and original radiations).
It was found that the equation for the precise expression of tho
Compton effect could be derived if the interaction of a photon and
an electron be regarded as an elastic collision of two particles, in
which the laws of conservation of energy and impulse are observed.
Below we will.show how the equation is derived; the derivation is
somewhat simplified.
Let us suppose that a photon of .energy hv (Fig. 3.2a) collides with
an electron, the energy and momentum of which are taken to be
1 This conclusion may seem strange to the beginner, but it merely means that
a motionless photon does not exist, i.e., that light is always in motion.
2 The momentum (or impulse) of a particle is the product of its mass and
velocity. The momentum, in contrast to energy, is a vector quantity; its direc
tion coincides with the direction of the velocity.
zero. After the collision the energy of the photon becomes hv' and
the scattered photon moves at an angle cp to the original direction.
The electron that has obtained from the photon a certain amount of
energy, the recoil electron, moves in a direction that is at an angle 0
to the direction of the original photon.
0>)
Fig. 3.2. Explanation of the Compton effect
(a) schematic representation of the motion of the photon and electron; ( b ) vector addition
of momenta of the recoil electron and scattered photon
According to the law of conservation of energy, the kinetic energy

of the recoil electron T is given by the equation
T = hv — hv' = —h (v' — v) = - M v (3.8)
The kinetic energy of a particle, equal to y mv2, where m and v are
the mass and velocity of the particle, respectively, is related to its
momentum p = mv according to equation
Combining equations (3.8) and (3.9) we find that the equation for
the momentum of the recoil electron is
pt= — 2 mehkv (3.10)
According to the law of conservation of momentum, the sum of
the vectors of the momenta of the scattered photon and recoil
electron is equal to the momentum of the original photon (Fig. 3.2b).
Applying the cosine theorem we obtain
Pt = P2ph (1) + Pin (2) - 2 p p* (1)Pph (2) cos q> (3.11)
where pph (1) and pph(2) are the magnitudes of the impulses of the
original and scattered photons. The difference between these magni
tudes is negligible and therefore it can be taken that
Pph (l) ^ Pph (2)
Ch. 3. W A V E P R O P E R T I E S OF PART I CLES 33
Then equation (3.11) can be written as

p5 = 2 ^ ( d ( 1 - cos<p)
And since
1 — cos cp= 2 sin2
it can be written that
P e = 4 Â (l)Sin2- |- (3-12)
The momentum of a photon, according to equation (3.7), is deter
mined by the equation
h hv
P X c
(3.13)
Substituting the expression for pph (i> from equation (3.13) into
equation (3.12) we obtain
2 , h 2v 2 . 2 cp
>?= 4 —C25—sin2--- (3.14)
Equating the right-hand sides of equations (3.10) and (3.14) we
find that
—meAv = 2 sin2—- (3.15)
If we differentiate the ratio v = c/X, we obtain

dv — — jp d k
Since Av is very small as compared to v, we can write that
A v « — -p-AX (3.16)
Substituting in equation (3.15) the value V by c/X and the value of Av

from equation (3.16), we obtain the equation for the Compton effect!
A?i = 2 —
mec
sin2-|-
2 v(3.17)'
The value hlmec in equation (3.17) has the dimension of length
and is equal to 0.0242 A. It is often called the Compton wavelength
of the electron.
Investigations have shown that equation (3.17) is in good agreem
ent with experimental data.
3.4. De Broglie Waves

Whereas the photoelectric effect and the Compton effect are clearly
indicative of the corpuscular (particle) nature of visible light and
X-ray radiation, interference and diffraction phenbmena are evidence
3 3 a « . 15648
of their wave nature. Therefore, the conclusion can be drawn that

the motion of photons obeys special laws in which corpuscular and
wave properties are combined. The unity of such seemingly incon
gruous characteristics is given by equation (3.6) which correlates the
mass of a photon and the wavelength of radiation.
In 1924 de Broglie (France) suggested that- the dual wave-particle
nature is true not only for the photon but for any other material
particle as well. The motion of any material particle can be regarded
as a wave process for which the following relation is valid:
mv (3.18)
similar to equation (3.6), where m and v are the mass and velocity
of the particle, respectively. These waves are called de Broglie
waves.
De Broglie’s wave hypothesis was supported by electron diffraction
experiments. It was found that when a beam of electrons passed
through a diffraction grating the diffraction picture observed on the
photographic plate was the same as that obtained on passing radiat
ion of a wavelength X calculated from equation (3.18). Metal crystals
were used as the diffraction grating. The atoms are regularly arranged
in the crystals, thus forming a natural diffraction space lattice. In
1927 Davisson and Germer (USA) were the first to carry out such
experiments; in the same year (1927) electron diffraction was also
observed by G. Thomson (England) and P. Tartakovski (USSR).
At the present time electron diffraction is widely used in the study
of the structure of matter (see Sec. 12.1); the electron-diffraction came
ra used for this purpose is now available in all physicochemical labo
ratories. Neutron diffraction is also used in structure studies. The
diffraction of helium atoms, hydrogen molecules, and other particles
has been studied.
Thus, the dual wave-particle nature of material particles is a fact
that is well founded on experimental data.
If we were to calculate the values of X with the aid of equation
(3.18) for various objects, we would find that for macro-objects the
values are vanishingly small. For example, for a particle of a mass
of 1 g moving with a velocity of 1 cm/s, X = 6.6 *10"27 cm. This
means that the wave properties of macro-objects are not revealed
in any way because if the wavelength is considerably less than the
size of the atom (10~8 cm), it is impossible to obtain a diffraction
grating or other device by which the wave nature of the particle
could be observed. It is quite another matter for micro-objects.
The motion of an electron accelerated by a potential difference of
1 V (u =■ 5.94-107 cm/s) is associated with X = 1.22-10~7 cm.
It can be proved that the velocity of propagation of de Broglie
waves w is related to the velocity of the particle v according to the
Ch. 3. W A V E P R O P E R T I E S OF PARTICLES 35
equation
c2
W —- -----
v
(3.19)
where c = velocity of light. Thus we see that only for photons the
velocity of propagation of de Broglie waves coincides with the velo
city of the particle. For all other particles, the velocity of propagation
of these waves is greater than the velocity of light; therefore, de
Broglie waves do not transmit energy, i.e., they cannot be regarded
as real vibration. The concept of de Broglie waves is merely a con
venient method for describing the motion of microparticles. As we
shall see later on, this motion cannot be clearly conceived inasmuch
as such concepts as trajectory and velocity are unsuitable in the
microworld. We shall also see that there is a more general way of
expressing the laws of motion of microparticles, which does not
include the parameter A.
3.5. Quantum Mechanics; the Schrodinger Equation

The mechanics describing the movement of microparticles was
initiated by the investigations of de Broglie. In 1925-1926 Heisenberg
(Germany) and Schrodinger (Austria) independently proposed two
variants of the new mechanics; subsequently it was found that both
variants lead to identical results. Schrodinger’s method proved to be
more convenient for carrying out calculations; the modern theory
of the structure of atoms and molecules is based on his method.
The mechanics of micro-objects is called quantum mechanics (or wave
mechanics); the mechanics based on Newton’s laws, applicable to
the motion of ordinary bodies, is called classical mechanics.
In contrast to the Bohr—Sommerfeld theory, quantum mechanics
is not an artificial combination of classical concepts and the quanti
zation rules; it is a harmonious theory based on a system of concepts
free of any contradictions. All the results obtained by quantum
mechanics are in complete agreement with experimental data.
In quantum mechanics the laws of motion of microparticles are
described by the Schrodinger equation which is of the same significan
ce for it as Newton’s laws are for classical mechanics. Like Newton’s
laws, this equation cannot be deduced from any more general assump
tions. It can be obtained from the analogy between the equations
of mechanics and optics.
The Schrodinger equation is a differential equation involving partial
derivatives', in problems met with in atomic and molecular theory*
the Schrodinger equation for one particle can be written as follows:
(3.20)
3*
where h = Planck’s constant

m = mass of particle
U = potential energy
E = total energy
z/, z = coordinates
The construction of the Schrodinger equation is given in Appen
dix III1.
The variable in this equation is known as the wave function.
The square of this function ty12*has a definite physical meaning cha
racterizing the probability of finding the particle in a given region
of space; to be exact, the value t|)2 dv gives the probability of finding
the given particle in an element of volume dv (see footnote 2 ).
In accordance with the physical meaning of the wave function,
it must be finite, continuous, and single-valued and become zero where
the particle cannot be found. For example, when considering the motion
of an electron in an atom, should become zero at an infinitely great
distance from the nucleus.
The quantum-mechanical solution of problems in atomic and mole
cular theory consists in finding the ^-functions (which have the above
mentioned characteristics) and the energies that satisfy the Schro
dinger equation. The solution of the Schrodinger equation, in most
cases, is a very complicated mathematical problem.
A special system of units is frequently used in the quantum-mechanical
treatment of atoms and molecules; the use of these units makes it possible to
write the equations used and obtained more simply. In this system, the unit
of length is the radius of the first Bohr orbit of the electron in the hydrogen atom:
K2
a0 = — = 0.529 A
me1
and the unit of energy is the absolute value of the potential energy of the elect
ron in this orbit:
me4 e2
E= — = 27.2 eV
a0
The charge and mass of the electron are taken as the units of electric charge
and mass. These units were proposed by the English scientist Hartree; they are
called atomic units (or Hartree units): when atomic units are employed, the
Schrodinger equation for one electron is
d2\|) d2ty
2 ( 3 + dy2 ' dz2
1 Equation (3.20) is applicable only in the case when the state of the system
does not change with time; it is the Schrodinger equation for stationary states.
In its general form, time is included in the equation.
2 The wave function can be expressed by a complex quantity; in this case
the probability of finding the particle in a region of space is determined by the
product ipty*, where if* is a complex conjugate quantity; the quantity is
called the modulus of the wave function and is denoted as |\|)|; hence, in the gene
ral case the probability of finding the particle in an element of volume dv is
given by |\|)|2 dv.
Ch. 3. W A V E P R O P E R T I E S OF PARTICLES 37
The system of concepts of quantum mechanics differs sharply from

that of classical mechanics. Quantum mechanics gives the probability
of finding a particle in a region of space and says nothing of the tra
jectory of the particle, its coordinates, and velocity at a given
moment; such concepts mean nothing in quantum mechanics. Howe
ver, the concepts of mass, energy, and angular moment of the particle
retain their importance.
One of the basic assumptions of quantum mechanics is the Heisen
berg uncertainty principle which states that it is not possible to
define simultaneously the position of a particle and its momentum,
p = mv. The more precisely the position of the particle is determin
ed, the more uncertain is its momentum; and, on the other hand, the
more exactly the momentum is known, the more indefinite is the
position of the particle. The equation expressing the uncertainty
principle can be written as
AxApx > h (3.21)
or
AxAvx > (3.22)
where Ax is the uncertainty of finding the position of the particle
along the x-axis at a given moment; Ap x and &vx are the uncertainties
in the momentum and velocity values in the direction of the x-axis.
Similar equations can be written for the y and z coordinates.
Many of the specific properties of microparticles can be explained
by the uncertainty principle. It often affords a rapid and simple
estimation of a given effect where calculation would be complicated.
This can be shown by the following example. Let us consider, with
the aid of the uncertainty principle, the motion of the electron in
the hydrogen atom.
Let us assume that the motion of the electron occurs in a region
with a radius r. Then, the uncertainty of its position can be taken
to be equal to r and, according to equation (3.21), the minimum
uncertainty of Ap in the value of the momentum of the electron p
is equal to hlr. It is obvious that the value of the momentum cannot
be less than the uncertainty of its value; therefore, the minimum
possible momentum value will be
p= ± (3.23)
The energy of the electron is equal to the sum of its kinetic energy
(which according to equation (3.9) is p2/2me) and its potential energy
(which at a distance r from the nucleus is —e2/r). Hence, substituting
the value of p from equation (3.23) into the expression for the kinetic
energy, we find that the total energy of the electron in the hydrogen
atom is
K2
E 2mer2 r
(3.24)
38 P A R T 1. A T O M I C S T R U C T U R E
This dependence of the energy of the electron on r is shown in

Fig. 3.3. As can be seen, the curve has a minimum;, the va:Uu* of r 0
at which E has a minimum value can easily be found since at the
point, corresponding to the minimum, dE/dr = 0. Differentiation
gives
wherefrom
Jl2
r0 = — 7 (3.25)
Substituting this value of r„ into equation (3.24), we find the equat
ion for the minimum energy of the electron in the hydrogen atom:
(3.26)
. The result obtained is very signi
ficant. According to the classical
conception the electron would have
a minimum energy when it fell
onto the nucleus; however, quantum
mechanics shows that the energy of
the electron is minimum, not when
it ‘rests on the nucleus’, but when
Fig. 3.3. Dependence of electron energy

on the radius of the region of space in
which the electron moves
it moves within a sphere with a radius r0, its exact position

within this region of space being unknown. When r < r 0, the energy
of the electron increases.
A comparison of equations (3.25) and (2.12) shows that the value
of r 0 is equal to the radius of the first Bohr orbit, and by comparing
equations (3.26) and (2.15) it is seen that Emin found by applying
the uncertainty principle coincides with the minimum energy value
of the electron in the hydrogen atom according to Bohr’s theory and
the study of the hydrogen spectrum. This interpretation of course is
approximate because the motion of an electron in an atom cannot
be confined to an exactly prescribed region. It explains, however,
why the electron does not fall onto the nucleus and makes it
possible to correctly estimate the minimum value of the energy of
an electron. This result can be obtained by solving the Schrodinger
equation for the hydrogen atom, but this would involve very compli
cated mathematical calculations.
Ch. 4. Q U A N T U M - M E C H A N I C A L E X P L A N A T I O N OF A T O M I C S T R U C T U R E 39
As we shall see further on (see p. 52) the radius of the first Bohr
orbit is equal to the distance from the nucleus at which the electron
of the hydrogen atom is most probably to be found (for the state
of minimum energy).
Since the value of h in the uncertainty equation is very small, the
uncertainties of the values of the coordinates and the momentum of
macro-objects are negligibly small and therefore the effect due to
them cannot be observed by any apparatus. When dealing with the
motion of macro-objects, it is necessary to speak of precise trajectori
es and to apply classical mechanics.
CHAPTER FOUR
QUANTUM-MECHANICAL EXPLANATION OF ATOMIC STRUCTURE
4.1. Solution of the Schrodinger Equation

for the One-Dimensional Square-Well Model
The solutions of the Schrodinger equation for problems met with
in the theory of atomic and molecular structure are very complicated
and cannot therefore be set forth in this book. However-^ to understand
the nature of the results of the
quantum-mechanical study of
the atom discussed below, it is
worth while to analyze the solu
tion of the Schrodinger equation
for at least one simple example.
Let us solve it for the hypo
IAc
i r
thetical square-well model.
First let us obtain the solution
for a one-dimensional particle-in-a-
x
Fig. 4.1. One-dimensional square po v=o
tential well x= 0 x= a*
box problem. In this model the particle (for example, an electron)

can move only in one direction, for instance, along the ar-axis from
x = 0 to x = a (Fig. 4.1)1. Within the limits of this region the poten
tial energy of the particle U is uniform; it is convenient to take it
to be equal to zero (the potential energy can be reckoned from any
1 Although the model is highly simplified, it is used for the solution of

actual problems. Thus, for example, the motion of ji-electrons in a system of con
jugated double bonds in organic compounds (see p. 194 and Appendix IX) can
be regarded approximately as motion in a one-dimensional square well.
arbitrary level). Outside this region fhe potential V acting on the

particle is infinitely high; that means that the particle cannot go
beyond the region of length 0 < x < a (this would require an infini
tely large increase in its energy).
The Schrodinger equation (3.20) for the one-dimensional square-
well model is
h2 d2\p
8n2m dx2
E\|? (4.1)
To solve this equation it is necessary to find the ^-function and
the value of energy E that would satisfy the equation; the magnitude
of the 'if-function should be finite, single-valued and continuous and
should be equal to zero when x = 0 and x = a (since the particle
cannot be located outside this region, the probability of finding it
beyond these points, determined by i|)2, is equal to zero; in order
that the function remain continuous, it must also be equal to zero
at points x = 0 and x = a).
The ^-function that satisfies the above conditions is
\1?=- A sin (4.2)
where n = 1, 2, 3, . . . and A is a constant value.
The value of n cannot be zero because that would imply the absen
ce of the particle in the well (\|?2 = 0 ).
That this function satisfies the Schrodinger equation can be shown
by substituting it into the left-hand and right-hand sides of equation
(4.1). Then after differentiating equation (4.2) we obtain
left-hand side:
K2 / Jl2JX2 mix rftti2, A nnx
5— x A Sin---
8ji2m \ "a2" a 8ma2 a
right-hand side:
nnx
EA sin a
It is evident that the two sides are equal if the energy of the particle
is determined by the relation
n2h2
E = 8ma2 (4.3)
where n = 1, 2, 3, . . .
Thus, we have found the ^-function and the energy value that
satisfy equation (4.1), i.e., we have solved the Schrodinger equation
for the one-dimensional square-well problem. Let us analyze the
solution obtained.
First of all* note should be taken of the sharp difference between
the result obtained and the picture observed in an analogous problem
for a particle for which the laws of classical mechanics hold. Obvi-
Ch. 4. Q U A N T U M - M E C H A N I C A L EX P L ANA T I ON OF ATOMI C S T R U C T U R E 41
ously, the energy of such a particle could have any value and the
probability of finding it would be the same for any point on the
x-axis.
On the contrary, as seen from equation (4.3), the energy of
a particle for which the laws of quantum mechanics hold can have
only a number of definite values characterized by the integral coeffi
cient n. The energy levels for a particle in a one-dimensional box
are shown in Fig. 4.2. Pay due attention to the fact that quantization
of energy is the inevitable result of the solu
tion of the Schrodinger equation, although the
E
equation itself does not contain integral
^-coefficients. n=6
This solution which shows that for micropar 36E,
ticles there are a number of permissible energy
values is characteristic not only of the motion
in the potential well: a similar result is n=E
obtained when considering any problem in Z5E1
which the microparticle is confined to a defi
nite region of space under the action of forces //--4
(see p. 45). Thus, quantum mechanics explains 16E;
n=3
9E1
n=Z
Fig. 4.2. Energy levels of a particle in a one-dimen n~1
sional square potential well (£1 is the zero-point
energy corresponding to n = 1) Eb
the existence of discrete energy levels for electrons in atoms and
molecules (this is indicated by spectra) and makes it possible to
theoretically calculate the magnitude of these energies.
Since in the equation for the energy of a particle in the potential
well n =£ 0, E also cannot be equal to zero. The minimum energy
(zero-point energy) corresponds to n = 1 .
The fact that the particle possesses a zero-point energy is one of the speci
fic characteristics of the microcosm. It is connected with the particle-wave natu
re of microparticles and stems from the uncertainty principle. We have already
seen (see p. 38) that the location of an electron in a particular region of space
results in the appearance in it of a certain impulse and, consequently, kinetic
energy which is the greater the more the motion of the electron is restricted.
The same can be said of any other microparticle. There is no state of matter in
which the kinetic energy of its particles is equal to zero. Even at absolute
zero, not only electrons, but even atoms as a whole, will be in constant motion,-
vibrating about the equilibrium position. The zero-point vibrations of atoms
affect many of the properties of substances. Their actual occurrence is confir
med by studies of diffraction of X-rays by crystals. These studies show that
even at temperatures close to absolute zero, there is a certain disorder in the
spacial distribution of atoms due to their zero-point vibrations.
The fact that atoms and other particles have a zero-point energy, as proved
by quantum mechanics, once more confirms the assertion of dialectical
materialism that matter cannot exist without motion.
AZ P A R T I. A T O M I C S T R U C T U R E
----------------------------------------------- 3----------------------------------------
Figure 4.3 shows the ^-function and ^-function for a particle in
a one-dimensional potential well where n = 1, 2, and 3. The diagram
showing the dependence of on x is similar to the picture of the
vibration of a string fixed at two ends, where only such vibration
is possible in which a whole number of half-waves go along the string.
As can be seen in Fig. 4.3,
the ^-functions also differ
considerably from the clas
sical pattern. From Fig. 4.3
it is obvious that the pro
bability of finding a par
ticle at various points of
the potential well is not the
same. Besides that, at va
lues of n > 1 , the proba
bility of finding the particle
at certain points inside the
well is equal to zero—which
is absolutely impossible
from the standpoint of clas
sical conceptions.
However, as seen from
equation (4.3), if the mass
of the particle m (and,
Fig. 4.3. if>- and i|)2-functions of

a particle in a unidimensional
square potential well
consequently, the value a) and its energy E are large enough,

the motion of the particle does not particularly differ from the classi
cal one; i.e., the permissible energy levels will lie so close to each
other that they cannot be distinguished experimentally, and it can
be considered that the particle can have any energy value. Thus, for
macro-objects, quantum mechanics leads to the same results as classi
cal mechanics does.
As concerns the constant A , it can be of any value from the point
of view of mathematical requirements. However, because of the
physical meaning of the \|)2-function, it is necessary to choose a defini
te value of A, namely such a value that the total probability of
finding the particle in the potential well be equal to unity. This
condition is expressed mathematically by the equation
\|)2 dx —■1 (4.4)

Such a value of A must be found that satisfies equation (4.4). By

substituting into equation (4.4) the expression for the ^-function
given in equation (4.2) and solving the integral we obtain
a ’- - Y ?
This mathematical operation, called normalization, is performed
in all cases when it is necessary to find the complete expression for
the wave function, i.e., to determine the constant value that it
always contains; this constant value is called the normalization
factor. In the general form, the equation is written as follows:
^ty2d v = l (4.5)
where dv = element of volume. Integration is carried out for the
entire volume from the value —oo to + ° ° for each of the coordinates.
Equation (4.5), the same as equation (4.4), shows that the total
probability of finding the particle is equal to unity, i.e., that the
particle can actually be found somewhere in space.
4.2. Three-Dimensional Square-Well Model

The existence of discrete energy levels of an electron in an atom
becomes clear from the above solution of the Schrodinger equation
for the one-dimensional square well.
To explain other characteristics of
atomic structure it is advisable to
consider the motion of a particle in
a three-dimensional square well.
In this problem the particle is
confined within a space inside a po
tential well, a cube with a side a. The
Fig. 4.4. Three-dimensional square poten

tial well
origin of the coordinates is in one corner of the cube (Fig. 4.4). The
potential energy of the particle within the square well is constant;
beyond it the potential is infinitely great and therefore the particle
can under no circumstances be outside the well.
As in the previous one-dimensional problem, the three-dimen
sional square-well model is a hypothetical one. However, there
actually exists a phenomenon for which these conditions are to
some extent true, namely, the motion of conduction electrons in
a piece of metal. These electrons move in all directions, but they

do not go beyond the piece of metal. Therefore, the three-dimensional
square-well model is used in the theory of the metallic state.
In this case it is necessary to solve the Schrodinger equation for
three dimensions. When solving such problems, the following pro
cedure is usually employed: the equation is analyzed and an attempt
is made to separate it into parts each of which contains only one
of the three coordinates. If this can be done, the sought function
is found more simply. Let us attempt to find the- T|?-function in this
way.
The potential energy U (x , y, z) can be regarded as the sum of
three terms each of which is a function of only one coordinate:
U (x, y, z) = U (x) + U (y) + U (z) (4.6)
The total energy is separated in a similar way. The velocity of the
particle v being a vector quantity can be resolved into its compo
nents along the coordinate axes, ux, vy, and vz, and the kinetic energy
can be expressed as the sum of the corresponding terms; accordingly,
in view of this and equation (4.6) we can write
E = EX+ Ey + Ez (4.7)
Let us now assume that the sought ^-function can be expressed
as the product of three functions each of which depends only on
one coordinate, i.e.,
'P (x, y, z) = X (x) Y (y) Z (z) (4.8)
and let us see if we can solve the Schrodinger equation by means of
the function thus divided. For the sake of simplicity we will denote
the functions X (x), Y (y), and Z (z) simply as X, Y and Z.
By substituting into equation (3.20) the new function given in
equation (4.8) and the expressions given in equations (4.6) and (4.7)
we obtain
+ vv> t p « i x
X Y Z = (Ex + Ey + Ez) X Y Z
X
On dividing this equation by X YZ we obtain
h* ( i d * X 1 d*Y ld * Z \
8ji2™ \ X dx* + Y dy% ' Z dz2 / +
+ U (x) b U (y) b U (z) - Ex + Ey + Ez
The resulting equation can be considered to be the sum of three
identical equations of the type
h? 1 d*X
8nzm X dx2 4-U(x) = Ex
or
h2 d2X
8n2m dx2
U (x)X -= E xX (4.9)
We already know the solution of equation (4.9); we obtained
it for the one-dimensional square-well problem (the function U (x)
according to the conditions of the problem can be taken to be equal
to zero).
Thus, such a division makes it possible to find the ^-function
and the energy of the particle E\ the wave function is expressed by
equation (4.8) in which
X {x )-Â Xs i n w h e r e nx ^ 1, 2, 3, . . .
Y (y) -- A y sin where ^ = 1, 2, 3, . . . (4.10)
Z (z) = Az sin — - where nz = 1; 2, 3, . . .
nth2 n\h2
E --- Ex -[- Ey -- ---- 1- ---- 1__ -— = {nl + nl f nl) h,z
Ez = 8ma2 (4.11)
8ma2 ' 8 m a 2
' 8 ma2
As for the one-dimensional square potential well, the values nx, ny
and nz can only be whole numbers {integers). Thus, passing from the
one-dimensional to the three-dimensional problem has resulted
in the appearance of three integral characteristics in the expression
for the wave function.
This result has a general meaning. The quantum-mechanical
treatment of various cases of motion of microparticles within
a confined region of space (say, in an atom, molecule, etc .)1 shows
that the wave function of a particle always contains dimensionless
parameters which can take a number of integral values. These values
are called quantum numbers. The number of quantum numbers
contained in the solution is equal to that of the degrees of freedom
of the particle. The number of degrees of freedom is the number of
independent components of motion of the particle. Thus, in the one
dimensional potential square well, the particle has only one degree
of freedom; in the case of translatory motion in space, it has three
degrees of freedom (motion is possible in the direction of each of the
three coordinates x , y and z)\ if, besides this, the particle can rotate
round its own axis, it has a fourth degree of freedom.
Whereas in the problem of the motion of a particle in the one
dimensional potential square well different values of the quantum
1 The expression ‘confined region of space’ implies that the particle is

confined to a given region by the action of some forces and the probability
•of finding it beyond this region is close to zero.
numbers correspond to different energies, in the three-dimensional

problem there are states characterized by different quantum numbers
but by one and the same energy. Thus, at nx = 2, ny = 1, and
nz = 1 the energy of the particle will be the same as when nx = 1 ,
ny = 2, and n z = 1. If one and the same energy corresponds to
several different states (characterized by different wave functions)
it is said that the given energy level is degenerate. Depending on the
number of states, degeneration may be two-fold, three-fold, etc.
In the simplified examples considered above we have come to
know some of the laws of quantum mechanics. Now we can investi
gate the motion of electrons in real systems, i.e.t in atoms of chemi
cal elements.
4.3. Quantum-Mechanical Explanation of Structure

of Hydrogen Atom
The hydrogen atom has the simplest structure; it has only one
electron moving in the field of force of the nucleus. In this case
the potential-energy function U
in the Schrodinger equation
(see p. 26) is expressed by
U= — — (4.12)
This seemingly insignificant com
plication of the equation as com
pared with the square-well prob
lem presents some mathematical
complexities that cannot be dealt
with in this book. We shall there
fore consider only the basic pecu
liarities of this solution and their
physical meaning.
Fig. 4.5. Polar system of coordinates
It is convenient to consider the movement of the electron in such

problems in polar coordinates, the centre of which coincides with
the nucleus of the atom (Fig. 4.5). Whereas in the Cartesian (rectan
gular) coordinates the position of a particle in space is given by the
coordinates x , y and z, in the polar coordinates it is defined by the
radius-vector r (the distance from the centre) and the angles 0 and <p.
In Fig. 4.5 it can be seen that the polar coordinates are related to
the rectangular coordinates as follows:
x = r sin 0 cos q), y = r sin 0 sin <p, z = rc o s 0 (4.13)
As in the solution of the three-dimensional square-well problem,

the ^-function should be presented as the product of three functions,
each of which contains only one variable:
^ (r, 0 , q>) = R (r) 0 (0) O ((p)
The expression R (r) is called the radial component of the wave
function; the product 0 (0 ) O (<p) is known as its angular component.
The presence of three degrees of freedom results in the appearance
of three values which can only be integers, i.e., three quantum
numbers denoted as n, I and m. These values are present in both
the radial and angular components of the wave function; in its
most general form the solution of the Schrodinger equation for the
hydrogen atom can be expressed as follows:
(4.14)
The quantum numbers n, I and m can have the following values:

n= 1 , 2, 3, 4, . . oo
*= 0, 1, 2, 3, . . . , ( 7 i- l) (4.15)
m —0 , =b 1 , ± 2 , ± 3 , . .. , ± I
As we shall see below, the quantum numbers n, I and m characte
rize the motion of the electron not only in the hydrogen atom, but
in any other atom as well.
As seen from equation (4.14), the quantum numbers n and I enter
into the expression of the /?-function; they therefore determine the
function of radial distribution of the probability of finding ari' electron
in an atom. These functions for the hydrogen atom are shown gra
phically in Fig. 4.6. Along the axis of ordinates are plotted the values
R 2 (r) multiplied by 4jtr2. These values characterize the probability
of finding the electron in a thin spherical shell of radius r; this
probability is proportional to 4jtr2R 2 (r) dr where dr is the thickness
of the shell.
From Fig. 4.6 it follows that in contrast to the Bohr—Sommerfeld
theory according to which the electron moves in exactly prescribed
orbits, quantum mechanics shows that the electron can be found
at any point in the atom although the probability of it being in
different regions of space is not the same. Thus, if we could observe
the electron in the atom, we would see that it is more often in certain
places and less often in others. Therefore, the modern conception
is that of an electron cloud the density of which at different points
is determined by the probability of finding the electron there. There
fore in the scientific literature instead of the term ‘orbit’, the term
now used is ‘orbital’ meaning the particular pattern traced out by an
Fig. 4.6. Radial distribution of the probability of finding the electron for diffe
rent states of the hydrogen atom
Fig. 4.7. Shapes of electron clouds for different states of electrons in atoms
electron in an atom (the electron distribution pattern). Each orbital

corresponds to a definite wave function ty.
The following designations are used to represent the electron
state of atoms: the quantum number n is designated by figures and Z,
by small letters according to the following scheme:
I ............................... 012345
Designation . . . . s p d f g h
The first four, letters coincide with the designations of spectral
series (see p. 22 ); the origination of these series is due to the transit
ion of electrons, corresponding to definite values of the quantum
number I. The last two letters, g and h, are the letters following /
in the alphabet. Thus, 1s is the designation for an electron for which
n = 1 and l'= 0 ; 2p, for an electron for which n = 2 and I = 1 ,
etc. The number of electrons in an atom with the given values of n
Table 4.1
The Wave Functions of the Electron in the Hydrogen Atom
Designation
of orbital Radial component Angular component
is 2e~
2~)/n
2s ~ (2— r)e-r/2 Same
21/2
1 P-r /2 1/3
2Px
21/6 21/ ji
- (*/')
Same
V3
2pz — (z/r)
2 "|/n
1/3
2py Same (y/r)
21/n
1 /1 5 [(l2_ j, 2)/r2]
U X2_?/2 — r*e~r/3
811/30 41/n
3dXz Same
21/2 ji
3dz Same
1/5— [(3z2- r 2)/r2]
^Z2
41/ a
*yz
3dy Same
21/ 2ji
3dxy Same
4"l/jt
4 3aK . 15648
and Zis indicated by a superscript. Thus, 2s2 shows that in the atom
there are two electrons with n = 2 and Z = 0.
Table 4.1 gives the wave functions for certain electron states of
the hydrogen atom. They are given in atomic units. Moreover, for
the sake of abbreviation, the trigonometric functions of the angles
are defined by the Cartesian coordinates x , y, and z and the distance r.
The motion of the electron in ions with a single electron (He+, Li2+,
etc.) is characterized by similar wave functions; to find the expres
sions for these ions the given wave functions, must be multiplied
by Z3/2 and r must be replaced by Zr. ‘
Figure 4.7 shows the electron clouds for different states of electrons.
It presents the shape of the surfaces defining the space in which
the greater part (~90% ) of the electron cloud is enclosed. The shape
of these surfaces is determined by the angular component of the
wave function 0 (0) O (<p). This pictorial representation of electron
clouds will be repeatedly used in the subsequent discussion.
4.4. Quantum Numbers of Electrons in Atoms

The quantum numbers characterize the motion of electrons not
only in the hydrogen atom but in any other atom as well. These
characteristics are very important for understanding the properties
of substances and the nature of chemical bonding. Therefore we shall
discuss their significance in greater detail.
The quantum numbers /z, Z, and m determine the geometric pattern
of the electron cloud. They are also associated with the physical
characteristics of the motion of the electron.
The quantum number n is equal to the number, of nodal surfaces
(nodes) of the orbital. A node is the locus of points at which oj) = 0 .
Obviously, if \|) = 0, ij)2 = 0; therefore, the density of an electron
cloud in a node is equal to zero. A surface infinitely removed from
the nucleus is also a nodal surface; as we already know, in this
case is always equal to zero.
The existence of nodal surfaces in the distribution of electron
density stems from the laws of the microcosm. The motion of the
microparticles is described by relations analogous to the equations
of wave motion. In every wave there are points where the amplitude
of vibration is equal to zero. If vibration is three-dimensional, the
sum of these points form a nodal surface. Nodal surfaces (nodes)
in atoms may be of two kinds: those that do not pass through the
centre of the atom (the nucleus) and those that pass through it. The
first are concentric spherical nodes the centres of which coincide
with the nucleus; the second kind are nodal planes or cones which
pass through its centre. The presence of spherical nodes is shown in
the radial component of the wave function—at certain distances
from the nucleus, \|? is equal to zero; this can be seen in Fig. 4.6.
Ch. 4. Q U A N T U M - M E C H A N I C A L E X P L A N A T I O N OF A T O M I C S T F UCTIJRE 51
The value I shows how many nodes of the wave junction oj the
electron pass through the nucleus. As has already been pointed out,
one of the nodes is always at an infinitely great distance from the
nucleus. It is therefore obvious that I can vary from 0 to /z — 1.
Fig. 4.8 shows the positions of nodes that pass through the centre
of the atom, for different states of the electron. It is worth comparing
this Figure with Fig. 4.7.
As has been noted above, according to quantum mechanics the
electron in an atom can be at any distance from the nucleus; however
z z z
Fig* 4*8. Position of nodal planes for different states of an electron
the probability of it being in different places of the atom is not

the same. Knowing the density distribution of an electron, we can
calculate the average distance of the electron from the nucleus, rav,
which characterizes the size of the orbital; rav can be found by inte
grating the function of radial distribution. The value rav is deter
mined by the values n and Z. For the electron in the hydrogen atom
and hydrogen-like ions, the relation between these values is express
ed as follows:
(4.16)
where Z = charge on the nucleus
a0 = radius of the first Bohr orbit
From equation (4.16) it can be seen that raU is approximately pro

portional to n2. Hence, it can be stated that n determines the size
of *the electron orbital.
It should be noted that the maximum probability of finding the
position of the electron in the hydrogen atom for the is , 2p, 3d, 4/,
etc. states coincides with the radius of the corresponding Bohr orbit
(see Fig. 4.6).
The energy of the electron in the hydrogen atom depends only
on the value n\ the solution of the Schrodinger equation gives the
relation
1 mee4
Y nVi2
(4.17)
As can be seen, it is the same expression as in the Bohr theory [see

equation (2.15)], but in contrast to the latter, quantum mechanics
reaches this result without having recourse to the arbitrary hypothe
sis that the electron can move in a definite set of orbits defined by
-a series of integers.
Since n determines the principal characteristic of the electron in
the hydrogen atom, its energy, this value is called the principal
quantum number. The quantum number I is called the orbital quantum
number; the latter determines the orbital angular momentum of the
electron M:
M = n V l ( / + 1) (4 . 18)
As we already know (see p. 26), the angular momentum is a vector
quantity. Its direction is determined by the quantum number m.
In other words, m characterizes the position of the orbital in space.
The direction of the vector can be given by its projection on one of
the axes, for example, on the z-axis (the projection of the orbital
momentum can be found only on one axis because in accordance with
the uncertainty principle the other projections cannot be found;
if we knew all three projections, we would know the trajectory of the
electron). The projection of the orbital angular momentum is deter
mined by the relation
M z = hm (4.19)
The quantum number m is called the magnetic quantum number1
because the projection of the orbital magnetic moment of the electron
depends on it (see Appendix VIII).
1 Since the absolute value of m cannot exceed I [see equation (4.15)], equati
ons (4.18) and (4.19) show that the projection of the orbital angular momentum
is always less than its full value. This also follows from the uncertainty princip
le. If the projection of the orbital momentum on the z-axis were equal to the
full value, we would know that its projections on the z-axis and y- axis are equal
to zero; hence, the value and exact direction of the vector of angular momentum
of the electron would be known, but this contradicts the uncertainty principle.
The quantum numbers n , I, and m in the solution of the Schrodin-

ger equation for the hydrogen atom do not completely describe the
motion of electrons in atoms. The study of the spectra and other
investigations have shown that there is an additional characteristic
resulting from a fourth degree of freedom, the so-called ‘spin’ of the
electron, or its rotation on its own axis.
The spin1 or self-rotation of the electron is due to its inherent
angular momentum. This characteristic of the electron is as fundamen
tal as its charge and mass12. As shown by experimental investigations,
the projection of the inherent angular momentum of the electron can
1 1
l\ave only two values, + y ^ and — h\ the plus and minus signs
correspond to different directions of rotation of the electron. Hence,
1 1
the spin quantum number s can have only two values, + y and — ^ *
i.e., these numbers differ by unity as do all quantum numbers. The
spin can be allowed for in the expression for the wave function by
introducing a factor (spin function).
The four quantum numbers n, Z, m and s completely describe
the motion of the electron in the atom. This motion can have no other
characteristics independent of the quantum numbers.
Since the energy of the electron in the hydrogen atom is determined
by the value n and does not depend on the other quantum numbers,
it is evident that there may be several electron states with the same
energy. These states are degenerate (see p. 46). Degeneration disap
pears when an external electric or magnetic field acts on the electron
in the atom. An electron in a state with the same n but with different
values of I, m, or s reacts differently with the external field; as
a result the energies of these states become dissimilar. This explains
the resolution of spectral lines when the source of radiation is placed
in an electric or magnetic field (Stark and Zeeman effects).
It is evident that all that has been said about the hydrogen atom
is completely applicable to the other one-electron systems, such as
H e+, Li2+. The energy of the electron in this case is equal to
1 mee*Z2
Y n*hz (4.20)
1 By spin or self-rotation is only meant that the electron possesses an inhe

rent angular momentum. The electron cannot he considered as a rotating
charged sphere; in particular, if the velocity of a point on the surface of such
a sphere were to be calculated by means of electrodynamic equations, the velo
city obtained would be greater than that of light. Spin, like the other character
ristics of the motion of microparticles, cannot be associated with any model
based on macro concepts.
2 It should be borne in mind that there are particles that have a spin equal
to a whole number (for example, zero).
4.5. Many-Electron Atoms

As in the hydrogen atom, in many-electron atoms the state of each
electron is determined by the values of the four quantum numbers n , Z,
m and s that can have the same values as in the hydrogen atom.
In many-electron atoms, the electron moves not only in the field
of the nucleus but also in the field of other electrons; due to this
factor the energies of electrons with
n= 1 Z J S 6 the same n but different I become
different (the reason for this will
be discussed below on p. 86 ).
Therefore, the energy of electrons
± in many-electron atoms is determin
ed by the values of the two quan
tum numbers n and Z, increasing
both with an increase in n and
Z. The dependence of the energy
on Zbecomes greater as compared
to its dependence on w, the greater
the number of electrons in the
atom. Thus, for the electron in the
sodium atom that is farthest remo
ved from the nucleus, the difference
<*> in the energies for levels with qua
ntum numbers n = 3, 1 = 0 (35)
and n = 3, Z = 1 (3/>) is equal
to 2.1 eV; this value approximates
the difference between the energies
of levels with n = 3,- Z = 0 (3s)
and n — 4, Z = 0 (45) which is
equal to 3.1 eV. For atoms that con-
Fig. 4.9. Sequence of energy levels in

many-electron atoms (schematic repre-
' sentation)
tain an even greater number of electrons, the influence of Z, in certain
cases, can be greater than that of n; this fact determines the structure
of a number of many-electron atoms.
In general, the energy levels of many-electron atoms can be describ
ed as follows. The energies of the levels ns, (n — 1) d, and (n — 2) /
do not differ considerably and are always less than that of the level
up. Thus, the sequence of energy levels, in the order of increasing
energy, is approximately as follows:
15 < 2 s < 2 p < 35 < 3p < 45 « 3d < 4p < 5s «
<C 5p < 65 ^ 5d « 4/ < Op
Ch. 4 . Q U A N T U M - M E C H A N I C A L E X P L A N A T I O N OF A T O M I C S T R U C T U R E 55
A schematic representation of the relative placement of the energy

levels in many-electron atoms is given in Fig. 4.9. In order not
to make the diagram too long, the levels corresponding to n = 1
and n = 2 are positioned higher than they are actually located. The
diagram is approximate (it holds for the first twenty elements)
because the placement of the levels varies noticeably from atom to
atom (see pp 82-83).
The state of electrons in many-electron atoms always satisfies
the quantum-mechanical rule formulated by Pauli (the Pauli exclus
ion principle). This principle states that in any atom or molecule no
two electrons can have the same four quantum numbers. The Pauli
exclusion principle restricts the number of electrons in an atom that
have a given value of n. Let us find the number for n = 1 and n = 2.
If n = 1, then I and m can only be equal to zero [see equation
(4.15)]. Consequently, electrons with n = 1 can differ only in their
spin quantum numbers. Hence, in an atom there can be only two
electrons with the principal quantum number n = 1 :
n I m s
1st e le c t r o n ................. . . 1 0 0 + 1/2
2nd electro n .......................... 1 0 0 —1/2
The same reasoning shows that in the case when n = 2, there can be
only eight different combinations of the four quantum numbers:
n I m s n I m s
2 0 0 + 1/2 2 1 0 + 1/2
2 0 0 - 1/2 2 1 0 - 1/2
2 1 -1 + 1/2 2 1 +1 + 1/2
2 1 —1 - 1/2 2 1 +1 - 1/2
In the same way it can be found that for n = 3 the maximum num
ber of electrons is equal to 18; for n = 4 it is equal to 32, etc. In the
general case the maximum number of electrons in an atom which can
have a given value of n is equal to 2n2. Table 4.2 gives the quantum
numbers for different electronic states.
Since the value n determines the average distance of the electron
from the nucleus, we will call the system of all the electrons in an
atom with the same n-value an electron shell. Electron shells are
designated by the following capital letters:
n .................................................. 1 2 3 4 5 6 7
Designation of s h e l l .................. K L M N 0 P Q
The system of electrons with the same /-value and the same rc-value
we will call an electron subshell; subshells are distinguished as
.s-subshells, p-subshells, etc.
Accordingly, the number of electrons in an electron shell cannot
exceed 2 n2; thus, in the first shell there cannot be more than two
CM
X
+ I CO
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+1 CM CM
•'th -r(
Hr I
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<M CM
+ 1 CO
+1 CM CM
+ 1 CO
CM CM
+7 CO
Quantum Numbers for Different Electron States
CM CM
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+ 1
+1 CM CM
X
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+ 1
CM CM
TT-t TH
+ I CO
+ I
CM CM
tH
+ 1 £
+1 CM CM
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given value of n
electrons with a
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SxJ O
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C C co
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can co Qj too
s
electrons; in the second shell, more than eight electrons, etc. The
maximum number of electrons in a subshell is equal to 2 (21 + 1).
Whereas there can be only two electrons (with opposite spins) in
the 5-subshell, the number of electrons in the p-subshell can be six.
Therefore, it is necessary to ascertain how the spins of the electrons
are oriented. Thus, for the nitrogen atom of the electron configurat
ion ls 22s22p3 (2 electrons in the first shell and 5 in the second) there
are two conceivable variants:
,----- A___ ,
s
2TfT T T
1i and
r
Each cell in these diagrams corresponds to a definite orbital.
On each orbital there can be two electrons with opposite spins. In the
first diagram all the p-electrons have different m-values; in the second
diagram, two p-electrons have the same m-value.
Which of these variants should be preferred? This question is
appropriate because for heavy atoms the number of conceivable
variants is considerable, since in the d-subshell the number of orbi
tals is equal to five and in the /-subshell, to seven; i.e., the first
subshell may have ten and the second subshell fourteen electrons.
Quantum mechanics and analysis of spectra show that filling of
the orbitals (the lowest energy state of the atom) proceeds as follows.
When electrons are being added to a subshell, the orbitals correspond
ing to different values of the magnetic quantum number are first
singly occupied by electrons with parallel spin before any become
doubly occupied. In other words, the subshells are filled with electrons
in such a way that the total spin is maximum1. This important conclu
sion is known as Hund's rule.
In Part Two of this book it will be shown that the periodicity
of the chemical and physical properties of the elements is due to the
gradual filling up of the electron shells; each subsequent shell is
essentially similar in construction to the preceding one.
4.6. Origination of Spectra

If the atom is not exposed to the action of any external force, its
electrons are in the lowest energy state or the ground state. If energy
is transmitted to the atom (as a result of collision with another atom,
the absorption of a light quantum, impingement of electrons, etc.)
1 The spin due to the inherent angular momentum of the electron is a vector
and is therefore designated by an arrow. The sum of the spins of two electrons
with opposite spins is equal to zero.
one or several electrons in the atom may pass to a higher energy level
and become ‘excited' . As a rule the atom persists in the excited state
for a very short period of time (of the order of 10~5 to 10“8 s), after
which the electron returns to the lowest energy level and the atom
returns to the ground state. If there are intermediate levels between
the lowest energy level and the one at which the electron is located,
the return to the ground state can take place in several stages.
When an electron returns from a higher energy level to a lower
one, the atom radiates a quantum of light the frequency of which
according to Planck’s equation (2.5) is determined by the relation
E 2 - E i = hv (4.21)
This frequency characterizes the respective spectral line. Thus, the
appearance of each spectral line is due to the transition of an electron
from one energy level to another (see Fig. 2.4). Therefore, the spectrum
of an element indicates the energy transitions of electrons that take
place when an excited atom returns to the ground state.
The transitions of electrons between the inner shells produce
X-rays of a wavelength much shorter than that of visible light. This
is so because the inner electrons are more strongly bound to the
nucleus; hence, their transitions are accompanied by greater energy
changes, which according to equation (4.21) results in radiation of
high frequency and, consequently, of short wavelength. X-ray spectra
consist of few lines; proceeding from one element to another their
frequencies vary continuously with an increase in the charge of the
nucleus (see p. 67).
The transitions of the outer electrons of atoms are accompanied by
smaller energy changes and produce spectra in the visible and ultra
violet regions.
The study of spectra makes it possible to deduce the electronic
structure of atoms of elements, i.e., to find the values of the quantum
numbers and the energy of electrons in atoms (generally, by the
‘electronic structure of an atom’ is meant its structure in the ground
state).
The determination of electronic structure from spectral a&ta is
often a very difficult task. For this purpose it is necessary to assign
the lines in the spectrum to definite series and to find out, with the
aid of the rules of quantum mechanics, what transition caused the
appearance of each spectral line. Because of the great number of
lines in spectra, that is not an easy task. However, as a result of the
painstaking work of a great number of investigators, the electronic
structure of most of the elements is now known. Mendeleev’s Periodic
Law has played a great part in the systematization and interpretat
ion of atomic spectra.
Figure 4.10 presents schematically the energy levels of the outer
electron of the lithium atom. The lines connecting the different
levels show the transitions of the electrons; the wavelengths of the

respective spectral lines in A are also given in the diagram. As can
be seen, for the outer electron in the lithium atom, the 25-state is
the one with minimum energy. The spectrum of lithium can be plott
ed with the aid of this dia
gram. The investigator who
studies spectra solves the
reverse problem: his task
is to construct an energy
level diagram (like the one
shown in Fig. 4.10) from
the spectroscopic data.
The energy levels and
electron density distributi
on in many-electron atoms,
the same as in the hydro
gen atom, can in principle
be calculated theoretically
by the methods of quantum
mechanics; however, this
involves complex calcula-
Fig. 4.10. Diagram showing the

origination of the lithium atom
spectrum. The energy levels
of the electron in the hydrogen
atom are given for comparison
tions because the Schrodinger equation must be solved for many

particles. The equation in this case is
/i=iV
2 r / ^ , d2\|) , d2,i|)\ 8jr2m / e*Z 8 j i 2 rn
\ rk
tf/, )*]-!■ ~ W E x 0
ft=l I- ' dx\ dyh dzh '
(4.22)
Here all the electrons in the atom are submitted to such mathemati
cal treatment (rk. is the distance of the Zcth electron from the centre
of the nucleus) and the term Uk takes into account the energy of
repulsion of the given electron from the other electrons; it is equal
to e2/rhi taken from i = 1 to i = N, where rki is the distance between
the /cth and rth electrons. Thus, even for the simplest many-electron
atom, helium, the sum of the second derivatives alone contains six
terms. The Schrodinger equation is not exactly soluble for such
problems; however, approximate methods of solution have been
found. These methods are very complicated and the calculations

involved are very laborious. At the present time, electronic computers
are used for this purpose.
4.7. Energy Characteristics of Atoms:

Ionization Energy and Electron Affinity
The behaviour of atoms in chemical processes depends essentially
on the strength with which the electrons are retained on their orbi
tals. Therefore, an important characteristic is the ionization energy,
i.e., the energy required to remove an electron from an atom in the ground
state. This concept also applies to molecules. The ionization energy,
like the energy levels in atoms can be determined from spectral data.
The short-wavelength limit in a spectral series, indicating the transi
tions of electrons to the ground state, corresponds to the emission of
energy when an electron outside the atom is transferred to the ground
state. It is evident that the same energy is required to remove an
electron from an atom. Hence, the ionization energy can be calculated
by Planck’s equation (2.5) from the frequency corresponding to the
short-wavelength limit of the given series; the spectral term corres
ponding to the lowest energy level is called the ground term.
The ionization energy can be determined by different methods,
in particular the electron impingement and photoionization methods.
The ionization energy is usually expressed in electron-volts and is
therefore often called the ionization potential, having in mind the
potential difference (expressed in volts) due to which the electron
acquires energy equal to the ionization energy.
The ionization energy of the hydrogen atom is easily calculated
theoretically; from equation (4.17) it is apparent that it is expressed
by the relation
1 mPel
~2~W (4.23)
Inserting into this equation the appropriate values of m e, e4, /z2,

we obtain I = 13.60 eV.
For many-electron atoms there exist several ionization energies:
7j, 12, . . ., corresponding to the removal of the first, second, etc.
electron; in every case
11 <Z 1 2 1 3 ^ • • •» (4 *2 4 )
because with an increase in the number of electrons removed, the

charge of the positive ion produced is increased and it attracts the
electron more strongly.
Table 4.3 presents the ionization energies of certain atoms. From
the Table it follows that the alkali metals have the smallest value
of I t and that for a given element the ionization potential changes
Table 4.3
Ionization Energies of Certain Elements *
J, eV
Atom
1 2 3 4 5
H 13.595
He 24.581 54.403
Li 5.390 75.619 122.419
Be 9.320 18.206 153.850 217.657
B 8.296 25.149 37.920 259.298 340.127
C 11.256 24.376 47.871 64.48 392.00
N 14.53 25.593 47.426 77.450 97.863
0 13.614 35.146 54.934 77.394 113.873
F 17.418 34.98 62.646 87.23 114.214
Ne 21.559 41.07 63.5 97.16 126.4
Na 5.138 47.29 71.65 98.88 138.60
Mg 7.644 15.031 80.12 109.29 141.23
A1 5.984 18.823 28.44 119.96 153.77
Si 8.149 16.34 33.46 45.13 166.73
P 10.484 19.72 30.156 51.354 65.007
S 10.357 23.4 35.0 47.29 72.5
Cl 13.01 23.80 39.90 53.5 67.80
A 15.755 27.62 40.90 59.79 75.0
K 4.339 31.81 46 60.90 —
Ca 6.111 11.868 51.21 67 84.39
!
* Higher potentials, beginning with the hlh, are not includod in the Table. A dash
signifies that no experimental data is available.
sharply from one value of I to another. Thus, for boron the removal
of the 4th and 5th electrons requires approximately a ten-fold con
sumption of energy as compared to the 1st, 2nd, and 3rd electrons.
This affords direct evidence that the electrons are grouped in shells.
In Table 4.3 sharp changes are indicated by step-shaped lines.
Figure 4.11 shows (the scale of the diagram is distorted) the energy
levels and values of I t (i = 1, 2, 3, 4, 5) for boron and its ions.
The increase in the binding energy between the Is- and 2s-electrons
and the nucleus with an increase in i is explained by the reduction
in size of the ion with an increase in its charge.
Although in principle any degree of ionization is possible, chemists
how ever are only interested in the first few ionization energies becau
se, taking into account that 1 eV is equal to 23.1 kcal/gram-atom,
it is obvious that the energies absorbed or emitted in chemical
processes are commensurable with the first ionization potentials
(see p. 135). Indeed, whereas chemical processes are accompanied
by the emission or absorption
of energy of the order of tens
and hundreds of kcal/mole,
the removal of seven electrons
from the fluorine atom, for
example,
F —> F7+
would require more than

15,000 kcal/gram-atom.
Fig. 4.11. Ionization energies and

energy levels of the boron atom
and ions (schematic representation)
The ionization energies are very important characteristics of atoms.

This will become evident below; here the significance of these values
can be illustrated by the following example. In 1962 Bartlett (Cana
da) synthesized the compound 0 2PtF(). By theoretical reasoning
Bartlett came to the conclusion that this compound consists of the
ions Og and [PtF0]~. The fact that the ionization energy of the 0 2
molecule and the Xe atom are almost the same (12.2 and 12.13 eV,
respectively) led him to conclude that it was possible to obtain
a similar compound with xenon. Bartlett actually synthesized
XePtF0. In this way the synthesis of compounds of noble gases (also
called rare and inert gases)1 was initiated. This is one of the remar
kable achievements of chemistry in recent years.
Electrons are retained in atoms by the field of the nucleus; this
field also attracts a free electron if it happens to be near the atom.
It is true that the free electron is repelled by the electrons of the
atom. However, theoretical calculation and experimental data have
1 None of these terms is perfect. We prefer the term ‘noble’ because ‘rare’
is not absolutely appropriate (about 1 per cent of argon is contained in air)
and ‘inert’ is contradicted by the readability of these elements. As we know,
noble metals also tend to react chemically.
Ch. 4. Q U A N T U M - M E C H A N I C AT, E X P L A N A T I O N OF A T O M I C S T R U C T U R E 63
shown that for many atoms the energy with which an additional
electron is attracted exceeds that with which it is repelled by the
electron shells; these atoms can add an electron and produce a stable
negative single-charged ion. The energy required to remove the addi
tional electron from such an ion is determined by the electron affinity
of the atom (also defined as the energy evolved when an electron is
added to an atom or ion). Like the ionization energy, electron affinity
is usually expressed in electron-volts.
Quantum-mechanical calculations showr that when two or more
electrons are added to an atom, the repulsion energy is ahvays greater
than the attraction energy, i.e., the electron affinity in this case is
alwrays negative. Therefore, single-atom multicharged negative ions
(0 2~, S2~, N3", etc.) cannot exist in the free state. As we shall see
later on (see p. 228) there are grounds for believing that neither do
such ions exist in molecules and crystals. Hence, the formulae
Ca2+S2“, Cu2+0 2“, etc. should be considered to be only a rough appro
ximation.
The electron affinities of all of the atoms are not as yet known.
Table 4.4 gives the electron affinities of certain elements. The halo
gen atoms possess the maximum electron affinity. The Table shows
that in passing from fluorine to iodine the electron affinity first
increases but then gradually decreases. The method used for cal
culation of electron affinity is discussed on p. 291.
Table 4.4
The Electron Affinity of Certain Elements
Atom E, eV | Atom E, eV I Atom E, eV Atom E, cV
H 0.747 c 1.24 Na 0.47 S 2.33

He 0.19 N 0.05 Mg - 0 .3 2 Cl 3.81
Li 0.82 O 1.47 At 0.52 Br 3.56
Be —0.19 F 3.58 • Si 1.46 I 3.29
B 0.33 No —0.57 P 0.77
i
PART II
MENDELEEV’S PERIODIC LAW

AND THE STRUCTURE OF ATOMS OF ELEMENTS
CHAPTER FIVE
INTRODUCTION
5.1. The Modern Formulation of the Periodic Law

The Periodic Law was set forth by D. Mendeleev in 1869 and was
formulated by him as follows: the properties of simple substances1
as well as the forms and properties of compounds of elements vary perio
dically with the atomic weights of the elements.
D. Mendeleev believed that the Periodic Law reflected the pro
found regularities in the internal structure of matter. In his book
“Fundamentals of Chemistry” he wrote: “...the periodic system has
not only embraced the interrelations of the elements but has perfect
ed the knowledge of the kinds of compounds formed by the
elements, shown that the physical and chemical properties of simple
substances and compounds vary periodically. Such relationships
make it possible to predict the properties of simple substances and
compounds that have not as yet been studied experimentally and
therefore pave the way to atomic and molecular mechanics”.
The Periodic Law is expressed by the periodic system of the elements.
As is known, mathematical functions can be expressed in three ways:
by equations, by curves, or by tables. In the case of the Periodic
Law, a table was the most appropriate way. Hundreds of periodic
systems have been proposed but only those that closely resemble
the table worked out by D. Mendeleev have been widely adopted.
The study of atomic structure has shown that the periodic system
can be presented as a table in which the elements are arranged in
a strictly definite order in accordance with the structure of the electron
shells of their atoms. The everlasting fame of Mendeleev, rests on his
setting forth a table that is simple enough yet covers all the basic
details of atomic structure; therefore it has remained practically
unchanged for over 100 years.
The electronic structure of an atom in the normal (ground) state
is determined by the number of electrons in the atom. When the atom
1 A simple substance is one that consists of atoms containing' nuclei

of identical charge, i.e., of atoms of one and the same element. Examples of
such substances are metallic sodium, diamond, graphite, argon, etc. A chemical
element consists of a combination of atoms with nuclei of the same charge;
these atoms can be present in simple substances or compounds.
is not excited the electrons occupy the orbitals in which their energy
is minimum. The number of electrons in an atom is equal to the
positive charge on the nucleus. Hence, the charge on the nucleus
determines the electronic structure of atoms and, consequently, the
properties of the elements. At the present time, therefore, the Perio
dic Law is formulated as follows: the properties of elements and their
compounds vary periodically with the nuclear charge of the atoms of the
elements.
As a rule, an increase in the nuclear charge (an increase in the
number of protons in the nucleus) is also accompanied by an increase
in the average mass of the isotopes that make up the element, i.e.,
in the atomic weight of the element. Because of this, D. Mendeleev
was able to work out a periodic system by arranging all the elements
in ascending order of atomic weights. Exceptions to this rule are
four pairs of elements: A and K, Co and Ni, Te and I, Th and Pa;
the atomic weight of the first element in each pair is a little greater
than that of the second one, although the nuclear charge of the
atom is smaller1.
The Periodic Law shows that the properties of the elements vary
periodically with the nuclear charge of their atoms; this is true for
a great number of diverse properties of the elements. Fig. 5.1a, b
presents curves showing the dependence of atomic volumes12*5 and
melting points on the atomic number of the elements; Fig. 5.1c, the
dependence of the first ionization energies on the atomic number.
These curves are periodic and have a number of maxima and minima.
Similar curves show the dependence on the atomic number of other
properties such as the coefficient of compressibility, the coefficient
of expansion, melting and boiling points, magnetic properties, dis
sociation energies, ionic radii and most important of all, chemical
properties (for example, the heat of formation of such compounds
as oxides).
The periodicity of properties can be obscure and even remain
undetected unless appropriate conditions are observed when studying
the elements. Thus, many physical properties (melting point, density,
hardness, etc.) depend on the structure of matter. Therefore, these
properties should be compared for identical structures; for instance,
atomic radii should be compared in analogous atomic surroundings.
There are only very few properties that do not vary periodically
with the atomic numbers.
1 The first element in each of these pairs contains a relatively large amount
of the heavier isotope. Thus, argon consisting of isotopes with mass numbers 36,
38, 40 contains 99% }gA; potassium consisting of isotopes with mass numbers
39, 40, 41 contains 93% f|K (only one isotope of cobalt, i?Co, is known, but
nickel is a mixture of isotopes in which ||N i predominates).
2 Atomic volume is the volume occupied by a gram-atom of an element; it
is equal to the quotient obtained on dividing the gram-atomic weight by the
density.
5 3 a « . 15648
.s’
Fig. 5.1. D epend en ce on a to m ic num ber ol

(a) atomic volume of elements; (b ) melting point of elements; ( o liist
ionization energy. Symbol x denotes transition elements; symbol y ,
lanthanides and actinides
Ch. 5. I N T R O D U C T I O N (57
At the first glance it may appear that the specific heats of elements
do not vary periodically. Indeed, the atomic heat C equal to the
product of the specific heat c (cal/g ‘degree) and the atomic weight A ,
according* to the rule of Dulong and Petit, is an approximately con
stant quantity:
C = cA « 6.3 (5.1)
Dulong and P etit’s rule was at one time of great importance for
finding the correct atomic weights of elements. It shows that the
specific heat decreases continuously with increasing atomic weight,

i.e., this property does not seem to show periodicity. As can be seen
in Fig. 5.2, except for the lightest elements, the points of the curve
C = f (Z) at T = 273°K fluctuate about the horizontal line corres
ponding to 6.3. The same Figure, however, shows that the curve at
50°K indicates that the atomic heats vary periodically with the
atomic numbers. The curve for t = 0°C ( T = 273°K) shows that
actually periodicity occurs at this temperature also, although it is
less pronounced due to the effect of the higher temperature (take
note of the position of the points for the alkali metals).
The frequency of radiation (or the wave number v, which is pro
portional to v) giving rise to X-ray spectral lines can be considered
to be a nonperiodic property. This value changes continuously with
an increase in the atomic number of the element according to the
equation
V 7 = A(Z — b) (5.2)
5*
68 PART II. MENDELEEV'S PERIODIC L A W , S TR U CT U RE OF A T O M S
where v = wave number of a particular (first, second, etc.) line

of the X-ray spectrum series
Z = atomic number of the element
A and b •= constants
Equation (5.2) is illustrated by Fig. 5.3. This regularity was esta
blished experimentally in 1913 by Moseley, an Englishman. Mose-
ey’s experimental data proved that the nuclear charge is numerically
Fig. 5.3. Dependence of v for lines of the X-ray spectrum on the atomic number
of the element
equal to the atomic number of the element and gave undeniable

verification of the arrangement of the elements in the periodic
system.
Equation (5.2) can easily be obtained theoretically. As we already
know (see p. 58), the X-ray spectrum arises from transitions of
electrons between the inner shells of atoms. For atoms and ions with
one electron, the spectral term is given by equation (2.3). Somewhat
modified, this formula is applicable to an electron on one of the inner
shells of the atom. Electrons that are located at a greater distance
from the nucleus than the one under consideration affect but slightly
the energy of the latter because they are bound to the nucleus much
less strongly. Their action on the electron is so small that can
be ignored. The electrons that are located between the given
electron and the nucleus will decrease its attraction to the nucleus.
This effect can formally be considered as a decrease in the charge
acting on the electron by a certain value b known as the screening
constant. The expression for the spectral term will then be
from which we can find the wave number:
When considering the transitions of electrons between identical

energy levels in different atoms, the value R (1 !ri\ — 1/ri?2) will be
a constant. If we denote this constant as A 2, we obtain the equation
v = A 2 (Z - b)2
which is the same as Moseley’s formula [see equation (5.2)].
X-ray spectra arise as the result of the transition of electrons
located near the nucleus. It may appear that Moseley’s law indicates
the absence of periodicity of properties of the inner electrons. Equat
ion (5.2), however, holds true only because here under consideration
is the change with increasing atomic number in the energetics of
an electron having the same set of quantum numbers. In the given
case the energy of the bond between the electron and the nucleus will
increase continuously with an increase in the nuclear charge. If,
however, we consider the high ionization energies, we see that they
are periodic functions of Z since in this case the electrons in question
have different sets of quantum numbers.
Hence, periodicity is characteristic of the electron shell as a whole
and not only its periphery.
5.2. The Structure of the Periodic System

Mendeleev’s system consists of several periods corresponding to the
periodic variation of the properties of the elements. This is presented
schematically in Fig. 5.4 in which are given the atomic numbers of the
first, next to the last, and last element of each period. Three of them
(I, II, III) are single-series short periods', the first contains only two
elements, the second and third each contains eight elements. The
remaining ones are long periods', two of them (IV and V) each contains
18 elements and one of them (VI) contains 32 elements. The seventh
period is incomplete. Omitting the first period, each pair of periods,
i.e., II and III, IV and V, VI and VII, are similarly constructed;
this is also confirmed by Fig. 5.1.
Let us now pass from the schematic representation to the detailed
structure of the periodic system (see the fly-leaf at the beginning
of the book).
Each period (except the first one) begins with a typical metal
(Li, Na, K, Rb, Cs, Fr, respectively) and ends at a noble gas (Ne,
70 P A R T II . M E N D E L E E V ' S P E R I O D I C L A W , S T R U C T U R E OF A T O M S
A, Kr, Xe, Rn) which is preceded by a typical nonmetal (F, Cl, Br,
I, At). These are the elements that are shown in Fig. 5.4 (the atomic
numbers of the noble gases are enclosed in double lines). In passing
from Li to F, from Na to Cl, etc. there is a gradual decline in the
metallic properties and an increase in
1 12 |
those characteristic of nonmetals.
The noble gas is the element that
3 9 |io | separates the typical nonmetal of the
11 17 [18]
given period from the typical metal
at the head of the next period. In the
19 first period, besides helium, there is
35 36 only one element, hydrogen; hence,
it may be expected that hydrogen has
37
some properties typical of metals and
53 54 some typical of nonmetals. This will
55
be confirmed below (see p. 103, 109).
The two-series fourth and fifth peri
ods, in contrast to the second and third
periods, contain transition groups comp
rising ten elements: after the second
element of Period IV, Ca, come the
85 86 10 transition elements (Sc — Zn) after
which come the other Gmain elements
87 of the period (Ga — Kr). Period V
is constructed similarly. Since the
elements of the transition groups are
all metals, the even series of Periods*IV
104
Fig. 5.4. Schematic representation of Men

deleev’s Periodic System of the Elements
IV and V contain only metals. In these two periods [the presence

of 1G elements between the typical metal and typical non-
metal (between K and Br and between Rb and I, respectively)
instead of 5 elements as in the short periods accounts for the fact
that the adjacent elements in the fourth and fifth periods differ
from each other much less than in the second and third periods.
As we shall see below this is explained by the fact that whereas
in the group Mg—S the number of outer electrons is not the same,
in the groups Sc—Zn and Y—Cd it is, apart from a few exceptions,
identical (two outermost 5-electrons).
The two following periods are characterized by the presence
of groups of ten and fourteen elements in addition to the main
elements. Immediately after the second element of Period VI, Ba,
there should come the ten transition elements La—Hg, but actually
Ch. 5 . I N T R O D U C T I O N 71
after the first transition element, La, there come the 14 elements
Ce—Lu. After Lu the transition group is continued and completed
(Hf—Hg) and after that come the other 6 main elements of Period
VI (Tl—Rn). The incomplete Period VII is constructed similarly:
the transition group here contains as yet only three elements which
are separated by the 14 elements Th—Lw. This group of elements
also contains only metals that resemble each other even more than
do the ten transition elements as is clearly shown in Fig. 5.1. Because
of this, all 14 elements can be regarded as occupying a single posi
tion in the periodic system (together with La and Ac, respectively).
If this is done, Periods VI and VII also become two-series periods,
i.e., similar to Periods IV and V. The similarity of these elements
to La and Ac, respectively, explains why they are called lanthanides
(Ce—Lu) and actinides (Th—Lw). The difference in the properties
of the lanthanides and actinides is attributed to some difference
in the (n — 1) d and (n — 2) / energy levels.
Although, as we have seen above, in passing from an 8-element
to an 18-element period and from an 18-element to a 32-element
period the increasing resemblance of adjacent elements mainly
concerns the middle of a given period, it is however true for the
period as a whole. Thus, for example, whereas there is hardly any
resemblance between C and N, the resemblance between Pb and
Bi is considerable.
The arrangement of the elements in periods (horizontal rows)
results in the formation of vertical rows of allied elements, i.e.,
of families or groups. Because of the presence of transition elements
in Periods IV, V, VI, and VII and actinides and lanthanides in the
last two periods, there are three types of subgroups.
The main subgroups are formed by the main elements of each period.
They are the longest ones; they begin with the elements of the second
period. The main subgroups are the Li, Be, B, C, N, O and F sub
groups; to them should be added the noble gases which form the zero
group (see footnote on p. 112).
The transition elements form the supplementary subgroups. They
are shorter than the main subgroups and begin with elements in
Period IV. There are ten of them corresponding to the number of
transition elements in this period: the Cu, Zn, Sc, Ti, V, Cr, Mn, Fe,
Co and Ni subgroups. The elements of the first seven supplementary
subgroups together with the elements of the respective main sub
groups make up the first seven groups. The elements constituting
the main and supplementary subgroups are arranged alternately
(see the fly-leaf at the beginning of the book) in order to show that
they form different families of closely related elements. The elements of
the last three supplementary subgroups constitute the eighth group.
Thus we see that the zero and eighth groups differ from the others:
the zero group does not contain elements of supplementary sub-
72 PA R T II. M E N D E LE E V 'S PERIODIC L A W , STRU CTU RE OF ATOM S
groups; the eighth group does not contain elements of the main
subgroups (see footnote on p. 112).
The shortest subgroups are those made up of two elements; one
lanthanide and one actinide. They begin in Period VI. There are
fourteen of them. They are all included in Group III. Hence, Group
III is the largest of all the groups; it contains 5 elements of the main
subgroup, 4 elements of the supplementary subgroup, and 28 elements
of the lanthanide and actinide series. In all, it contains 37 elements.
Within each group the properties of the .elements of the main and
supplementary subgroups differ from one another, but the degree
of difference varies from group to group. In the first group the diffe
rence in the properties of the elements is considerable; in the follo
wing groups at first it is less, then greater, and in the seventh group
very great. Thus, the copper subgroup includes the low-activity
metals Cu, Ag, Au, that differ sharply from the active metals of the
lithium subgroup (in particular from K, Rb, Cs); the elements of
Group III are relatively close to each other in their properties; while
the elements of the Mn subgroup differ greatly from the halogens.
Table 5A
Mendeleev’s Periodic System of the Elements
(long-form table, first variant)
Periods
VIIA 0
2
IA IIA IIIA IVA VA VIA H He 1
3 4 9 6 7 8 9 10
Li Be B c N 0 F Ne 2
ii 12 13 14 15 16 17 18
Na Mg
IIIB IVB VB VIB VIIB VIIIB IB IIB A1 Si P $ Cl A
3
19 20 21 22 23 24 29 26 27 28 29 30 31 32 33 34 35 36
K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr 4
37 38 39 40 41 42 43 44 49 46 47 48 49 50 91 92 53 34
Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Cd In Sn Sb Te 1 Xe 5
Ag
93 96 37 72 73 74 73 76 77 78 79 80 81 82 83 84 85 86
Cs Ba L a! H f Ta w Re Os lr Pt Au Tl Pb Bi Po At Rn
6
Hg
87 88 8 9 ! 104 105
Fr Ra Ku
7
\ c:
s !! d P
— t—
i--------------------------------------------------
98 99 60 61 62 63 64 65 66 67 68 69 70 71
Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm V b Lu
90 91 92 93 94 95 96 97 98 99 100 101 102 103

Th Pa u Np P u A m Cm B k Cf Es Fm M d No LW
/
Csi
-c>
74 P A R T 11. M E N D E L E E V ' S P E R I O D I C L A W , S T R U C T U R E OF A T O M S
Nevertheless, when noting the degree of difference, it should always

he remembered that all the elements of a given group have characte
ristics in common; this is discussed in detail in courses of inorganic
chemistry (see also p. 112).
As has already been mentioned, there are many different representations of
the periodic system of the chemical elements. Tables 5.1 and 5.2 present two
long-form variants. These, in addition to the short-form table discussed above,
are the most widespread.
In Table 5.1 the elements of the supplementary subgroups are positioned
separately between the beryllium (I IA) and boron (111A) subgroups as 10 verti
cal columns and the lanthanides and actinides are placed outside the Table as
two horizontal series. In Table 5.2, in contrast to the preceding one, the lan
thanides and actinides are included within the Table and are arranged as four
teen vertical columns between the scandium (I I IB) and titanium (IVB) subgro
ups. In this book we shall use the short-form table.
5.3. Predicting the Properties of Substances

with the Aid of the Periodic Law
Mendeleev’s Periodic Law can be used to determine^ unknown pro
perties of elements and their compounds. A great number of such
properties were first determined by D. Mendeleev himself; he even
calculated the properties of elements then not discovered yet. As is
known, the predictions made by D. Mendeleev were fully verified.
The history of natural science knows no other examples of such bri
lliantly confirmed prognosis. To predict the properties of elements
and their compounds D. Mendeleev made use of the following proce
dure: he found the unknown property as the arithmetic mean of the
properties of the elements surrounding the given element in the
periodic system: to the right of, to the left of, above and below it.
This procedure can be called the Mendeleev method.
Thus, for example, the elements next to selenium are arsenic
(to the left) and bromine (to the right) which form H3As and HBr,
respectively; evidently, selenium can form H2Se and the properties
of this compound (melting and boiling points, solubility in water,
density in the liquid and solid states, etc.) will be close to the ari
thmetic mean of the corresponding properties of H 3As and HBr.
In the same way the properties of H2Se can be determined as the
average of the properties of I i2S and TI2Te, i.e., of similar compounds
of sulphur and tellurium, the elements placed in the periodic system
above and below selenium. Obviously the results obtained will
be more accurate if the properties of H2Se are calculated as the
average of the properties of the four compounds, H 3As, HBr, H2S,
and TIoTe. At the present time this method is widely used to estimate
the properties of uninvestigated substances.
There are however other methods of using the Periodic Law to
determine unknown physical and chemical constants. It is worth-
while to dwell on the use of the periodic law in the methods of compara
tive calculation worked out and studied by M. Karapetyants.
Fig. 5.5. Dependence of the distances Fig. 5.6. Comparison of interatomic

It—S and R —O on the atomic number distances R—O and R—S
of the element
Fig. 5.7. Comparison of ionization potentials of sodium and potassium
In these methods, the same as in Mendeleev’s method, the physi

cochemical constants are found by comparing the known values
of properties. Let us consider two examples of how the Periodic
76 P A R T I I . M E N D E L E E V ' S P E R I O D I C L A W , S T R U C T U R E OF A T O M S
Law is used in the method known as the first method of comparative

calculation. There are six methods in all.
In Fig. 5.5 the distances between the atoms of sulphur and the
elements of the fourth group, C, Si, Pb, combined with it in gaseous
CS2, SiS2, and PbS2 are plotted against the atomic numbers of the
elements. As can be seen, there is no simple relationship by which
the unknown distances Ge—S and Sn—S can be determined. Simi
larly, in the compounds C02, S i02, Ge02, Sn02 and P b 0 2, the
dependence of the distances R —O (where R —atom of the given
element) on the atomic numbers of the elements is complex.
However, if the idstances R —S and R —O are correlated as is shown
in Fig. 5.6, the points form a straight line from which the unknown
distances Ge—S and Sn—S can be found.
Figure 5.7 shows how the unknown fifth ionization potential
of the potassium atom can be found by a similar method.
As can be seen, in both cases the properties of closely related
{allied) substances are compared. The Periodic Law indicates which
inorganic substances can be compared in this way (see also Figs 7.2,
8.1, 11.7, 11.8 and others).,
CHAPTER SIX
THE PERIODIC SYSTEM OF THE ELEMENTS

AND THEIR ATOMIC STRUCTURE
6.1. Filling of Electron Shells and Subshells

Let us consider the connection between the electronic structure
of atoms and the position of the elements in the periodic system.
The electronic structure of atoms' of the elements is presented in
Table 6.1.
Hydrogen is the first element in the periodic table. The minimum
energy of its single electron corresponds to the if-shell {n = 1), i.e.,
the Instate. Helium has two electrons (Is2) of opposite spin.
Beginning with lithium the L-shell (n= 2) is formed since in accor
dance with the Pauli exclusion principle the first shell cannot be
occupied by more than two electrons of opposite spin. In neon the
second shell is completely filled; the s- and p-subshells are fully
occupied. The third shell is progressively occupied from sodium to
argon in the same way as the second shell.
Although in the third shell there are still 10 vacancies because
the 3d-subshell remains entirely unoccupied, in potassium and
calcium (the elements following argon) the fourth shell begins to be
filled and only beginning with scandium the 3d-subshell begins
to be formed and is completed in copper. In the transition elements
Sc—Zn, the d-subshell is not filled consecutively: in the Cr and
Table 6.1
Electronic Structure of Atoms of Elements in the Ground State
Symbols
Jcom p le te ly f i l l e d e le c lr o n s h e ll
[
* tr a n s itio n e le m e n ts
▼ la n th a n id e s a n d a c tin id e s
1 H 1s
2 He F * 1 1st p e r io d
3 Li [~Kl2s
4 Be
5 B m 2p
6 C .. 2p->
m
7 N
m
.. 2 p- ’ *
8 0 m
.. 2p* ■ *c
i CM
9 F » 2p -
m
10 Ne [ K w m
11 Na Ij l L L |3s
12 Mg | K | L j3s2
13 A1 M L | 3p
14 Si M L | 3P2
15 P \K\ L I •• 3p3
16 S M L | « 3p4 -g
17 Cl M L | 3p5 ro
18 A M L | ■ 3p6
19 K M L | n i• 4s
20 Ca L * L L | ,, 4s2
*21 sc r* T L | „ ,, 3d „
* 2 2 Ti \ * \ L | „ .. 3d2
1£
*23 V |* | L | „ 3d3
—
1 c cL
*24 Cr M L | „ „ 3 d 5 4s X
T?
*25 Mil L*L L | ,, „ 3d5 4s2
Continued
♦26 Fe | ff | L 13s23pS3d6 4 s2
♦27 Co | ff | L | .. „ 3d 7
*28 Ni | ff | L | „ .. 3d8
♦29 Cu | ff | L l:3s23p63dic‘|4s
*30 Zr\ \ K \ L | M j 4 s2
31 Ga I L | M | '• 4p
b
32 Ge | ff | L | M I • V -C
33 As | 1C| L 1 M I •• V
34 Se | ff | L 1 M I •• V
35 Br [ ff | L I M 1 •• V
36 Kr |ff | L I M 1» v
37 Rb | ff | L I M 5s
38 Sr | ff | L 1 M 1 .. .. 5s2
*39 Y | ff | L 1 M 1 .. .. 4d M
*40 Zr | ff | L | M I .. „ 4d2 ..
♦41 Nb | ff | L 1 M | » „ 4d4 5s
♦42 Mo | ff | L | M | .. .. 4d* 5s
♦43 Tc | f f | I I M | 4d5 5s2
*44 Ru | ff | L | M j .. .. 4d 7 5s
♦45 Rh | K | L | M | .. 4d8 5s
*46 Pd | f f | L | M | „ .. 4d'°
♦47 Ag | K | L I M 5s
♦48 Cd | ff I L | M l M .1 5s2 .c
■♦3
to
49
in W\ L | M 5p
50 Sn | ff | L 1 M 5p2
51 s b | ff | L | M 5P3
52 Te | ff | L | M .. 5p"
53 1 \K\ L 1 M 5ps
54 Xe | ff | L | M .. 5p6
C o n tin u e d
55 Cs 1K | L | M 1. 24p
|4s * 64
4a,10 5s25p6 6s
56 Ba 1K 1 L | M 6s 2
*57 La 1K 1 L | M 1 ■■ 5d
▼58 Ce 1 K | I | M •»f2
▼59 Pr 1K 1 L | M 4f3
▼60 Nd 1A 1 L | M ¥
▼61 Pm 1A | L | M ¥
▼62 Sm 1K 1 L | M V
▼63 Eu 1 A | L | M 4f?
▼64 Gd 1K 1 L | M | ........... <f7 „ 5d
▼65 Tb 1 K 1 L | Af 4f 9
▼66 Dy 1A | L | M ¥°
▼67 Ho 1K 1 L | M 1 ........... 4f"
▼68 Er 1K 1 L | M 4f '2
▼69 Tm 1K 1 L | Af 4/’3 „ ••
▼70 Yb 1 A | L | Af 4f"
▼71 Lu 1 K 1 L | M |4s24p64d rrqp 5d
*72 Hf 1A | I | Af 1 A 1 .. .. 5d2 ,,
*73 Ta 1K 1 L | Af 1 A 5d3 .. •§0
*74 W 1A | L | Af 1 A 5d4 „ 8.
*75 CD
Re 1K 1 L | Af 1 A | .. „ Sd*
*76 Os 1 A | L | M 1 A 5de ..
*77 lr 1 A | L | M 1 A l„ .. 5d7
*78 Pt 1 K 1 L | M 1 A j .. 5d9 6s
*79 Au 1 K 1 L | M 1 A ] .. .. 5d'° 6s
*80 Hg 1 K 1 L | M 1 A 6s2
81 T1 I A | Z. | Af 1 A ” 6P
82 Pb 1 K 1 L | Af 1 A .. 6p2
83 Bi 1 A | L | Af 1 A 1 .. •• „ 6p3
84 Po 1 A | L | Af 1 A •• 6p4
85 At 1 K 1 L 1 Af 1 A | „ .. 6p5
86 Rn 1 K 1 L | Af 1 A - 6p6
80 PART II. M E N D E LE E V 'S PERIODIC L A W , STRU CTU RE OF ATOM S
C o n tin u e d
87 Fr 1 K 1 L | Af | N 15s25p65d10 6s26p6 7s
88 Ra t K 1 L | Af | N | „ „ .. •• 7s2
*89 Ac 1 K 1 L | M | N 6d ..
▼90 Th 1 K 1 L j M | N .. 6d2 ..
▼91 Pa i * l L | M | N 5f2 „ 6d ..
▼92 U 1 K 1 L | M | N 5f3 .. 6d ..
▼93 Np 1 K 1 L | M | N | ............ 5fS
▼94 Pu 1 K 1 L | M | N 5 /6
▼95 Am 1 K 1 L | M | N 5/?
▼96 Cm 1 K 1 L | M | N 1 •• ,, .i 6>d ..
■g
▼97 Bk 1 K 1 L 1 M | N I ............. 5f9 ,,
▼98 Cf 1 K 1 L | M | N 5f10 CL
_c
- 11
▼99 Es 1 K 1 I | M | N 5f
▼100 Fm 1 K 1 L | M | N 1 ............. 5f'2 ••
▼101 Md 1 K 1 L | M | N 5f13 ” »
▼102 No 1 K 1 L | M | N Sf"
▼103 LW 1 K 1 L | M | N " .. 6d ..
*104 Ku 1 K 1 L | M | N - .. 6
*105
Cu atoms, an outer 5-electron ‘descends’ onto the preceding d-sub-

shell. The energies of the 3d- and 45-states in the elements Sc—Zn
are so close to one another that the addition of electrons can cause
a change in the distribution of electrons on these energy levels.
As a result of the ‘descent’ of an electron in the chromium atom,
the d-subshell is half-filled (configuration db) and in copper is comple
tely filled (configuration d10). Similar irregularities in the building
up of the d- and /-subshells are also observed in the following periods
(see Table 6.1).
After zinc, down to krypton, the fourth shell (4/?-subshell) conti
nues to be filled. Thus, after the two-element period and two eight-
element periods there follows a long period containing 18 elements.
The electron subshells of the following 18 elements (Rb—Xe)
are filled in the same way as the electron subshells of the 18 elements
of Period IV (K—Kr): after Rb (55) and Sr (5s2) the 4d-subshell
is filled in the transition elements from Y (4d) to Cd (4d10) with
several ‘descents’ of outer 5-electrons. Then, although the 4/-subshell
Ch. 6. P E R I O D I C S Y S T E M OF E L E M E N T S A N D THEIR S T R U C T U R E 81
is not occupied at all, electrons are added to the p-subshell of the

fifth shell.
Further building-up (aufbau) is accompanied bv two deviations
from the consecutive order of filling the subshells within one period:
although after Cs (65) and Ba (6s12*) comes La (5d), filling of the
incomplete fifth shell is temporarily stopped after this element.
The 4/-subshell is then formed, i.e., from Ce (4/2) to Lu (4/14) elect
rons fill the fourth shell that was previously left unfilled. After
n 1 Z 3 4 5 6 7 8
s s s s 5 5 s s
Fig. 6.1. Sequence of filling
of electron subshells
lutecium, from Hf (5d2) to Au (5d10), the 5cJ-subshell is filled comple

tely, and beginning with Hg and ending with Rn filling of the sixth
shell is continued. In this way, the sixth period containing 32 ele
ments is completed.
The seventh period is formed similarly to the sixth one: after
the 7s-elements (Fr, Ra) begins the Ac (6<2) group of transition ele
ments which is interrupted by the 14 actinides, from Th to Lw
(5/14). Only after that the group of transition elements is continued;
kurchatovium Ku (Z = 104), is already known and according to
published data an element with an atomic number of 105 has been
discovered. The seventh period is not complete1.
The sequence in which the shells and subshells are filled is shown
in Fig. 6.1.
Irregularities in the sequence of filling the electron shells, observed
once in the fourth and fifth periods and twice in the sixth and seventh
periods, are explained by the fact that it is energetically more
favourable to temporarily leave a given subshell unfilled. Thus,
the reason for the sequence 3p6 4s2 ->3d10 (instead of 3p6 - ^ d 10 -*•
— 4s2) is that it is very difficult for a new electron to occupy the
1 The completion of this period is connected with the upper limit of the
synthesis of transuranian elements (the heaviest naturally occurring element is
uranium). The following must be kept in mind: the lifetime of the new elements
should he greater than the time it takes for nuclear transitions to occur (esti
mated to be of the order of 10~20 s); with increasing atomic number, the size of
the electron shells decreases, so that at Z « 137 the radius of the first shell
would be so small that the electrons on it would be instantly captured by the
nucleus; calculations show that the nuclei of elements with atomic numbers
higher than-114-116 would undergo instantaneous s p o n t a n e o u s f i s s i o n . It is true
however, that lately it has been supposed that there exist ranges of stability
for very heavy nuclei with’ a surplus of neutrons.
6 3aK. 15648
\FF
Fig. 6.2. Dependence of the energy of electrons in different shells and subshells
on the atomic number of the elements. The unit of energy used here is 13.6 eV
(the energy of the electron in the hydrogen atom in the ground state)
Ch. 6 . P E R I O D I C S Y S T E M OF E L E M E N T S A N D THEIR ST R U C T U R E 83
third shell due to the repulsion of the eight electrons already there.
An increase in the nuclear charge, however, makes filling of the
previously unfilled subshells more favourable as it compensates the
repulsive forces of the electrons. With increasing nuclear charge
the degree of screening (shielding) of the nucleus by electrons is also
increased and that is why a large number of shells are filled irregu
larly; the effective nuclear charge becomes smaller and the prece
ding electrons are repulsed by those that follow to a greater extent.
The ‘descent’ of electrons to lower electron subshells in Cr, Nb
and other elements (see Table 6.1) is also energetically more
favourable.
It should also be borne in mind that the energy of the electrons
of each subshell depends not only on-the number of the subshell
hut also on the nuclear charge; in other words, the energy of a given
level varies from atom to atom. This is schematically shown in Fig. 6.2
where the atomic number is plotted (to logarithmic scale) on the
axis of abscissae and the square root of the energy of the electron
(with a minus sign) is plotted (also to logarithmic scale) on the axis
of ordinates1. The curves show that there is a decrease in the energy
of each level with an increase in the nuclear charge. The sharp drop
in the d-curves is caused by the appearance of the transition elements
(the diagonal shading above the axis of abscissae); and the still
steeper drop in the /-curve, by the appearance of the lanthanides
(the crosshatching above the axis of abscissae) and the actinides.
The irregularities in the sequence of filling of the electron subshells
should not be regarded as a violation of the Pauli exclusion prin
ciple; this principle only gives the maximum number of sets of quan
tum numbers, but gives no indications as to the sequence of filling
the energetic levels corresponding to these sets.
Thus, each successive element in the periodic system differs from
the preceding one in that it contains one more electron. This electron
may begin a new shell (Li, Na, . . .), may occupy an outer shell that
already exists (Be, Mg, . . .), or may occupy a previously formed
inner shell (Sc, Ti, . . .).
The filling of the electron shells and subshells of atoms of the"
elements in the periodic system makes evident the following general:
principles:
1. Each period begins with the formation of a new electron shelL
A period is a consecutive series of elements the atoms of which
differ in the number of electrons in the outermost shells. Each period
ends in a noble gas the outer shell of which (besides helium) consists
of 8 electrons and is denoted as ns2np 6 (where n > 1).
1 The logarithmic scale is used because it makes it possible to plot a large

range of values of the independent variable and function on relatively short
regions of the axes.
6*-
2. The elements of the main and supplementary subgroups dif

fer in which electron subshells are being filled. In all the elements
of the main subgroups, either the outer ras-subshells (Groups I and II)
or the outer rap-subshells (Groups III to VII) are being filled; the
former are called s-elements, the latter p-elements. In the elements
of the supplementary subgroups (except for Mn, Zn, Tc, Ag, Gd
and Hg) the inner (ra — 1) d-subshells are being filled. The elements
of these supplementary subgroups form three groups of ten transition
elements: 21 (Sc) — 30 (Zn); 39 (Y) - 48 (Cd); 57 (La), 72 (Hf) —
80 (Hg). A fourth group begins with 89 (Ac) in the incomplete seventh
period. The transition elements are called d-elements.
3. In the lanthanide series, 58 (Ce) — 71 (Lu), and the actinide
series, 90 (Th) — 103 (Lw), the (ra — 2) /-subshells, the 4/- and
5/-subshells respectively, are being filled. Therefore, these elements
are called f-elements.
Because of the above-specified characteristics of electronic struc
ture the properties of the elements in the periodic system vary in
-accordance with the principal regularities given below.
To clearly understand these regularities it should be borne in mind
that in passing from one shell to the next one the change in the
energy of the outer electrons (those determining the chemical pro
perties of the element) decreases with an increase in ra (see Fig. 6.2).
1 . The elements of the first period, hydrogen and helium, in
which the first electron shell of the atoms is being filled, are unique
in many of their properties, i.e., they have some properties that
no other element possesses (the properties of the H + ion, of liquefied
He, etc.).
2. The elements of the second period, in which the second shell
of the atoms is being filled, differ considerably from all the other
elements. This is explained by the fact that the energy of the electrons
in the second shell is much lower than that of the electrons in the
following shells (see Fig. 6.2) and that there cannot be more than
8 electrons in the second shell.
3. The transition elements, in which the inner d-subshells are
being filled, within a given period differ from each other much
less than the elements of the main subgroups do, in which the outer
shells are being built up.
4. The differences in the properties of the lanthanides, in which
the (ra — 2) /-subshell (namely, the 4/-subshell) is being filled,
are insignificant1.
The actinides are also similar to each other in many of their pro
perties; in them the (ra — 2) /-subshell is also being filled. The diffe
rences in the properties of the actinides, however, are greater than
1 For this reason it is very difficult to separate the compounds of these ele
ments from one another.
those of the lanthanides because the 5/-subshell being filled is farther

from the nucleus than the 4/-subshell of the lanthanides, i.e., the
former is ‘more outer’ than the latter. Because of this, the difference
between the energies of the electrons in the 5/- and 6 d-subshells in
the atoms of the actinides becomes small (see Fig. 6.2); it is much
less than the difference between the energies of the electrons in the
4/- and 5d-subshells of the lanthanides. Hence, the addition of elect
rons to the 5/-subshell in the actinides causes approximately the
same change in the properties as does the addition of d-electrons
in the transition elements.
6.2. Variation of Ionization Energies

The ionization energy is a very important characteristic of atoms.
As we shall see below, the nature and stability of chemical bonds
depend on it as do the reducing properties of atoms, because an atom
gives up an electron the more readily the smaller the ionization
potential.
The dependence of the first ionization energies on the atomic
number is shown in Fig. 5.1 c. It has already been noted that the
ionization potential varies periodically. It is now necessary to exa
mine this dependence more closely.
The ionization energy is equal in magnitude but opposite in sign
to the energy of the most weakly bound electron when the atom
(or ion) is in the ground state. Therefore, in order to find out how
the ionization energy varies, it is necessary to consider the factors
that determine the energy of electrons in atoms in greater detail.
If the electron under consideration were the only one in the atom,
the energy of the electron, in accordance with (4.20), would depend
only on the nuclear charge Z and principal quantum number n. The
greater the value of Z and the smaller the value of n, the lower the
energy level in a one-electron system and the stronger is the bond
between the electron and the nucleus. The presence of other electrons
in the atom (in addition to the one in question) greatly affects this
simple relationship. The effect of the other electrons can be ascertai
ned with the aid of two correlated conceptions; the concepts of
screening (shielding) of the nuclear charge and penetration of electrons
to the nucleus.
The screening effect, already referred to on p. 68 , consists in a re
duction of the effective positive charge of the nucleus that acts on
the given electron due to the presence of other electrons between the
electron under consideration and the nucleus. This effect can be taken
into account by introducing the screening constant. The concept
of screening affords a formal means of allowing for the mutual
repulsion of electrons. It is evident that screening increases with
an increase in the number of electron shells surrounding the nucleus.
86 PART II. MENDELEEV'S PERIODIC LAW , STRUCTURE OF A T O M S
The penetration effect arises from the fact that according to quan
tum mechanics an electron can occupy any point in the atom. Hence,
some of the time even the outermost electrons are in the near-nuclear
region where the screening effect is negligible; it can be said that
the outer electron penetrates through the inner electron shells to
the nucleus. It is obvious that the penetration effect strengthens the
bond between the electron and the nucleus. At the same value of
n, the smaller the value of I the greater the part of the electron
cloud concentrated in the vicinity of the. nucleus; hence, 5-electrons
penetrate to a greater degree, ^-electrons to a lesser degree, and
d-electrons to a still smaller degree. This explains the already familiar
sequence of energy levels of 5 - , p-, d- and /-electrons; at equal values
of n and Z, the 5-state has the lowest energy, the p-state has a higher
energy, etc.
In addition to the above factors, the repulsion of the electrons
in one and the same shell also affects the strength with which the
electrons are bound in an atom; this effect is also sometimes called
a screening effect. The repulsion is stronger when two electrons
of opposite spin occupy the same orbital.
The information given above can be used to explain how the
ionization energies vary in the periodic system. Let us consider
the first ionization energies.
The first ionization energies of atoms of the alkali metals ( /4, eV)
are: 5.39 (Li), 5.14 (Na), 4.34 (K), 4.18 (Rb), 3.89 (Gs). They are
the lowest ionization energies. This is explained by the strong scree
ning effect on the nuclear charge of the electron shells of the atoms
of the noble gas, that precede the outer electron. The decrease in the
ionization energy from lithium to caesium is due to the increase
in the distance of the electron from the nucleus as the size of the
atom grows.
Let us now see how the ionization energy varies in the second
period. The elements of this period have the following values of
I u eV: 5.39 (Li), 9.32 (Be), 8.30 (B), 11.26 (G), 14.53 (N), 13.61 (0),
17.42 (F), 21.56 (Ne). As can be seen, in passing from Li to Ne the
ionization energy increases. This is explained by the increase in the
nuclear charge, the number of electron shells remaining the same.
The values given above show, however, that Ii does not increase
uniformly; there is even a slight decrease in 1^ in the elements
following beryllium and nitrogen, i.e., in boron and oxygen, res
pectively. This occurs in conformance with the electronic structure
of the atoms. Beryllium has the configuration ls22s2; i.e., the outer
5-subshell is filled and therefore in boron which follows it an electron
occupies the p-subshell and because the p-electron is less strongly
bound to the nucleus than the 5-electron is, the first ionization
energy of boron is less than that of berrylium. The structure of the
outer electron shell of the nitrogen atom in accordance with Hund’s
rule is expressed schematically as follows:
"□yimr
from which it can be seen that there is one electron in each p-orbital.
In the element that follows nitrogen, i.e., in oxygen, a second electron
is added to a p-orbital which is already occupied by one electron
IE mm
O s
The two electrons on one and the same orbital strongly repulse each
other, and therefore it is easier to tear away an electron from an
oxygen atom than from a nitrogen atom.
The same sequence is observed in all the periods^ the alkali metal
that begins the period has the lowest ionization energy and the noble
gas that ends it has the highest ionization energy. In passing from
one transition element to another the ionization energies vary rela
tively slightly; they are greater than for the metals of the main
subgroups because the outer ^-electrons penetrate under the ‘screen1
of the other electrons.
Thus we see that the variation of the ionization energies is readily
explained on the basis of the electronic structure of the atoms of the
elements.
6.3. Secondary Periodicity
It would be wrong to suppose that the properties of elements
(and their compounds) within the subgroups always vary monoto-
nically with the atomic number. Let us plot, for example, the sum
of the first four ionization potentials of the Group IV elements as
ordinates and their atomic number as abscissae. In the resulting
graph (Fig. 6.3) it can be seen that the points for the C, Si, Ti, Zr
and Hf atoms can be connected by a continuous curve. For the
group C, Si, Ge, Sn, Pb, however, the slope of the curve is not mono
tonic (hence, for instance, the sum of the potentials for Sn cannot
be found as half the sum of the values for Ge and Pb). The same
result is obtained on plotting a curve showing the dependence on
the atomic number of the amount of energy evolved when oxides
of the RO 2 type are formed by the Group IV elements. Here also
the properties do not vary monotonically. This has come to be known
as secondary periodicity. It was discovered by Y. Biron, a Russian,
in 1915 and has since been established for many properties. Secon
dary periodicity was explained by S. Shchukarev in 1940. It is
88 PART II. MENDELEEV'S PERIODIC L A W , ST R U C T U R E OF A T O M S
associated with the filling of the respective d- and /-subshells, lea

ding to the strengthening of the bond between the outer s- and p-elec-
trons and the nucleus. This affects the s-electrons most strongly,
the p-electrons to a smaller extent, and the d-electrons even less.
Secondary periodicity is therefore most noticeable in the properties
of compounds of the elements of the main subgroups in which their
valency is equal or close to the group number.
Let us examine another example. In the fourth period in passing
from K to Cu and from Ca to Zn there is an increase in the first
(/,w v . IV
Fig. 6.3. Dependence of the sum of the first four ionization potentials of the
Group IV elements on their atomic number
ionization energy by 3.4 and 3.3 eV, respectively. This is due to

the penetration of the 4s- and 4s2-electrons under the screen of the
3d-electrons that progressively fill the third shell and to the result
ing increase in the effective charge on the nucleus, attracting the
outer electrons. A similar picture is observed in the fifth period.
In the sixth period, in passing from Cs to Au and from Ba to Hg
there is a much greater increase in I t (by 5.33 and 5.22 eV, respec
tively) which is explained by the penetration of the 6s- and 6s2-electrons
under the double screen of the 5d- and 4/-electrons; additional
/-strengthening occurs. That is why the elements that come after
the lanthanides have particularly high ionization energies. The
strengthening of the bond of the s-electrons resulting from their
penetration under the d- and /-orbitals gives rise to the conside
rable differences in the properties of the elements of the two subg
roups in the first and second groups (see p. 72).
Ch. 7. F OR MS A N D P R O P E R T I E S OF C O M P O U N D S 89
Of course the prediction of the properties of uninvestigated ele

ments and their compounds by Mendeleev’s method, -based on ver
tical and horizontal interpolation, reduces possible error due to
secondary periodicity.
CHAPTER SEVEN
ELEMENTARY PRINCIPLES OF FORMS AND PROPERTIES

OF CHEMICAL COMPOUNDS
Now that we know the modern theory of the electronic structure
of atoms of elements, we can pass over to the discussion of its influen
ce on the form and properties of compounds of the elements. Prior
to doing so, however, it is necessary to acquaint ourselves with
certain basic concepts current in this field.
7.1. Oxidation State

One of the basic concepts in inorganic chemistry is the concept
of oxidation state1. By the oxidation state or oxidation number is meant
the charge of the atom of an element in a compound found on assuming
that the substance is made up of ions.
We shall denote the oxidation number by a superscript, as Arabic
numerals with a + or — sign in front of the numeral (for example,
C1+7)12*. When there are grounds for believing that there are actually
ions in a compound or solution, in denoting their charge the + or
—sign is usually placed after the numeral (the charges 1 + and
1 — are denoted simply as + or —), for example, Ba2+, Na+, etc.
The following rules are helpful in determining the oxidation state:
1. The oxidation number of the atoms of simple substances is
equal to zero.
2. In neutral molecules the algebraic sum of the oxidation numbers
is equal to zero; for ions the sum is equal to the charge of the ion.
3. The oxidation number of the alkali metals is always equal
to -J- 1.
4. Hydrogen in all its compounds, except the metal hydrides
(NaH, CaH2, etc.), has an oxidation number equal to + 1 ; in the
metal hydrides the oxidation number of hydrogen is equal to — 1 .
5. The oxidation number of oxygen is equal to — 2. Exceptions
are peroxides, i.e., compounds that contain the —O—O— group
in which the oxidation number of oxygen is —1 , and certain other
substances (superoxides, ozonides, oxygen fluorides).
1 Synonyms for ‘oxidation state’ are ‘oxidation number’, ‘electrochemical
valency*.
2 The oxidation state is also denoted in the literature by Roman numerals
placed in parentheses after the symbol of the element, for example, Mo(VI).
The oxidation numbers of elements in various compounds is rea

dily found with the aid of the above rules. Thus, for example, in the
compounds Na^SOg2 and Na^SO*2 the oxidation numbers of sul
phur are equal to + 4 and + 6 , respectively; manganese in KMn04
exhibits the oxidation state of + 7 , etc. There are cases when the
oxidation number is a fraction; thus, in H20 the oxidation number
of oxygen is equal to —2, in H 20 2 to —1, but in K 0 2 and K 0 3 it is
equal to —1/2 and —1/3, respectively.
It should be pointed out here that the concept of oxidation state
is a formal one and it usually fails to give an idea of the actual
charge of the given atom in a compound. In many cases the oxida
tion number is not equal to the valency of the given element. For
example, in methane (CH4), methyl alcohol (GH3OH), formaldehyde
(Cli20), formic acid (1IGOOH), and carbon dioxide (C02), the
valency of carbon is equal to four but the oxidation numbers of
carbon are equal to —4, —2, 0, + 2 , and + 4 , respectively.
The concept of oxidation state is however very useful for classi
fying substances and for writing chemical equations. Thus, for
instance, on determining the oxidation number of phosphorus in the
compounds H P+50 3, H3P +50 4 and H 4P250 7, we see that all these
compounds are closely, related and should differ considerably in
their properties from the compound H3P +30 3 in which the oxidation
state of phosphorus is different.
The concept of oxidation state is particularly widely used when
studying oxidation-reduction reactions, i.e., chemical reactions in
which the oxidation numbers of the elements change. Such reactions
are discussed at length in the course of inorganic chemistry. Here
we shall only note that in the process of oxidation there is an increase
in the oxidation number, whereas in the process of reduction a dec
rease in the oxidation number occurs. Hence, substances in which
there is an increase in the oxidation number of the element are called
reducing agents, while substances in which a decrease in the oxidation
number of the element takes place are called oxidizing agents. Atoms
in the higher oxidation states exhibit only oxidizing properties;
in the lower oxidation states, only reducing properties. In the
intermediate oxidation states, the atom can be both an oxidizing and
a reducing agent.
An example of an oxidation-reduction reaction is
2KMn+70 4+ S° = 2Mn+40 2+ K2S+«04
The oxidizing agent in this reaction is KMn04; the reducing
agent, sulphur. As a result of the reaction, manganese is reduced;
its oxidation number decreases from + 7 to + 4. Sulphur is oxidized;
its oxidation number increases from 0 to +6.
The ability of a substance to react as an oxidizing or reducing
agent can be characterized quantitatively by the change in the
Ch. 7. F O R M S A N D P R O P E R T I E S OF C O M P O U N D S 91
thermodynamic potential AG (Gibbs’ function or free energy), which

takes place during a given oxidation-reduction reaction; the values
of AG have been determined for a great number of processes. Never
theless, oxidizing and reducing agents are characterized as ‘strong’,
‘moderate’, or ‘weak’, without indicating the exact value of AG.
In this book we shall use the above terms.
7.2. Atomic and Ionic Radii

The concept of atomic and ionic radii is often used in chemistry
and allied fields. These values are conventional; they are calcu
lated from the interatomic distances which depend not only on
the nature of the atoms but also on the type of chemical bond
between them and on the state of aggregation of the substance.
Atoms and ions cannot be considered to be incompressible spheres
resting motionless in contact with one another. We know (see
pp 41,168) that the nuclei of atoms in molecules and crystals are
always vibrating, even at absolute zero. In many cases the electron
density practically falls to zero at distances less than the atomic and
ionic radii; on the other hand, the distance at which an atom or ion
acts on other particles can be much greater than its conventional
radius.: Finally, the sizes of atoms and ions depend on their intera
ction with neighbouring particles.
When considering simple substances and also organic compounds,
use is usually made of the concept of atomic radii, rat; when studying
inorganic compounds, of ionic radii, rion.
Atomic radii are subdivided into atomic radii of metals, covalent
radii of nonmetal elements, and atomic radii of the noble gases.
The structures of most of the metals are now well known. The
atomic radius is found by dividing the distance between the centres
of any two adjacent atoms in half1. The magnitudes of the atomic
radii of metals are given in Table 7.1. In a given period the atomic
radii of the metals decrease because, the number of electron shells
being the same, the nuclear charge increases and consequently the
electrons are attracted more strongly to the nucleus; thus, (ra<)Na =
= 1.89 A; (rat)Mg = 1.60 A; (rat)Ai = 1.43 A. The rat in the
groups of ten transition elements decreases relatively slowly, espe
cially in the two series of three elements in Group VIII; thus, whe
reas (rat)sc =1.64 A and (rat)Ti =1.46 A, rat for Fe, Co, Ni are
equal to 1.26 A, 1.25 A and 1.24 A, respectively. In the lanthanide
and actinide series, rat decreases even more slowly; thus, from Ce
(1.83 A) to Lu (1.74 A ) rat decreases only by 0.09 A.
1 The methods of determining the interatomic distances in crystals are des

cribed on pp 273-277.
Table 7.1
Atomic Radii of Metals *
Metal ra f A Metal Ta V A Metal r a t< A Metal ra f A
Li 1.55 Cu 1.28 Cs 2.68 Pr 1.82

Be 1.13 Zn 1.39 Ba 2.21 Eu 2.02
Na 1.89 Rb 2.48 La 1.87 Gd 1.79
Mg 1.60 Sr 2.15 Hf 1.59 Tb 1.77
A1 1.43 Y 1.81 Ta 1.46 Dy 1.77
K 2.36 Zr 1.60 W 1.40 Ho 1.76
Ca 1.97 Nb 1.45 Re 1.37 Er 1.75
Sc 1.64 Mo 1.39 Os 1.35 Tm 1.74
Ti 1.46 Tc 1.36 Ir 1.35 Yb 1.93
V 1.34 Ru 1.34 Pt 1.38 Lu 1.74
Cr 1.27 Rh 1.34 Au 1.44 Th 1.80
Mn 1.30 Pd 1.37 Hg 1.60 Pa 1.62
Fe 1.26 Ag 1.44 Tl 1.71 U 1.53
Co 1.25 Cd 1.56 Pb 1.75 Np 1.50
Ni 1.24 In 1.66 Ce 1.83
* The table is taken from data of G. Boky. Proceedings of the Academy of Sciences
of the USSR, 69, 459, 1953.
In the main subgroups the atomic radii increase from top to

bottom with an increase in the number of electron shells.
In the supplementary subgroups there is also an increase in rat
in passing from the first element to the second one, but there is even
a slight decrease in passing from the second to the third element.
Thus, in the titanium subgroup, rat is respectively equal to 1.46 A,
1.60 A, 1.59 A. This is explained by the lanthanide contraction
(see p. 97).
Table 7.2 gives the covalent radii of nonmetals. They are also
found by dividing in half the interatomic distances in the molecules
or crystals of the corresponding elementary substances. As in the
case of the atoms of metals, in the groups of the periodic system,
the atoms of nonmetals with higher atomic numbers have larger
radii. This is due to the increase in the number of electron shells.
In the periods, the dependence of the atomic radii of nonmetals on
the atomic number is more complicated. Thus, in the second period
rat first decreases, then increases once more; this is explained by the
change in the strength of the chemical bond (see p. 210).
The atomic radii of the noble gases He, Ne, A, Kr and Xe are
equal to 1.22, 1.60, 1.91, 2.01 and 2.20 A, respectively. The above
Table 7.2
Covalent Radii of Nonmetals
Element H B C N O F Si P
r, A 0.37 0.80 0.77 0.55 0.60 0.71 1.18 0.95
Element S Cl Ge As Se Br Te I
r, A 1.02 0.99 1.15 1.25 1.16 1.14 1.35 1.33
values were obtained from the interatomic distances in the crystals

of the given substances at low temperatures. Here is also observed
an increase in rat with an increase in the atomic number. The atomic
radii of thfi noble gases are considerably greater than the radii of
atoms of the nonmetals in the respective periods (see Table 7.2).
This is so because in the crystals of the noble gases the interaction
between atoms is very weak (see p. 265), whereas in the molecules
of the nonmetals the covalent bond is strong.
Of special importance to the inorganic chemist are the ionic
radii: we shall therefore discuss them in detail. If a crystal consists
of ions (for example, Na+Cl“, Ca2+Fg), the internuclear distance
can be regarded as the sum of the ionic radii rion. But in order to
find the value of one of the radii it is necessary to know the value
of the other radius in addition to their sum. On the basis of experi
mental and theoretical investigation it has been assumed that the
ionic radii of 0 2“ and F “ are equal to 1.32 A and 1.33 A, respecti
vely. With the aid of these values the radii of other ions are found
from the interionic distances in crystals. Their values are presented
in Table 7.3; Fig. 7.1 shows the relative sizes of individual
ions.
A comparison of rat and rion shows that the cation radius rcat
is smaller than rat; thus, rMn = 1.30 A, whereas rMn2+ = 0.&0 A.
The value of rion differs from that of rat the more, the greater the
charge on the ion; thus, rMn2+ = 0.80 A, while rMn4+ = 0.60 A;
t cr3+ = 0.63 A, while rCr6+ = 0.52 A. This is explained by the
fact that the conversion of atoms to cations results in contraction
of the electron shells, the more so the greater the number of electrons
missing.
An examination of Table 7.3 and Fig. 7.1 shows that the values
of the radii of elemental ions vary in accordance with the following
four rules.
1. For ions with the same charge and of similar electronic structure
the radius is the larger the greater the number of electron shells in
the ion.
2. The radii of ions that contain the same number of electrons
(isoelectronic ions) decrease with an increase in the charge on them.
Thus, in the group S2~, Cl“, K +, Ca2+ the radii are equal to 1.74,
1.81, 1.33, 0.99 A, respectively. This decrease is greater for positive
•
n b+
o 9
p6 + s 6+ C l7+
0
Qo
v B* C r6+ Mn7+
o ©
As5+ Se6+
© o 0
Tc 7+
Nb6t Mo6*
0
S b 5+
Q
Te6* I7*
ions. This is so mainly for two reasons: in the first place, with an
increase in the charge on the ion the electrons are more strongly
attracted to the centre of the ion; in the second place, ions with
a greater charge react more strongly with ions of the opposite sign
which results in a decrease in the interionic distances, and conse
quently, in the ionic radii. In the case of negative ions, on the contra
ry, as the charge increases the electrons are more strongly repulsed
from the centre of the ion; the effect of the second factor, however,
remains the same and as a rule it exceeds the repulsion of the elect
rons from the centre of the ion.
Table 7.3
Ionic Radii *
Ion r, A Ion r, A Ion r, A Ion

r,A
Li+ 0.68 Mn7+ 0.46 Cd2+ 0.97 Lu3+ 0.85

Be2+ 0.35 Fe2+ 0.74 In3+ 0.81 Hf4+ 0.78
B3+ 0.23 Fe3+ 0.64 Sn2+ 0.93 Ta5+ 0.68
C4+ 0.16 Co2+ 0.72 Sn4+ 0.71 W 6+ 0.62
N3+ 0.16 Co3+ 0.63 Sb3+ 0.76 Re7+ 0.56
N5+ 0.13 Ni2+ 0.69 Sb&+ 0.62 Os6+ 0.69
02“ 1.32 Cu+ 0.96 Te2” 2.11 Ir4+ 0.66
F- 1.33 Gu2+ 0.72 Te4+ 0.70 Pt2+ 0.80
Na+ 0.97 Zn2+ 0.83 Te6+ 0.56 Pt4+ 0.65
Mg2+ 0.66 Ga3+ 0.62 I- 2.20 Au3+ 0.85
Al3+ 0.51 Ge2+ 0.73 15+ 0.62 Hg2+ 1.10
Si4+ 0.42 As3+ 0.58 17+ 0.50 T1+ 1.47
p3+ 0.44 As5+ 0.46 C3+ 1.67 Tl3+ 0.95
p5+ 0.35 Se2" 1.91 Ba2+ 1.34 Pb2f 1.20
S2- 1.74 Se4+ 0.50 La3+ 1.14 Pb4+ 0.84
S4+ 0.37 Se6+ 0.42 Ce3+ 1.07 Bi3+ 0.96
S«+ 0.30 Br- 1.96 Ce4+ 0.94 Bi5+ 0.74
Cl- 1.81 Br5+ 0.47 p r3+ 1.06 p 0 6+ 0.67
Cl*+ 0.34 Rb+ 1.47 Nd3+ 1.04 At7+ 0.62
C17+ 0.27 Sr2+ 1.12 Pin3+ 1.06 Fr+ 1.80
K+ 1.33 Y3+ 1.06 Sm3+ 1.00 Ra2+ 1.43
Ca2+ 0.99 Zr4+ 0.87 Eu3+ 0.97 Ac3+ 1.18
Sc3+ 0.81 Nb5+ 0.69 Gd3+ 0.97 Th4+ 1.02
Ti4+ 0.68 Mo6+ 0.62 Tb3+ 0.93 Pa4+ 0.65
V5+ 0.59 Tc7+ 0.56 Dy3+ 0.92 U6+ 0.80
Cr3+ 0.63 Ru4+ 0.67 Ho3+ 0.91 Np4+ 0.95
Cr«+ 0.52 Rh3+ 0.68 Er3+ 0.89 Pu4+ 0.93
Mn2+ 0.80 Pd2+ 0.80 Tm3+ 0.87 Am3+ 1.07
Mn4+ 0.60 Ag+ 1.26
* The cation radii are taken from data of Arens (1952); the anion radii, from data
of Goldschmidt (1926) (Landolt-Bornstein, Zahlenwerte und Funktionen, Berlin, 1955,
Band I, Teil 4, Seite 523-525). The ionic radii given here correspond to the coordina
tion number 6.
3. Ions of the noble-gas type, i.e., with outer subshells of the

noble gases (s- and p-subshells) have larger radii than ions with
d-electrons in the outer shell, fo r example, the radii of the K + and
R b+ ions are equal to 1.33 k and 1.47 A, whereas the radius of the
Fig. 7.2. Comparison of radii
(a) of ions of the metals of the main subgroups of Groups I and II; (b) of isoelectronic ions
of the alkali metals and the halogens
Cu+ ion is equal to 0.96 A. The reason for this is that within each
period the nuclear charge increases in passing from the s- and p-ele-
ments to the d-element; thus, ZK = 19, but ZCu = 29. In each
period, the radii of ions of d-elements with the same charge also
decrease with an increase in Z; thus, rMn2+ = 0.80 A, while r Ni2+ =
= 0.69 A. The decrease in the ionic radii is called d-contraction',
it is particularly noticeable in the Group VIII elements.
4. With increasing atomic number of the elements, there is a si
milar decrease in the radii of ions formed by the lanthanides (the
radius of the Ce3+ ion is equal to 1.07 A, whereas that of Lu3+ is
0.85 A). This is known as lanthanide contraction. In the lanthanide
ions the number of electron shells is the same. An increase in the
nuclear charge increases the attraction of the electrons to the nucle
us, and, as a result, the radius of the ions decreases.
Figure 7.1 also shows that the ionic radii vary periodically. Con
sequently we would obtain a curve for rion similar to the one in
Fig. 5.1 a. The variation of rion can be expressed quantitatively
by the method of comparative calculation. This is shewn for two
cases in Fig. 7.2. In Fig. 7.2a are correlated the values of the ionic
radii of the metals of the main subgroups of Groups I and II of the
periodic system of the elements; in Fig. 7.2b, the values of rion of
isoelectronic ions of the alkali metals and the halogens. The value
of rAt- can be found from the curve in Fig. 7.2b.
As has already been stated above, the concept of ionic radii in
many cases is conventional; the value of a given rion in different
compounds is only approximately constant. Moreover, the term ionic
charge actually applies only to single-charged and double-charged
ions since ions with a greater charge practically do not exist in
crystals. In compounds containing elements in oxidation states
higher than + 2 , the bond is generally not ionic and therefore the
concept of ionic radius in these cases is also formal, like the concept
of oxidation state. Nevertheless, the change in ionic radii characte
rizes the change in interatomic distances and this makes it possible
to understand many properties of substances containing elements
in given oxidation states.
The variation of ionic radii of elements in accordance with their
position in the periodic system is also very important for under
standing certain properties of compounds discussed below. Besides,
it should be taken into consideration that multicharged ions actually
do exist in solutions.
7.3. Coordination Number
Each atom or ion in a crystalline substance is always surrounded
by other atoms, ions, or molecules. In the polyatomic ions of acids
containing oxygen, such as the anions (S04)2“, (P 0 4)3", (C1Q4)“,
7 3 a « . 15648
the nonmetal atom is surrounded by oxygen atoms. Investigations

have shown that the number of atoms or ions surrounding the central
atom or ion is not arbitrary; it is, as a rule, a definite number which
depends both on the nature of the central atom (ion) under consi
deration and on the surrounding atoms (ions). The number of particles
(ions, atoms, or molecules) that directly surround a given atom (ion)
is called the coordination number. Thus, in the ions (S04)2", (P 0 4)3-,
(C104)", the coordination number of the sulphur, phosphorus, and
chlorine atoms is equal to four; in the ions (S03)2“, (C03)2“, (N03)“t
the coordination number of sulphur, carbon and nitrogen is equal
to three.
For most of the metals, the coordination number is twelve, which
corresponds to close packing (see p. 281). The radii of atoms and ions
depend on the coordination number. Thus, when the coordination
number n is reduced from 12 to 8, 6 and 4, the rat corresponding
to n = 12 should be multiplied by coefficients equal to 0.97, 0.96
and 0.88, respectively. For ions, in passing from the coordination
number 6 to 12, 8 and 4, the rion should be multiplied by 1.12,
1.03 and 0.94, respectively.
The coordination numbers 3, 4 and 6 are most frequently met with
in compounds. For example, in the crystals of sodium chloride,
in which the Na+ and Cl“ ions are arranged alternately, the coordi
nation number for both ions is identical and is equal to six (see
p. 283). For ions that have a similar electronic structure, the coordi
nation number generally increases with increasing ionic size; this
can be illustrated by the anions of oxygen-containing acids of ele
ments of the main subgroup of Group IV. As the size of R +4 increases
in the ions (C+40 3)2", (Si+40 4)4~ and [Sn+4(OH)J2~, the coordination
numbers of R +4 increase accordingly and are equal to 3, 4 and 6.
7.4. Compounds Containing R —H and R —O— Bonds

Later on we shall see how the properties of substances depend on
the oxidation state, ionic (atomic) radii and on the coordination
number. But let us first investigate some important classes of che
mical compounds.
Among the compounds that play a very important role are sub
stances in which atoms of elements are bound to hydrogen or oxygen.
The importance of these substances is due to the great quantity
of oxygen and hydrogen present on our planet and, consequently,
to the very frequent occurrence of their compounds. Therefore, we
shall confine ourselves to discussing these compounds of the elements
of the periodic system, all the more so that the principles characte
ristic of these two types of compounds are largely applicable to other
classes of compounds.
We shall, moreover, confine ourselves to discussing only those of
the afore-mentioned substances which can exist in the presence

of water.
Examples of compounds containing the R —H bond are CH4,
SiH4, PH 3, H20, HC1. These compounds formed by the union of
hydrogen with other elements are called hydrides. The R —0 — bond
is present in NaOH, KOH, Ca(OH)2, T10H and other bases (hydro
xides), and also in oxygen-containing acids (H3B 03, H 3P 0 4, H N 03,
etc.) and their salts.
In the laboratory as well as in the industry, chemists often have
to deal with hydrides, hydroxides, and oxy-acids. The acid and
basic properties of substances are of great importance in laboratory
research and industrial processes.
7.5. Acids, Bases, and Amphoteric Compounds1

An acid is a substance containing hydrogen, which dissociates in
aqueous solution with the formation of hydrogen ions, H +. A base is
a substance containing the hydroxyl group OH, which dissociates
in water to form hydroxyl ions, OH". Some substances containing
II+ and OH" ions do not dissociate as readily as others. The facility
with which substances dissociate into ions is characterized by the
degree of dissociation in solutions. The degree gf dissociation a is the
ratio of the number of dissociated molecules to the total number of mole
cules in solution. It is expressed by a fraction (0.1, 0.2, etc.) or in per
cent (10%, 20%, etc.) and depends on the concentration of the dissol
ved substance and the temperature. The degree of dissociation also
depends on the nature of the solvent. Here we shall consider only
the degree of dissociation of acids and bases in aqueous solutions and
discuss the effect of only one variable, the nature of the dissolved
substance.
Depending on the degree of dissociation, an acid may be strong,
weak, or moderate. The same terms apply to bases. An acid is conven
tionally considered to be strong if a > 3 0 % in a 0.1 N solution;
weak, if a < 3% in a solution of the same concentration; moderate,
if in a 0.1 N solution 30% > a > 3 % . The same pertains to bases
also.
iSome substances can dissociate both as acids and bases; such sub
stances are called amphoteric. Some of them are: Al(OH)3, Zn(OH)2,
(lr(OH)3. Both the acid and basic properties are usually feebly mar
ked. They react as weak bases with strong acids and as weak acids
with strong bases.
There are many acids in which the atoms of an element alternate
with atoms of oxygen. Acids containing the —R —O—R— chain
1 The definitions of acids and bases given here are somewhat simplified and
incomplete. More general and exact definitions are set forth in courses of inor
ganic chemistry.
7*
are called isopolyacids; for example, H2S20 7 (pyrosulphuric or disul-

phuric acid), H 4P 20 7 (pyrophosphoric acid), (H P03)3 (trimetaphos-
phoric acid). These acids have the following structure:
0
HO. / \ .0
OH OH OH OH \p p /'
I | 1 1 0^ | | Ô H
Ti
TJ-
O
1
1
= S —0 -- s = o , 0 0
II
0
0
ll
II II l l \ /
0 0 OH OH p
/ \
Acids containing the —R t—0 —R2— chain, i.e., derivatives
of oxy-acids in which the O2- ions are completely or partially
replaced by acid radicals of other acids are called heteropoly acids.
An example of such a compound is phosphotungstinic acid containing
the —P—0 —W— chain. A great many silicon, phosphorus and
boron isopolycompounds are known. The very large variety of sili
cates is due to the formation of compounds of the above type. Almost
all the silicates, both naturally occurring and synthetic,* contain
the —Si—0 —Si— chain.
7.6. Dependence of the Strength of Acids

and Bases on the Charge and Radius of the Ion
of the Element Forming Them
Prior to discussing the compounds of elements and their properties, ;
it is necessary to determine the factors on which the character of
Fig. 7.3. Kossel diagram
dissociation of substance depends; to find out, for example, why

Ca(0H)2 is a base whereas B(OH)3, which has a similar formula,
is an acid. .
For this purpose it is worthwhile to examine a simplified diagram
which, nevertheless, gives a clear idea of the essence of the problem.
This diagram was proposed by Kossel, a German.
The Kossel diagram (Fig. 7.3) presents the atomic group R —O—H.
The R +n and O-2 ions are shown as spheres with radii corresponding
to the radii of the ions; their interaction is determined by Coulomb’s

law. The radius of the hydrogen ion (proton) is very small in compa
rison with the radius of the R +n and oxygen ions. It is assumed,
therefore, that the distance between the hydrogen ion and the centre
of the oxygen ion is equal to the radius of the oxygen ion.
The character of dissociation of compounds containing the
R —0 —H group depends on the relative strength of the R —0 and
0 —II bonds. The strength of a chemical bond is determined by the
energy required to break the given bond (see p. 133). It is necessary
to distinguish bond breaking that yields neutral atoms (or groups
of atoms) from bond breaking that results in the formation of ions.
In the present case we are considering the dissociation of substances
into ions. If the R —0 bond is the stronger one, the 0 —II bond is
broken on dissociation, i.e., the hydrogen ion is formed and the
compound behaves like an acid. Conversely, if the 0 —II bond is the
stronger one, it is the R —0 bond that is broken on dissociation and
the OH" ion is formed, i.e., the substance behaves like a base. If the
R —0 and 0 —II bonds are approximately equal in strength, the
compound is amphoteric.
According to the Kossel diagram the R —O bond becomes stronger
with increasing charge and decreasing radius of the ion of the
element R1. On the other hand, an increase in the charge and decrease
in the radius of the ion of an element weakens the 0 —II bond be
cause the proton is more strongly repulsed by the ion of the element.
Hence, the higher the oxidation state of an element and the smaller
the ionic radius of the element, the more pronounced are the acid pro
perties of the compound. Consequently, strong oxy-acids are formed
by elements in the top right-hand corner of the periodic system.
On the contrary, the lower the oxidation state and the greater the
ionic radius, the more pronounced are the basic properties of the
substance. Consequently, strong bases are formed by the elements
positioned in the bottom left-hand corner of the periodic system.
In compounds containing the R —II bond, by similar reasoning
we can conclude the following. If this bond, as a first rough appro
ximation, is considered to be ionic, it will be the stronger the higher
the absolute value of the oxidation number of the element and the
smaller the radius of its ion. Hence, a decrease in the oxidation
1 When discussing the dissociation of substances it is necessary to take into

consideration not only the interaction of the ions with each other, but also
their reaction with water molecules, i.e., their hydration (see p. 297). To a first
approximation this effect can be allowed for by introducing the dielectric con
stant into the formula for Coulomb’s law (see p. 301). As a result of hydration,
the interaction of ions in aqueous solution is approximately 80 times weaker
than in vacuum. The relationships indicated by the Kossel diagram hold true
if the peculiarities of the reactions of the RO~ and H+ and R+ and Oil - ions,
respectively, with water molecules are ignored.
number of an element (absolute value) and an increase in the radius

of its ion lead to an increase in the acid properties of hydrogen
compounds.
It should be stressed once more that the Kossel diagram is a very
rough simplification. The 0 —H bond is not ionic and the distance
between the centres of the oxygen and hydrogen atoms is never
equal to 1.32 A because the hydrogen ion penetrates into the electron
shells of oxygen (see p. 228). Moreover, in the upper oxidation
states, the bond between the element R and oxygen is likewise not
ionic, and the oxidation number, as stated above, does not correspond
to the charge of the ion of the element. Nevertheless, in most cases,
the Kossel diagram enables qualitatively correct conclusions to be
drawn on comparing like compounds, say, the hydroxides of the
elements of one and the same group in the periodic system. This
rough diagram can be used with such unexpected results because
even in the case of bonds that differ greatly from ionic ones, the
strength of the bonds increases with decreasing interatomic distances
(and, consequently, with decreasing ‘ionic radii’ calculated from
them) and with increasing oxidation numbers. The oxidation number
usually indicates approximately the number of electrons of the
given atom that take part in the formation of a chemical bond.
Therefore the use of the Kossel diagram is very convenient for
a first general orientation in the diversified data of inorganic
chemistry.
CHAPTER EIGHT
ELECTRONIC STRUCTURE AND PROPERTIES

OF ELEMENTS AND THEIR COMPOUNDS
Let us examine the acid-basic and oxidizing-reducing properties
of the elements and their compounds in the various groups of the
Mendeleev Periodic System, confining ourselves to the most cha
racteristic oxidation states.
8.1. First Group

The first group consists of the lithium subgroup (Li, Na, K, Rb,
Cs and Fr) and the copper subgroup (Cu, Ag, Au).
Lithium, sodium, potassium, rubidium, caesium and francium
exhibit the oxidation state of +1 in their compounds. The atoms
of these elements readily give up the single electron in their outer
shell and are therefore strong reducing agents, the strength of which
increases in passing from lithium to francium. Francium is the
strongest reducing agent of all the elements because its atoms are
larger than those of the other elements of the subgroup. The alkali
metals dissolve in water to form R1+—0 —H compounds which are
Ch. 8. E L E C T R O N I C S T R U C T U R E A N D P R O P E R T I E S OF E L E M E N T S 103
strong bases that are readily soluble, i.e., alkalies. The reasons for
this are the small charge and large radii of the ions.
Hydrogen is often included in the first main subgroup because,
like the alkali metals, it is an 5-element; however, notwithstanding
the characteristics that they have in common (similar spectra, forma
tion of the univalent positive ion R +, reducing properties, mutual
replacement of metals and hydrogen), there are essential differences
between metals and hydrogen: the proton is incomparably smaller
than the cations of the alkali metals and it always penetrates deeply
into the electron shells of the atom with which it is united; the
ionization energy of hydrogen is almost three times greater than the
approximately equal first ionization energies of the alkali metals;
the behaviour of hydrogen is similar to that of metals only in aqueous
solutions; it acts as a reducing agent only at high temperatures.
On the other hand, hydrogen in many of its properties'resembles
the halogens (see p. 109). That is why it is more appropriate to inclu
de hydrogen in the fluorine subgroup.
The Cu, Ag and Au atoms also have one electron in the outermost
shell. The single-charged ions of these elements, however, are smaller
than the ions of the alkali metals. Hence, the R-Ô bond is stronger
than in compounds of the elements of the main subgroup. Indeed,
their hydroxides are weaker bases than the hydroxides of the alkali
metals. A second difference is that the shell next to the outermost
one is an 18-electron shell (s2ped10), i.e., it contains cZ-electrons that
are less strongly bound to the nucleus than are the 5- and p-electrons
of the same (n — 1) electron shell of the alkali metals. Therefore,
the elements of the supplementary subgroup can have an oxidation
number greater than + 1. As a matter of fact in aqueous, solutions
these elements exhibit the following oxidation states: Cu+1, Cu+2,
Ag+1, Au+3. The compound Cu(OH)2 is a weak base and exhibits
slight amphoteric properties; Au(OH)3 is an amphoteric compound
in which the acid properties predominate because of the strengthening
of the R —0 bond caused by the increase in the charge and the corres
ponding decrease in the size of the R +3 ion.
8.2. Second Group

The oxidation state characteristic of all the elements of this group
is + 2. The elements of the main subgroup (Be, Mg, Ca, Sr, Ba and
Ra) have two 5-electrons in their outermost shell. The reducing
properties of the elements of this subgroup are weaker than those
of the alkali metals (the atoms of which are larger in size), although
due to the increase in their atomic radii Ca, Sr, Ba, and Ra are
strong reducing agents. The Be2+, Mg2+, Ca2+, Sr2+, Ba2+ and Ra2+
ions, like the ions of the lithium subgroup, have the configuration
of noble gases, but they differ from R + ions in their charge and radii.
Because of their greater charge and smaller radii, their hydroxides

are weaker than those of the alkali metals. The increasing of ionic
radii in the Be2+—Ra2+ subgroup explains why Be(OH)2 is an ampho
teric compound, Mg(OH)2 is a weak base, Ca(OH)2 is a strong base
and Ba(OH)2 is a very strong base that readily dissolves in water;
it is an alkali and hence it is called caustic baryta.
The elements of the supplementary subgroup are weaker reducing
agents than those of the main subgroup because their atoms have
relatively small radii. The radii of Zn2+, Cd2+ and Hg2+ ions are
also smaller than those of the neighbouring elements of the main
subgroup due to the effect of the 18-electron subshells. Hence, the
hydroxides of these metals are weak bases. The compound Zn(OH)2r
like Be(OH)2, is amphoteric; Cd(OH)2 is only slightly amphoteric;
HgO is not amphoteric at all1. Since the elements of the zinc sub
group are the last d-elements (they conclude the group of the transi
tion elements), they exhibit some similarity to the elements follo
wing them, i.e., to Ga, In, Tl.
8.3. Third Group

The elements of the boron subgroup (with the exception of thal
lium) exhibit the characteristic oxidation number of + 3 and form
R(OH)3 compounds. Just as the elements of Group II are less alkaline
than those of Group I, so is there a further weakening of basic pro
perties in passing from Group II to Group III: Li(OH) is a base;
Be(OH)2 is an amphoteric compound; B(OII)3 is an acid. Thus,
in the third group for the first time we meet an element that forms
an acid (in this respect boron differs from all the other elements
in Group III) and with isopolyacids which are also characteristic
of boron. The basic properties become more pronounced as the
ionic radii of the elements increase from Al(OH)3 to Tl(OH)3: the
degree of dissociation of Ga(OH)3into OH" and H +ions is practically
equal; the basic properties of In(OH)3 predominate; Tl(OH)3 is only
slightly amphoteric. It can be noticed that the basic properties
increase very slowly in these compounds. The reason for this is that
whereas the atoms of the elements of the third main subgroup have
structural identity (their outer electron shell has the configuration
s2p), the B3+ and Al3+ ions differ greatly from the Ga3+, In3+ and
Tl3+ ions. The former have the outer shells of noble gases; the latter
have 18-electron shells containing 10 d-electrons. Consequently,
the ionic radii after aluminium increase to a lesser extent and becau
se of this the basic properties of the compounds increase slowly.
Here, as in the preceding group, a diagonal similarity is observed:
the amphoteric hydroxides of A1 and Be closely resemble each other
in their properties.
1 Hg(OH)2 is unstable and decomposes into HgO and H20.
Ch. 8. E L E C T R O N I C S T R U C T U R E A N D P R O P E R T I E S OF E L E M E N T S 105
We have already stated that thallium is an exception among the

elements of the boron subgroup. Its characteristic oxidation number
is + 1 and TlOH is a strong base. This is explained by the greater
stability of compounds of T1 in which the atom retains the electrons
on the s-orbital. Therefore, it is the /^-electron that first of all deter
mines the valency of T1 (and also of the following elements of Period
VI as we shall see below). This is not characteristic of In, and even
less so of Ga. Hence, Ga+ is a very strong reducing agent, while
Tl3+ is a strong oxidizing agent. The fact that TlOH is a strong base
is explained by the considerable size of the T l+ ion and its small
charge.
The elements of the supplementary subgroup (Sc, Y, La and Ac)
also exhibit the characteristic oxidation number of + 3 and form
R(OH)3 compounds. The electronic structure of the Sc, Y, La and
Ac atoms is not similar to that of B and Al; these atoms have 2 5-elec
trons in their outermost shell. The B3+, Al3+, Sc3+, Y3+, La3+ and
Ac3+ ions, however, all have the same electronic structure, that
of atoms of the noble gases. Therefore in passing from B to Ac, the
properties of their compounds vary more continuously than in pas
sing from B to Tl; in particular, in the group B(OH)3—Al(OH)3—
Sc(OH)3—Y(OH)3—La(OH)3—Ac(OH)*3, the basic properties grow
more rapidly: whereas Sc(OH)3 is a weak base with hardly any signs
of amphoteric properties, La(OLI)3 is already a strong base. Because
of the large radii of the La3+ and Ac3+ ions, the R —0 bond is weak,
which makes their hydroxides approximately as strong as the hydro
xides in the main subgroup of the second group.
The elements of the lanthanide series closely resemble one another
in their properties. For the most part, they exhibit the characteristic
oxidation number of + 3. Their hydroxides, R(OH)3, as a i*ule, are
not amphoteric; in passing from Ce to Lu the basic properties decline,
which is associated with a decrease in the ionic radii (lanthanide
contraction). Since rCe4+ < rCe3+, the basic properties of Ce(OH)4
should be weaker than those of Ce(OH)3. As a matter of fact Ce(OH)4
is a base that is slightly amphoteric.
The actinides exhibit various oxidation states from + 2 to + 7 .
As the atomic number increases, the oxidation state of + 3 becomes
the predominant one. We shall discuss the oxidation states of only
the three most important elements: Th, U and Pu. For Th the
characteristic oxidation number is + 4. The compound Th(OH)4
is a nonamphoteric base; this is explained by the large radius of the
Th4+ ion. The oxidation state most characteristic of U is -f6. It
forms the compound U 02(0H)2, uranyl hydroxide. This is an ampho
teric compound; it can react both with acids and alkalis. Uranyl
hydroxide contains the (U02)2+ ion which is also contained in many
other uranyl compounds where the oxidation number of uranium
is + 6 , for instance, uranyl chloride, U 02C12, uranyl nitrate,
-106 P A R T II. MENDELEEV'S PERIODIC L A W , STRUCTURE OF A T O M S
U 02(N03)2. The formation of similar groups is characteristic of

many elements in the upper oxidation states. For example, the
formula of sulphuric acid can be written as S 0 2(0H)2; the compounds
S 0 2F2, S 0 2C12, and others are analogues of the uranyl salts. The
formation of compounds containing (R 02)2+ is less characteristic
of sulphur than of uranium because the radius of S+6 (rs+6 = 0.30 A)
is much smaller than that of U +6 (ru+6 = 0.80 A); hence, S+6 usually
retains four oxygen ions rather than two oxygen ions; the (S04)2-
ion is more stable than the (U 04)2" ion. Uranyl hydroxide reacts
with alkalis to form uranates, M2 (U 04)2", and diuranates,
M2 (U20 7)2-; it reacts with acids to form uranyl salts. The charac
teristic oxidation state exhibited by Pu is + 4 . The compound
P u(OH)4 is a weaker base than Th(OH)4 due to actinide contraction.
8.4. Fourth Group
Whereas the elements of the boron subgroup mainly exhibit the

oxidation number of + 3 and only in rare instances +1 (because
of which the oxidation-reduction processes in the transition
from one state to another were not typical), the elements of
the carbon subgroup (G, Si, Ge, Sn, Pb), in accordance with
the structure of the outer electron shell, exhibit two oxidation
states + 2 and + 4. In the first state the element has reducing
properties; in the second state, oxidizing properties. In passing
from C to Pb, the oxidation state of + 2 becomes more characteris
tic, and, as a result, substances containing R +2 are more stable.
For C and Si the oxidation state of + 2 is exhibited only in very few
compounds (for example, CO and SiO). The characteristic oxidation
state of these elements is + 4. Ge(OH)2, Sn(OH)2 and Pb(OH)2
are amphoteric compounds in which the basic properties increase
from Ge to Pb; acidic ionization predominates for Ge(OH)2, basic
ionization, for Pb(OH)2. Compounds containing the Ge2+ ion are
strong reducing agents; those containing Pb+4 are strong oxidizing
agents.
Since rR+4 <^r+ 2, the R +4—O bond is stronger than the
R +2—O bond. That is why compounds containing the R +4—O—H
group exhibit acid properties. Carbon in the oxidation state of + 4
forms carbonic acid, H2C03; this is a very weak acid. The acids
H 4S i0 4, H 2[Ge(OH)6], H2[Sn(OH)0] and H2[Pb(OH)?] are even
weaker ones. With increasing size of R+4, the coordination num
ber in this group of compounds increases from 3 (for C+4) to 6
(for Pb+4).
All these substances are unstable; they decompose with the forma
tion of water. Their salts, however, are quite stable. The acid
properties disappear very slowly for the same reason that the basic
Ch. 8 . E L E C T R O N I C S T R U C T U R E A N D P R O P E R T I E S OF E L E M E N T S [ 07
properties increase very slowly in the third group: the Ge4+, Sn4+
and Pb4+ ions have 18-electron shells and consequently their ionic
radii increase slowly. Here there is also observed diagonal simila
rity. Thus, Si+4 resembles B+3. Silicon and boron resemble each
other, in particular, in that they both form a large number of iso
polycompounds.
Boron forms a number of hydrides, which is not characteristic of
the other elements in the third group; but hydrides are formed
by all the elements in the fourth group. Their stability, however,
sharply decreases in the carbon subgroup. Thus, carbon forms
a great number of hydrides, silicon forms a much smaller number
of such compounds; germanium, only a few; tin, only two; lead,
only one, PbH4, which is, moreover, very unstable. Because of
the high oxidation number of these elements, none of the hydrides
are acids.
The elements of the supplementary subgroup (Ti, Zr and Hf)
also exhibit the characteristic oxidation state of + 4. The compounds
corresponding to the formula R(OH)4 are amphoteric; their acid
properties diminish from Ti to Hf.
8.5. Fifth Group

The elements of the main subgroup (N, P, As, Sb and Bi) with
outermost electron shell of the configuration s2p3 exhibit three oxi
dation states: + 5, + 3 and —3.
The oxidation state of -f 5 becomes more stable in passing from
nitrogen to phosphorus, after which it becomes less stable. From
the formulae H N 03, H 3P 0 4 (also H 4P20 7, (H P03)3, and. a great
number of other isopolycompounds), H 3As04 and H[Sb(OH)6]
it can be seen that the coordination number increases from 3 to 6
in passing from nitric acid to antimonic acid. HN03 is a strong
oxidizing agent.
In passing from nitrogen to bismuth, the oxidation state of + 3
becomes more and more stable and the basic character of the com
pounds increases: H N 02 and H3P 0 3 are acids; H3As03 is an ampho
teric compound in which the acid properties predominate; Sb(OH)3
is an amphoteric compound in which the basic properties predomi
nate; Bi(OH)3 is a nonamphoteric hydroxide. These compounds
can exhibit both oxidizing and reducing properties.
The hydrides of these elements have the formula R"3H3; they
are not acids. These compounds are reducing agents; the reducing
properties increase in the main subgroup from N to Bi.
V, Nb and Ta, constituting the supplementary subgroup, exhibit
the characteristic oxidation number of + 5 . They form weak acids
having the formula H R 03; the strength of the acids declines from
V to Ta.
8.6. Sixth Group

In the main subgroup the elements S, Se, Te and Po exhibit the
characteristic oxidation states of + 6, + 4 , and —2, respectively
in conformance with the electronic configuration of their outermost
shell. Oxygen exhibits the oxidation state of —2 (exceptions are
discussed on p. 89).
Compounds containing R +6 have the following formulae: H2S 0 4,
l l 2Se04 and H6Te06. Thus, in passing from selenium to tellurium,
the coordination number increases from 4 to 6 due to the increase
in the ionic radii from S+6 to Se+6 to Te+6. H 2S 0 4 and H2Se04 are
strong acids of almost equal strength; H6TeOe is a weak acid. The
oxidizing activity increases in passing from H2S 0 4 to H2Se04 and
decreases in passing on to H6Te06. Thus, we see that the group of
acids II2S 0 4—H2Se04—H6Te06 presents a typical example of secon
dary periodicity. That is why H 2Se04 does not have properties that
are the average of the properties of H2S 0 4 and H6Te06.
Compounds containing R +4 have the formula H2R 0 3 and are acids
of moderate strength. In the group of acids H2S 0 3—H2Se03—H2Te03
there is a decline in the acid properties and tellurous acid shows
amphoteric properties. Whereas II2S 0 3 is a rather strong reducing
agent, H2T e03 exhibits oxidizing properties.
The hydrides R “2H2 can dissociate with the formation of the
hydrogen ion; there is an increase in the acid properties as we pass
from H2S to II2Te, corresponding to an increase in the ionic radii.
The compounds H2S, H2Se, and H2Te are strong reducing agents;
their reducing power increases from H2S to H 2Te.
The elements of the supplementary subgroup (Cr, Mo, W) form
compounds corresponding to oxidation states from + 2 to + 6; the
most characteristic oxidation state is + 6 , and for chromium + 3
as well. Simple compounds have the formula H 2R 0 4. They are all
acids. The strength of the acids declines from H2Cr04 to H2Mo04
to II2W 04. Besides, for the oxidation state of + 6 there are known
a great number of isopolycompounds (for example, H2Cr20 7). The
acids H2Cr04, H2Cr20 7 and their salts are strong oxidizing agents.
In this subgroup, compounds containing R +6 are progressively weaker
oxidizing agents. The ionic radius of Cr3+ (0.63 A) is close to that
of Al3+ (0.51 A). That explains why Cr(OH)3, like Al(OH)3, is an
amphoteric compound.
8.7. Seventh Group

The main subgroup of this group contains the halogens, i.e., the
elements F, Gl, Br, I, At. Free halogens are oxidizing agents. The
first element of the halogen subgroup, fluorine, is the strongest
of all known oxidizing agents. It differs considerably from the other
Ch. 8. ELECTRONIC STRU CTU RE AN D PRO PERTIES OF ELEM EN TS 109
elements of the subgroup and exists solely1 in the oxidation state

of —1. The aqueous solution of HF is a moderate acid.
The oxidation state of —1 is also characteristic of chlorine, bro
mine, and iodine; these elements, however, can exhibit other oxida
tion states as well. In aqueous solution HR compounds are acids;
the strength of the acids increases from HC1 to HI with increasing
ionic radius, rR_. The HR acids react as reducing agents, the reduc
ing power increasing from HC1 to HBr to HI.
Besides the oxidation state of —1, chlorine, bromine, and iodine
form compounds in which they exhibit positive oxidation states.
The most important of these compounds are the following:
+1 + 5 + 7
HCIO HCIO, HCIO,
HBrO HBrO,
HIO h io 3 h 5i o 4
The stability of the acids increases with an increase in the oxidation
number and in passing from Cl to I. The strength of the acids declines
from top to bottom; it increases from left to right. This increase
is especially noticeable for the acids formed by chlorine. Indeed,
whereas HIO is amphoteric, HC104 i£ the strongest of all the known
acids.
As has been stated above, hydrogen can also be classed together
with the halogens since it can form H" ions which like the halogen
ions (F", Cl", Br", I", At") are isostructural with the atoms of the
noble gases (He, Ne, A, Kr, Xe, Rn, respectively). There are other
resemblances between hydrogen and the halogens: the gaseous state
of hydrogen, the fact that its molecule contains two atoms, the ease
with which hydrogen is replaced by halogens in organic compounds,
the closeness of the energies of decomposition of H2 and Hal2 mole
cules, the comparability of the ionization potentials of hydrogen
and the first ionization potentials of the halogens, etc. Of course
it should be kept in mind that hydrogen differs from the halogens
{because the latter being p-elements, form compounds in which the
oxidation number is higher than unity). However, the properties
of hydrogen bear a greater resemblance to those of the halogens than
to the properties of metals (see p. 103). There is another argument
in favour of this statement: namely, the results of comparative cal
culation. This is illustrated, for example, in Fig. 8.1 which correlates
the melting point (m.p.) and melting heats (A #m) in the halogen
group; the point for hydrogen lies on one straight line with the points
for the halogens. Consequently it is more proper to consider hydrogen
to be a partial analogue of fluorine and to place it above F in the
table of elements, placing it above Li only in parentheses.
1 This is true for the compound formed with oxygen; therefore, 0 +2Fa is not
.fluorine oxide—it is oxygen fluoride.
The elements of the supplementary subgroup (Mn, Tc, Re), with

the electron configuration (n — 1) s2pGd*ns2, exhibit many oxidation
states, from + 2 to + 7. The oxidation numbers characteristic of
manganese are + 2 , + 4 , + 6 and + 7. The compounds corresponding
to these oxidation numbers are Mn+2(OH)2, Mn+4(OH)4, H2Mn+60 4
and IlMn+70 4. In accordance with the Kossel diagram, the acidity
of the compounds increases with increasing oxidation number:
Mn(OH)2 is a base; Mn(OH)4 is a very weak base which is slightly
A Hm , kcal/mole
Fig. 8.1. Comparison of the

melting points and melting heats
of the halogens and hydrogen
amphoteric (this compound readily decomposes to form Mn02 and

H20); H2Mn04 and HMn04 are acids. HMn04 readily decomposes
into Mn20 7 and H20; H2M n04 is also an unstable compound, but
many salts of these acids are stable. In the group Mn+2—Mn+4—
Mn+6—Mn+7, the oxidizing properties grow and the reducing proper
ties decline; Mn+2(OH)2 is readily oxidized, i.e., it is a reducing
agent; the salts of HMn04 are very strong oxidizing agents. The
most characteristic oxidation number of Tc and Re is + 7. The cor
responding compounds have the formula H R 04. The strength of the
acids and their oxidizing power decline in the groups H2Mn04—
H2T c04—H 2R e04 and HMn04—H Tc04—H Re04.
8.8. Eighth Group

This group has no main subgroup. It can be subdivided into two
parts: elements of the iron family (Fe, Co, Ni) and the family of the
platinum metals (Ru, Rh, Pd, Os, Ir, Pt). The oxidation state for
these elements varies from + 2 to +8.
The oxidation numbers characteristic of Fe, Co and Ni are + 2
and +3; in passing from Fe to Ni, compounds in which the oxidation
Ch. S. ELECTRONIC STRUCTURE AND PROPERTIES OF E L E M E N T S 1 1 1
number is + 3 become progressively less stable. In passing from Fe

to Ni, there is a gradual decline in the basic properties of the hydro
xides corresponding to the small decrease in the ionic radii.
Compounds with the formula R(OH)2 are nonamphoteric bases; those
with the formula R(OH)3 are very weak bases with extremely feeble
amphoteric properties. Compounds containing Fe2+ are moderate
reducing agents. The reducing power decreases from Fe+2 to Co+2
to N i+2. The oxidizing power gradually increases from Fe+3 to Co+3
to N i+3.
Iron is also known to exist in the oxidation state of + 6 ; the com
pound H2Fe04 corresponds to this state. Although H2Fe04 is an
unstable acid, many of its salts have been obtained. In the group
Fe+2—Fe+3—Fe+6 there occurs an increase in the oxidizing power,
especially in passing from the oxidation state of + 3 to the oxidation
state of + 6 .
The platinum metals exist in a large number of oxidation states.
The oxidation states of + 2 and + 4 are characteristic of platinum.
The hydroxide Pt(OH)2 is nonamphoteric due to the large radius of
the Pt2+ ion. The compound Pt(OH)4 is amphoteric; it reacts with
both acids and alkalis. Only Ru and Os are known to exist in the
highest oxidation state of -(-8 which they exhibit in the compounds
R u 0 4 and 0 s 0 4. There are no acids or bases in this oxidation state
because the compounds R u 0 4 and 0 s 0 4 are coordinately oxygen-
saturated (consider the group of compounds H2W 04—H iRe04—
H 0O s O 4).
Not only do the properties of the elements vary progressively
within each of the two families, they also vary in the three subgroups,
i.e., in the three vertical groups. Thus, in the group Fe+6—R u+6—
Os+6 there is a sharp decline in oxidizing properties: K2Fe04 is
a very strong oxidizing agent (Fe+6 —>-Fe+s); K2R u04 is readily
reduced to R u02; K20 s 0 4 is relatively easily oxidized to 0 s 0 4.
Whereas Fe04 does not exist at all and R u 0 4 is easily decomposed
on heating (producing an explosion), 0 s 0 4 boils without decomposing.
8.9. Zero Group

The zero group contains only a main subgroup consisting of the
noble gases He, Ne, A, Kr, Xe and Rn. Up to 1962 it was believed
that the atoms of the noble gases could not form stable molecules
with the atoms of other elements, i.e., they exhibited a zero oxida
tion state. This opinion has now been disproved (see p. 62); at the
present time there are already known tens of compounds of xenon
and several compounds of krypton and radon (compounds of the
other noble gases have not been obtained). The stability of compounds
of the noble gases increases with increasing atomic numbers. The
xenon compounds have been studied better than those of the other
112 PA R T II. M E N D E LE E V 'S PERIODIC L A W , STRU CTU RE OF ATOM S
noble gases. It has been found that xenon exhibits the oxidation
states of + 2 , + 4 , + 6 and + 8 as, for example, in the compounds
XeF2, XeOF2, X e03 and Na4Xe06-8H20 , respectively. These com
pounds are relatively stable and can exist at room temperature1.
8.10. Some Conclusions

Generalizing the data on the acid and basic properties and the
oxidizing and reducing properties of compounds containing R —0 —H
and R —H bonds, we can draw the following conclusions. If we
examine any given oxidation state, we see that in each subgroup,
in passing from top to bottom, the basic properties of the elements
are progressively more pronounced and their acid properties decline.
On the contrary, with increasing oxidation numbers of a given ele
ment, there is a gradual decrease in the basic properties and an
increase in the acid properties.
As the number of the group grows there is a greater tendency
to exhibit various oxidation states, including negative ones,
the latter becoming more characteristic of the elements in the
main subgroups with an increase in electron affinity. The greater
variation of oxidation states is manifested in the greater proba
bility of the occurrence of a larger number of different oxidation-
reduction reactions.
We have already noted (see p. 72) that in passing from Group I
to Group VII, the difference in the properties of the elements of the
main and supplementary subgroups first diminishes, then increases,
and finally becomes considerable. But this difference is great for
the lower oxidation state and smaller for the upper states. Thus,
chlorine (Is2 2s2 2p6 3s2 3p5) and manganese (Is2 2s2 2p6 3s2 3p6 3d5 4s2)
have nothing in common and the properties of compounds of Cl+
(Is2 2s2 2p6 3s2 3p4) and those of Mn+2 (Is2 2s2 2p6 3s2 3p6 3d5) are not
similar, but compounds containing Cl+7 (Is2 2s2 2p6) and Mn+7
(Is2 2s2 2p6 3s2 3p6) should have similar properties; this we know
to be a fact.
Thus, although the Kossel diagram is a rough approximation,
and the more so the higher the oxidation state of the elements, it
has been, however, of great value not only in systematizing the
voluminous data on the properties of the most important compounds
but also in predicting their behaviour. That is why the material set
forth above may be regarded as inorganic chemistry in miniature,
its outline so to say.
1 Since compounds of the noble gases have been obtained, recently the zero
group has often been called Group VIIIA in order to emphasize the fact that
the heavier elements in this group form compounds in which the oxidation
state is the highest, i.e., + 8.
Ch. 9. SIGNIFICANCE OF PERIODIC LAW 113
CHAPTER NINE
SIGNIFICANCE OF THE PERIODIC LAW
Friedrich Engels called the discovery of the Periodic Law by
D. Mendeleev a scientific feat. Now when the Periodic Law is ac
cepted by us since our school days as one of the fundamental laws
of nature, it is difficult to overestimate and adequately appreciate
the genius of Mendeleev’s generalization.
At the time the Periodic Law was formulated only 63 elements
were known and their atomic weights and valencies in many cases
had been determined incorrectly. Nevertheless, on the basis of the
Periodic Law, D. Mendeleev revised the valencies and corrected
the values for the atomic weights of many elements; he placed a
number of elements in the periodic table regardless of the concep
tions of their relationships generally accepted at that time, not
being tempted to correct certain seeming (as we now know) devia
tions. Moreover, he predicted the discovery of many new elements
and even indicated the properties of the main compounds of some
of them.
It is indisputable that the aim of true science is to know in order
to predict. The significance of the Periodic Law, however, is not
confined to the possibility it affords of estimating the values of the
great number of physical and chemical properties of the elements
and their compounds, so necessary for theoretical and practical
purposes.
If the Periodic Law had not been discovered, it would have been
impossible to determine the atomic structure of elements. Just like
in the ancient Greek myth Thesius found his way out of the Mino
taur’s labyrinth by medns of Ariadne’s thread, so with the aid of
the Periodic Law it became possible to understand the structure
of atoms, i.e., to solve a problem more intricate than the legendary
labyrinth. ✓
It should be noted that although information about atomic struc
ture is of major importance for science, it does not substitute the
Periodic Law. The Periodic Law has made possible the prediction
and calculation of such properties of elements and their compounds,
which .as yet cannot be calculated theoretically on the basis of data
on the electronic structure of atoms and molecules. Obviously, with
the advancement of science, there will be greater possibilities for
theoretical calculation but it is also evident that it will lead to the
study of a still greater number of substances and properties; there
fore, apparently there will always be some difference between what
can be calculated by means of the theory of atomic and molecular
structure and what can be found with the aid of the Periodic Law.
According to the principles of dialectical materialism, chemistry
cannot be reduced to physics. All the laws of physics are also observ-
} 3aK . 15648
114 P A R T I I . M E N D E L E E V ’S P E R I O D I C L A W , S T R U C T U R E OF A T O M S
ed in chemistry; the latter, however, is governed by certain laws of

its own, the most important of which is the Periodic Law.
Though the Periodic Law and Periodic System of the Elements
were set forth and established when the atom was considered to be
indivisible, they constitute a fundamental generalization. Many and
varied are the complex regularities of nature that are expressed
in this simple law. Subsequent scientific progress, as the discovery
of new elements; the determination of their properties and the pro
perties of compounds formed by them; the discovery of radioactivity,
isotopes, the complex structure of atoms, artificial radioactivity
and many other advances in science have only corroborated the
Periodic Law, disclosed new aspects of the law, broadened and dee
pened its content. .
The Periodic Law is the quintessence of the science of chemistry,
the basis for correlating and comprehending the enormous number
of facts that constitute the inexhaustible source of new discoveries
and generalizations. Niels Bohr wrote that the Periodic System is
“the guiding star for investigators in the fields of chemistry,
physics, mineralogy, technology”. It has greatly influenced the
development of geology, geochemistry, nuclear physics, astrophysics,
cosmogony. The Periodic Law is one of the general laws ol nature
by which science is constantly being enriched. In this lies its great
service to science.
The philosophical significance of the Periodic Law cannot be-
overestimated. It vividly shows the interrelationship and interde
pendence of chemical phenomena. All the elements are connected
like the links of a chain. Each element can be studied only in con
nection with others and the entire Periodic System, only in .the-
light of the properties of each element.
It is hard to name another law which would illustrate more dis
tinctly how quantitative changes (increase of Z) result in qualitative
changes, viz., the appearance of new properties of elements and their
compounds.
When studying the Periodic System of the Elements, we are
convinced over and over again that it contains vivid examples con
firming the law of unity and conflict of opposites, constituting"
the gist of dialectics: the periods combine elements with opposite
properties; and the same elements, depending on the conditions, can
react in different ways. For example, sulphur can act as an oxidizing"
agent and as a reducing agent; arsenic in some of its properties resem
bles metals, whereas in others, nonmetals, etc.
Numerous examples can be cited that confirm the third law of
dialectics, the law of negation of negation. We shall mention only
one of them—the transition from one period to another as the result
bf the formation of a new electron shell, i.e., a repetition of the
course of development of passed stages on a new and higher basis.
PART III
THE STRUCTURE OF MOLECULES

AND THE CHEMICAL BOND
CHAPTER TEN
INTRODUCTION
10.1. Molecules, Ions and Free Radicals

The atoms of elements can form three types of particles involved
in chemical processes—molecules, ions and free radicals.
The molecule is defined as the smallest neutral particle of a given
substance possessing its chemical properties and capable of independent
existence. Molecules may be monoatomic, diatomic, triatomic and
so on, that is, polyatomic. In ordinary conditions- the noble gases
consist of monoatomic molecules; on the other hand, the molecules
of high-molecular compounds contain thousands of atoms.
The ion is a charged particle comprising an atom or group of che
mically bound atoms with an excess of electrons (anions) or an insuf
ficiency of electrons {cations). The positive ions in a substance always
exist together with negative ions, and since the electrostatic forces
acting between ions are great it is impossible to bring about a signi
ficant excess of ions of the same sign in a substance.
The free radical is a particle possessing unsaturated valences (the
term is defined on p. 203, on the basis of electronic conceptions).
CH3 and NH 2 are examples of such particles. Under ordinary condi
tions free radicals cannot, as a rule, exist for any considerable length
of time, but they play a very important part in chemical processes,
many reactions being impossible without their participation. At very
high temperatures, e.g., in the solar atmosphere, the only diatomic
particles that can exist are free radicals (CN, OH, CH and some
others). Many free radicals are present in a flame.
Free radicals of more complex structure are known which are
relatively -stable under ordinary conditions. One such radical is
triphenylmethyl and it was the discovery of the latter that marked
the beginning of the study of free radicals.
In 1900 Gomberg of USA tried to prepare hexaphenylethane
C6H5 c 6h 5
C6H5 - C ---------- G — C6H5
[ I
c 6H 5 c 6h 5
by reacting triphenylchloromethane, (C6H5)3CC1, with zinc, copper or finely
dispersed silver. Through the union of the metal with the chlorine in the tri-
8*
116 P A R T I I I . S T R U C T U R E OF M O L E C U L E S A N D C H E MI C AL BOND
phenylchloromethane molecule one valence should be freed and the radicals

(C6H5)3C — formed should unite to give hexaphenylethane (Wurtz — Fittig syn
thesis).
However, the properties of the substance Gomberg obtained were not like
those of hydrocarbons. It formed colourless crystals which yielded bright yellow
solutions when dissolved in organic solvents, and these solutions readily reacted
with iodine, atmospheric oxygen and other substances. Gomberg accordingly
concluded that in solution hexaphenylethane dissociated, thus forming the
free radical triphenylmethyl
(C6H5)3C— C(C6H5)3 ^ 2(G6H5)3C—
Later investigations showed that this reaction actually takes place, though not
to such a degree as Gomberg supposed; the dissociation of hexaphenylethane
into triphenylmethyl becomes considerable only at temperatures of 50-100°C.
The relative stability of triphenylmethyl is due, among other
things, to spatial restriction; the large size of the phenyl groups C6H6
hinders union of the radicals to form hexaphenylethane. Other cau
ses will be discussed below (see p. 193). Other stable free radicals
similar to triphenylmethyl are also known.
10.2. Development of Conceptions

of the Chemical Bond and Valence
The first attempt to elaborate a theory of the chemical bond
dates back to the beginning of the nineteenth century when Bergman
(Sweden) and Berthollet (France) advanced the idea that the tendency
of particles to unite with other particles 1 was due to the force of
gravity acting between them. But it was found that chemical affi
nity is not proportional to the mass of the atoms combined in a mole
cule—the mercury atom is two hundred times heavier than the hydro
gen atom but water is incomparably more stable than mercuric oxide.
(Give other examples, comparing the stability of compounds of
a given element with that of compounds of other elements in the
same subgroup of the periodic system.) Furthermore, the force of
gravity acts at any distance, but chemical forces are effective for
only 0.5-3 A. The former is small and diminishes in inverse propor
tion to the square of the distance; the latter are colossal—roughly
K P times greater than the force of gravity—and in many cases
diminish with distance to a degree considerably greater than two.
The force of gravity is never saturated—matter forms enormous accu
mulations of atoms, such as planets, for example. The hydrogen
atom, on the other hand, after combining with another hydrogen
atom cannot combine with a third—chemical forces are characterized
by saturability. Moreover, unlike gravitation, chemical forces are
usually associated with a definite direction in space (see p. 122 ).
4 This tendency is usually called chemical affinity, a term that goes back to
the alchemic period of chemistry.
Ch. 10. I NTRODUCTI ON 117
Whereas the force of gravitation, according to Newton’s law, acts

on all bodies, the effect of chemical forces is specific—a chlorine atom
is coupled to a sodium atom by a strong bond and to a chlorine atom
by a weaker bond; it is not coupled to a helium atom by any bond.
If one takes into consideration the effect of external conditions on
the strength of the chemical bond, temperature, for instance, it
becomes clear why the Bergman—Berthollet gravitational theory was
untenable.
It gave way to the electrochemical theory formulated by the Swedish
chemist, Berzelius (1810). According to this theory the atom of every
element has two poles, positive and negative, the former being
predominant in some elements, and the latter predominant in others.
From the standpoint of the Berzelius theory, the union of electro
positive magnesium with electronegative oxygen was explained by
the attraction of the predominant poles, since these are of opposite
signs. If there is partial compensation of the charges, the reaction
product does not lose them completely. This explained the formation
of more complex molecules, such as magnesium carbonate through
the combining of ‘positive’ MgO with ‘negative’ CO2. The Berzelius
theory was a development of Davy’s idea (1806) that the chemical
bond springs from the mutual attraction of bodies bearing charges
of unlike signs. On the face of it, the electrochemical theory seems
plausible and would seem to be confirmed by the process of electro
lysis-electrolysis would seem to restore the polarity lost by the
atoms through the formation of compounds. But Hegel, criticising
the Berzelius theory, wrote that such an approach took no notice
of changes in specific gravity, cohesion, shape, colour and so on,
which take place during a chemical process, nor of acid, corrosive,
alkaline and other properties, everything disappearing in the abstrac
tion of electricity. “Let people stop upbraiding philosophy “for
abstraction from the particular and for empty abstract conceptions”,
if physicists chose to forget about all the enumerated properties
of corporeity for the sake of positive and negative electricity.”
Actually the electrochemical theory soon disappeared from scienti
fic use since it was refuted by the existence of stable molecules
consisting of atoms of like polarity, e.g. H 2 and Cl2, and by the
carrying out of processes by Dumas (1834) in which elements of
unlike polarity, according to the Berzelius theory, replaced one
another in compounds.
In the forties of last century, the French chemists, Dumas and
Gerhardt, advanced the theory of types. According to this theory the
chemical properties of substances were associated with the analogy
in the composition of their molecules, and were practically indepen
dent of the nature of the atoms. This was an attempt to elaborate
a theory of chemistry based entirely on facts relating to the compo
sition of substances. Organic compounds were considered to be deri-
118 P A R T III. STRUCTURE OF M O L E C U L E S A N D C H E M I C A L B O N D
vatives of a few inorganic substances. Thus, ethyl alcohol,

C2H 5OH, and diethyl ether, 0211.5—0 —02115, were thought to be of
the H20 type, and were regarded as substitution products of water
in which one or both hydrogen atoms were replaced by the C2H 5
group. In the same way, CH 3NH 2 and (CH3)2NH were thought
to be of the NH 3 type and so on. Thus a systematization of informa
tion was achieved, but on a strictly formal basis. Things went so far
that in order to make the numerous newly discovered compounds fit
into the procrustean bed of types, different formulae had to be ascribed
to the same substance, depending on the reactions into which it
entered. In his manual of organic chemistry, Gerhardt wrote that by
confining a substance, so to say, to a single formula, one often concea
led chemical relationships which another formula would have made
understandable. Attempts to elucidate the structure of molecules were
rejected on the grounds that this was beyond human comprehension—
the mind could not fathom the infinitesimal. It was held that studying
the matter in chemical processes only made it possible to establish
its past and future, and not its present; the nitration of C6H6 was
the past of nitrobenzene, while reduction of the latter to C6H 5NH 2
was its future.
Soon •the results of an investigation were published which led
to the idea of definite structure of molecules. In 1852 the English
chemist Frankland on the basis of a study of the formation of certain
metallo-organic compounds—GH3Na, (CH3)2IIg, (CH3)3A1, (CH3) 4Sn,
etc.—introduced the conception of atomicity (valence) which expres
ses numerically the capacity of an atom of a given element to combi
ne with a definite number of atoms of a different element. If the
valence of hydrogen is taken as unity, it can be considered that
the valence of another element is the number of atoms of hyd
rogen (or some other monovalent element) with which it will com
bine.
The magnitude of valence depends on the state of the given ele
ment, the nature of the element with which it reacts and the condi
tions of the reaction. All elements can be divided into two catego
ries—those with a constant valence, such as sodium, and those with
a variable valence, such as phosphorus or sulphur.
10.3. A. Butlerov’s Theory of Chemical Structure

In 1861 A. Butlerov propounded a theory which can be formulated
essentially as follows:
(a) the atoms in a molecule are combined in a definite order;
(b) atoms combine in accordance with their valence;
(c) the properties of a substance depend not only on the nature
of the atoms and their number, but also on their arrangement, i.e.,
on the chemical structure of the molecules.
Ch. 10 . I N T R O D U C T I O N 119
This theory explained the multiformity of organic substances,

and dealt a decisive blow to the agnostic theory of types. Butlerov
showed that the internal structure of molecules is cognoscible and
can be intentionally reproduced. On the basis of Butlerov’s theory
the structure of molecules can be established by the study of chemical
transformations—his theory indicated chemical methojds of investi
gating the structure of a substance. Thus, the empirical formula of
ethyl alcohol is C2H 60. Taking into account the valence of the ele
ments contained, only two structural formulae are possible
H H H H
H— i —O—H and H - C - O —C - H
k H H H
A study of the chemical reactions of ethyl alcohol shows that its
molecule has the first of the above structures—when alcohol is
reacted with metallic sodium only one hydrogen atom is replaced
2C2H60 + 2Na = 2C2H5NaO + H2
This is in accordance with the first formula in which one atom of
hydrogen is different from all the others—it is joined to oxygen,
not carbon. During the century which has passed since the formulation
of the theory of chemical structure, chemists have worked persistently
along this line, thus establishing the structural formulae of thousands
of organic and elemento-organic compounds. Proving some of these
formulae required tremendous effort. For instance, a large number
of investigators in different countries had worked on the chemistry
of quinine for over 60 years before the structure of the quinine
molecule was finally elucidated.
Later study of molecular structure, employing physical methods
developed in the 20 th century, some of which will be considered
below, confirmed the arrangement of atoms in molecules found by
means of the Butlerov theory. Thus the Butlerov theory laid the
foundation of present-day principles of molecular structure.
The theory of chemical structure introduced the extremely fruitful
conception of the mutual influence of the atoms in molecules. It was
found that in a molecule it is not only the directly bound atoms
that act on one another—there is a mutual influence of all the atoms.
It goes without saying that the effect of the mutual influence of
atoms in the molecule which are not directly bound with one another,
the so-called induction effect, is relatively small. Nevertheless, in
some cases it is significant. Thus, if all the hydrogen atoms in one
CH3
120 PART III. STRUCTURE OF M O L E C U L E S A N D C H E M I C A L B ON D
of the methyl groups of tertiary butyl alcohol are replaced by fluo

rine, the alcohol acquires acidic properties. This is explained by the
fact that fluorine, which possesses great electron affinity, strongly
attracts electrons, and its introduction into a molecule causes
a displacement of electrons along the chain of atofns
F CH 3
X I
----C<— O*---- H
F CH,
This displacement is dampened as the distance from the atom causing

the induction effect increases.
Lastly, the Butlerov theory explained isomerism, a phenomenon
discovered by Liebig and Wohler in 1823, which in turn played an
important part in the formation of the theory of chemical structure*
Isomerism is the existence of several compounds of the same compo
sition but differing in properties because of differences in the struc
ture of their molecules. The idea of the existence of such substances
goes back to the investigations of M. Lomonosov. In his work “On
Metallic Glance” (1745) he wrote that one of the possible causes
of changes in the properties of a substance could be the shifting of
the arrangement of its parts.
There are two types of isomerism—structural and spatial or stereo
isomerism.
10.4. Structural Isomerism
Structural isojnerism is caused by a difference in the sequence
of the bonds between the atoms in a molecule. There are several
kinds of structural isomerism.
If molecules differ in the arrangement of the carbon atoms forming
their framework, this is termed skeletal isomerism. Here is an example
CH3 — CH2— CH2—CH2— CH3 and CH3—CH —CH2—CH3
< k
Normal pentane 2-Methylbutane

The number of isomers quickly mounts as the number of carbon
atoms in the molecule increases. For C6H 14 there are six isomers,
but for C20H 42 there are 366,319! And compounds of the formula
CiooH2o2 are known. Isomerism involving changes in the carbon
skeleton can be demonstrated with other examples. Here is one
CH2— c h 2 c h 2—c h 2
H2c ' ^CHjj and H3C —H C ^
\ / \
h 2c —c h 2 c h 2—c h 2
Cyclohexane Methylcyclopentane
Molecules having the same carbon skeleton but differing in the

position of functional groups1 are termed position isomers
CH3—CH2—CH2OH and CH3- C H - C H 3
In
Normal propyl alcohol Isopropyl alcohol
c i/^ Cl —^ V - Cl
and
Cl1
V
Orthodichloro Metadichloro
benzene benzene
It should be noted that on the basis of his theory Butlerov foretold

the existence of tertiary butyl alcohol and was the first to synthesize
this isomer of normal butyl alcohol; reduction gave isobutane, the
only possible isomer of butane.
A third form of structural isomerism is metamerism, in which mole
cules having the same empirical formula differ in the composition
or structure of radicals attached to a non-carbon atom. Thus
CH3— 0 — CH2— CH2— CH3 and CH3—CH2—0 —CH2— CH3
Methylpropyl ether Diethyl ether
have the same empirical formula but the radicals attached to the
oxygen atom differ.
There are also structural isomers which differ from one another
in the character of functional groups
CH.COOH and CH2(0H) — c / °
XH
Acetic acid Glycolic aldehyde
A special kind of structural isomerism is dynamic isomerism or
tautomerism, in which the two isomeric forms are easily transformed
from one to the other and are in a state of equilibrium. This pheno
menon logically follows from Butlerov’s conception of the dynamic
interrelationships of the atoms in a molecule, and was first foretold
and explained by him (1862). The keto-enol equilibrium is an exam
ple of tautomerism
CH3— C— CH3 T t CH3— C= CH2
II I
O OH
Keto form Enol form
1 Functional groups are groups of atoms, such as —OH, —N 0 2, —NH2v

—COOH and —S 0 3H, which to a considerable degree determine the chemical
properties of a compound, and consequently, the class of compounds to which
it belongs.
122 PART I I I . S T R U C T U R E OF M O L E C U L E S A N D C H E M I C A L BO N D
It should not be thought that structural isomerism is limited to

organic compounds. It is also encountered among inorganic compo
unds, as illustrated by the following examples
H O -N = N — OH and 0 2N - N H 2
Hyponitrous acid Nitramide
O
II .0
F — C1 = 0 and F—O — C1C
II
o
Perchloryl fluoride Chloryl oxyfluoride
Developments in atomic physics during the last several decades

have led to the discovery of a new type of isomerism, which may
be termed isotope isomerism. This can be illustrated by the molecules
CH COOH and CH COOH which are tagged with radioactive car
3 3
bon—the former in the methyl group, and the latter in the carboxyl
group.
It was only Butlerov’s work that made it possible to explain all
forms of structural isomerism with a single theory. Before that,
isolated instances of isomerism “wandered homelessly”, as Liebig,
put it, through the realm of science.
Isomerism is a manifestation of one of the forms of transition
of quantity to quality—when the number of atoms in a molecule
reaches a certain value, a variation in their arrangement becomes
possible.
10.5. Spatial Isomerism
Butlerov’s theory was significantly extended in 1874 when the
Dutch chemist van’t Hoff and the French chemist Le Bel working
independently of one another, suggested that the four valences of
carbon are directed towards the vertices of a regular tetrahedron,
at the centre of which the carbon atom is situated (Fig. 10.1)1.
In this case the angles between all the bonds are equal to 109.5°.
This means that if the centre of the carbon atom in a molecule is
connected by straight lines to the centres of the other atoms surround
ing it, these lines representing the bonds will be directed towards
the vertices of a tetrahedron with the carbon atom at the centre.
This was the origin of the conception of the spatial structure of the
molecule which developed into the branch of chemistry known as
stereochemistry.
On the basis of van’t Hoff and Le Bel’s suggestion it was concluded
that a particular class of isomerism—optical isomerism should exist.
1 A regular tetrahedron is a three-faced pyramid whose faces are regular tri

angles.
Let us consider a carbon atom joined to four unlike atoms or groups

which can be designated by the latters A, B, D and E (Fig. 10.2).
Such an atom is said to be asymmetric. As can be seen in Fig. 10.2
two structures are possible in this case, as in a and 6, which are
mirror images of one another. Since all the distances between the
atoms are equal to both structures, as are the angles between the
bonds, the chemical properties of the two isomers, according to the
Butlerov theory, should be identical.
The existence of certain compounds in several forms which could
mot be distinguished by their chemical properties was known long
Fig. 10.1. Spatial arrangement

of valences of the carbon atom
before the appearance of the works of van’t Hoff and Le Bel This
phenomenon was discovered in. 1848 by Pasteur (France). When
studying tartaric acid
COOH — CH(OH) — CH(OH) — COOH
he found that this compound existed in two forms which had iden
tical chemical properties but differed in the asymmetry of their
crystals—the crystals of one form were as if the mirror images of the
crystals of the other form (Fig. 10.3). Such crystals are said to be
enantiomorphous and the compounds forming them are termed optical
antipodes.
Van’t Hoff and Le Bel explained the existence of optical antipodes
as being due to the presence of asymmetric carbon atoms in their
molecules; consequently there could be isomers of the structure
shown in Fig. 10.2. Thus, in the tartaric acid molecule
HO OH 0
0\'cJL‘L y
H o / A A x OH
carbon atom 2 is asymmetric. It is bound to H and three different
radicals, —OH, —COOH and —CH(OH)—COOH. For that reason
there should be two optical isomers1.
1 Carbon atom 3 is also asymmetric but since the atoms and radicals bound
to atoms 2 and 3 are identical, the isomers resulting from the asymmetry of
atom 3 are the same as the isomers resulting from the asymmetry of atom 2.
124 PART III. S T R U C T U R E OF MO L E O UL E S A N D C H E MI C AL BOND
E E
Fig. 10.3. Crystals of mirror isomers of sodium ammonium tartrate

To show the asymmetry of the crystals two faces, denoted a and bt are hatched
It should be noted that there would be no optical isomerism if all

four valence lines of the carbon atom were in the same plane. In
that case the distances between the atoms in the structures
A B A B
\ /
C
,
and
\ /
C
/ \ / \
D E E D
(for example, between atoms D and A) would be different and the

isomers would have different chemical properties; there would be
a different type of isomerism which will be discussed below.
Optical isomerism is characteristic of all compounds containing
asymmetric carbon atoms. Here are some examples of such compounds,
the asymmetric atoms being indicated with asterisks

CH3 OH
CoH5—C* —OH CH3— C* —COOH
H1 a
Isobutyl alcohol Lactic acid
OH OH OH OH
1 1 1 1
Hv X — c* —C* — C*—-C* — CHoOH
0^ 1 1 1 1
H H H Ii
Hexose
The presence of each additional asymmetric carbon atom in a

molecule doubles the number of possible isomers, i.e., if there are n
asymmetric atoms the number of isomers is 2n. Consequently for
COOH COOH
Fig. 10.4. Optical isomers of alanine
hexose, the last compound shown above, 16 isomers are possible,

and all have been found. Four occur in nature, and the others have
been prepared.
As stated above the chemical properties of optical antipodes are
identical. Such isomers differ from one another only in the symmetry
{or asymmetry, to be more exact) of their crystals and in the direction
of rotation of the plane of polarized light.
If polarized lig h t1 is passed through an optical isomer in the crystalline,
liquid, gaseous or dissolved state, the plane of polarization is rotated through
a certain angle, depending on the^ number of molecules of the substance in the
path of the light beam. Substances which rotate the plane of polarization are
said to be optically active. When polarized light is passed through the other
isomer, the plane of polarization is rotated through the same angle, but in the
opposite direction. One of the isomers is dextrorotatory (+ ) and the other,
laevorotatory ( — ).
The configuration of optically active organic compounds is distinguished

by the prefix D- or L- (from the Latin words dexter , right, and laevus, left).
Thus, the D- and L-forms of amino-acids are considered as derivatives of the
D- and L~isomers of the amino-acids of alanine (Fig. 10.4).
1 The polarization of light is discussed in Appendix IV.

126 P A R T III. S T R U C T U R E OF M O L E C U L E S A N D CH EMI C AL BOND
In chemical reactions in which optically active compounds are not involved

the end product, if it contains an asymmetric carbon atom, will always be
a mixture of equal quantities of the two stereo-isomers. Three methods of divi
ding optical isomers are known, all of which were first proposed by Pasteur.
1. When a mixture of optical isomers is crystallized they form crystals of
different asymmetry; by carefully examining these crystals it is possible to
separate the crystals of one isomer from those of the other isomer.
2. Microorganisms for which the given substance is a nutrient, usually decom
pose only one of the optical isomers, leaving the other untouched. The explana
tion is that only certain forms of optically active compounds can serve as buil
ding material for living organisms.
Fig. 10.5. Structure of addition products of optical isomers with another opti
cally active compound
3. When optical isomers are made to combine with another optically active
compound the products will not be identical in properties. This is due to the
fact that in this case the distances between the atoms are different. As can be
seen in Fig. 10.5, when two particles with asymmetric atoms are combined in
a molecule, the interatomic distances, e.g. the distance DD', cannot be identi
cal for the different isomers. Compounds thus prepared can be separated, and
after that the optical isomers can be isolated.
It is noteworthy that the proteins of all living organisms consist only of the
L-isomers of amino-acids, but the causes of this phenomenon are unknown.
Since only certain forms of optically active substances are utilised by organisms,
the other forms are useless for them. This explains the fact that microorganisms
decompose only one from a pair of optical isomers, the other being left untouched.
Another form of spatial isomerism not associated with the presence
of mirror-like arrangements of the atoms in the molecules, which
cannot be superimposed, is geometric isomerism. It is caused by
unlike arrangement of the atoms and is characteristic of unsaturated
and cyclic organic compounds. It is the different arrangement of the
substituents attached to the carbon atoms in relation to the double
bond in unsaturated compounds, or in relation to the plane of the
ring in cyclic compounds that gives rise to this form of isomerism.
In the first case the isomers are compounds containing a like substi
tuent combined with both of the carbon atoms joined by a double
Ch. 11. BASIC CHARACTERISTICS OF C H E M I C A L BON D 127
bond. The substance in which the like substituents are on one side
of a plane drawn through the double bond is called the cis isomer.
In the trans isomer the substituents are on different sides of the
plane. In cyclic compounds the difference consists in whether the
like substituents are arranged on the same side of the plane of the
ring (cis form) or on different sides (trans form). Thus in the cis form
the like substituents are closer together than in the trans form. This
is illustrated by the following molecules
H H H Cl
\ /
C= C and C^C
/ \ /
Cl Cl Cl H
Cis-l,2-dichloro- T r a n s - 1,2-di-
ethylene chloroethylene
Cl
h 2c and H,C'< * lr £ s ?
^ c n r a
Cis-1,2-d ichlorocy clo- Trans-1,2-d ichlorocyclo-
pentane pentane
It should be emphasized that cis—trans isomerism is not found
in acyclic compounds in which the carbon atoms are joined by single
bonds because of the possibility of rotation round the single bond
(for example, the CH3 group in ethane).
Unlike optical isomers, geometric isomers differ in their physical
and chemical properties. Thus melting points of the cis- and trans-
dichloroethylenes differ by over 30 °C.
It follows that often it is not enough to know the sequence of the
atomic bonds—one must also know their spatial arrangement, i.e.,
the configuration of the molecules.
It should be noted that spatial isomerism, like structural isomerism-
is also characteristic of inorganic compounds, such as complex com,
pounds (see pp 235-236).
CHAPTER ELEVEN
BASIC CHARACTERISTICS
OF THE CHEMICAL BOND-LENGTH, DIRECTION, STRENGTH
The basic parameters of molecules are the length of the bonds
between the atoms (the internuclear distance), the angles formed
in the molecules by the lines connecting the centres of the atoms
in the direction of action of the chemical bonds between them (valen
ce angles), and also the energies of the bonds, which determine their
128 P A R T I I I . S T R U C T U R E OF M O L E C U L E S A N D C H E M I C A L BON D
strength1. For a complete characterization of a molecule it is neces

sary to know the distribution of electron density and the electronic
energy levels.
11.1. Length of Bonds12
Bond length d can be estimated approximately on the basis of the
atomic or ionic radii or from a rough determination of the size
of the molecules by means of the Avogadro number (see p. 15).
Thus, the volume of the water molecule
^h2o = 0.023-10-23 ==29.7*10 24 cm3*
from which
dH 2 1^29.7.10-24« 3 -10“8 cm = 3 A
Actually the length of the bonds is of the order of 1 A. For an
approximate estimation of d one can also use the formula
dA-A + ^B-B
â- b 2
based on the assumption that each atom makes a constant contri
bution to the interatomic distance.
Methods to be taken up later enable a precise determination of
bond lengths. Thus, the following values of d were found by means
of these methods: for H2, 0.74; N2, 1.09; 0 2, 1.21 A. The regular
change in the atomic (ionic) radii in the periodic system entails
a regular change in internuclear distances d. Thus, for the molecu
les HX we find
H - F ......................0.92 A H—B r ....................... 1.42 A
H—C l ......................1.28 A H—I ........................... 1.62 A
If the hydrogen in this series is replaced by a different element,

say carbon, the character of changes in d for the same X remains the
same.
The parallelism of changes in d in diatomic molecules makes it
possible to employ the method of comparative calculation for their
quantitative estimation (see p. 75)3.
An analysis of experimental data shows that for a constant valent
state the internuclear distance for a given type of bond remains
1 The strength of the bond can be characterized by energy or by the force
required to break the bond, but these characteristics are not equivalent. The
term strength is used here as a synonym of energy.
2 Here and below, what is meant is the internuclear distance corresponding
to the stable state of the molecules where the forces of attraction are balanced
by the forces of repulsion, and potential energy is minimum (see p. 167).
3 In this way it has been possible to determine the value of d for over 100 bonds
for which there is no experimental data.
Ch. 11. BASIC CHARACTERISTICS OF C H E M I C A L BON D 129
practically constant in different compounds. Thus, in alloaliphatic

compounds dc-c lies within the range from 1.54 to 1.58 A, and in
aromatic compounds, from 1.39 to 1.42 A.
Transition from the single bond to a multiple bond is marked
by a shortening of internuclear distances which can be associated
with strengthening of the bond. Whereas dc-c ~ 1.54, dc=c ~ 1.34
and d c=c ~ 1-20 A.
11.2. Valence Angles
The values of valence angles depend on the nature of the atoms
involved and the character of the bond. Whereas all diatomic mole
cules of the A2 or AB types can be represented as
— O
A B
various configurations are possible for triatomic, tetratomic and

more complicated molecules.
A triatomic molecule can be linear or nonlinear
• O •
B A B B B
dC. BAB = 180° ^BAB<180°
The first type is found in molecules containing some elements of

Group II, for example, BeCl2, ZnBr2, Cdl2, and in a number of
Fig. 11.1. Structural variants of molecule AB3
other molecules (C02, CS2). Some molecules with unlike internuclear

distances, HCN for one, also have this configuration. The second
type includes many compounds of p-elements of Group VI (S02,
H 20 , etc.).
In series of similar molecules, /_ BAB changes regularly, an exam
ple being the series H20 (104°28'), H2S (92°), H2Se (91°), and H2Te
(89°30').
9 3aK . 15648
130 PART III. STRUCTURE OF M O L E C U L E S A N D C H E M I C A L BON D
A tetratomic molecule, AB3, can be of planar or pyramidal confi

guration (Fig. 11.1). The first type of molecule is characteristic
of some compounds of elements of Group III (BC13, AlBr3); some
ions also have a planar configuration (NO3, CO2"). C1F3 is an example
of a T-shaped molecule.
g 106 C 1.21 g 1.06 g

(a)
I---------------------------- \
Fig. 11.2. Structure of molecules of
(a) acetylere: (b ) hydrogen peroxide
(interatomic distances are shown over
the bonds)
A three-dimensional arrangement of the atoms in molecules is

encountered more often. NH3, PC13 and other compounds of p-ele-
ments of Group V have a pyramidal configuration. Here too there is
a regular change in the angles, as seen in the series NH 3 (107°20'),
Fig. 11.3. Structural variants of molecule AB4
PH 3 (93°20'), AsH 3 (91°50') and SbH 3 (91°20'); PC13 (101°), AsCl,

(97°), SbCl3 (96°) and BiCl3 (94°); PF 3 (104°), PC13 (101°) and P I 3
(98°).
Valence angles in tetratomic molecules can also have other values.
Acetylene and hydrogen peroxide are examples (Fig. 11.2).
AB4 molecules may have the configurations shown in Fig. 11.3.
The first type is found comparatively rarely. An example of such
a square, planar particle is the (PdCl4)2" ion. A tetrahedral arrange
ment of thfc atoms is more frequent, and is typical of compounds
of carbon (see p. 123) and its analogues of Group IV. The (S04)2" ion
likewise has a tetrahedral structure.
Ch. 11. BASIC CHARACTERISTICS OF C H E M I C A L BO N D 131
TeCl4 is an example of a molecule of the type
The following structures are characteristic of AB5 molecules:
PC15 is an example of the first structure (trigonal bipyramid), and

IF5, an example of the second.
Diagrammatic representations of AB6 and AB7 molecules (octa
hedron and pentagonal bipyramid, respectively) are as follows
132 PART III. S T R U C T U R E OF M O L E C U L E S A N D C H E M I C A L BO N D
SF6 is an example of the first structure, and IF7, an example of the

second.
In aliphatic organic compounds the length of the C—C bonds
is ~1.54 A, and the valence angles between the C—C bonds are
Fig. 11.4. Structure of normal pentane molecule
109.5°. Hydrocarbons of the methane series have the structure of

a zigzag chain. A representation of the normal pentane molecule is
given in Fig. 11.4.
In saturated cyclic hydrocarbons the length of C—C and C—H
bonds is the same as in the paraffin hydrocarbons but the valence
angles are distorted, which produces tension in the cycle. The cyclo-
pentane molecule has tetrahedral angles—four carbon atoms are

in one plane, while the fifth lies by approximately 0.5 A higher
(Fig. 11.5).
The cyclohexane molecule also has a non-planar structure with
tetrahedral valence angles. In this case two configurations are
possible—‘armchair’ and ‘bath’ (Fig. 11.6), of which the former has
been considered the more probable. Recent investigations have
demonstrated that in ordinary conditions the cyclohexane molecule
does have the shape of an armchair.
It is evident from the carbon compounds cited above that the
valence angles of an element can he different in different compounds.
Methods employed in determining the values of valence angles
are the electron-diffraction (see p. 138-145), the X-ray diffraction
(see p. 273-277) and the spectral (see p. 145-149), as well as measure
ment of dipole moments (see p. 153-157).
11.3. Strength of the Bond

The strength of the chemical bond can be expressed as the amount
of energy expended in breaking it, or the value which when summed
up for all the bonds gives the energy of formation of the molecule
from atoms (mean bond energy). The bond-breaking energy (bond-
dissociation energy) is always positive; the bond-forming energy is
the same in magnitude but is negative.
For diatomic molecules the bond energy is equal in magnitude
to the dissociation energy. For polyatomic molecules with a single
type of bonds, e.g. for ABn molecules, the mean bond energy is equal
to one-rcth of the total energy of formation of the compound from
the atoms. Thus, the energy absorbed in the process
CH4 G+ 4H
is equal to 397 kcal/mole. But in the methane molecule all four
C—H bonds are equivalent. Hence the mean energy of this bond
E'c- h = = 99 kcal/mole
This calculation has been employed to determine the scale of the
value of E for other atoms: 104 kcal/mole for hydrogen, and
118 kcal/mole for oxygen. Applying each of these values of E to
a single molecule, we would obtain a quantity of the order of ~
~ 1 0 "19 cal.
But let us assume a process of consecutive removal of B atoms from
the ABn molecule. Such dissociation of the molecule will cause
a change in the nuclear and electron configuration of the system and
consequently a change in the energy of interaction of the atoms
forming the molecule. Whereas the H —G—H angles in CH4 are equal
134 PART iii . structure of molecules and chemical bond
to 109.5°, they are approximately 120° in CH3—the tetrahedral CH3

group in methane is converted into the almost planar methyl radical.
For this reason the energy involved in the consecutive removal
of B atoms from the ABn molecule will be different for each B atom.
Different cases are possible. If breaking one bond requires a certain
weakening of another bond, the energy required for consecutive
removal of the B atoms will decrease. The H20 molecule is an exam
ple. Removal of the first hydrogen atom requires 118 kcal/mole
but removal of the second atom requires only 102 kcal/mole, a quan
tity characterizing the strength of the OH radical. If breaking one
bond involves strengthening of another, the sequence will be rever
sed. Thus removal of the chlorine atoms from A,1G13 requires 91, 95
and 119 kcal/mole. More complicated cases are also possible. The
consecutive removal of the four hydrogen atoms from methane
involves a loss of energy equal to 102, 88, 124 and 80 kcal/mole,
respectively. Nevertheless, for any substance the mean arithmetical
quantity coincides with the mean bond energy. Thus, for CH4 we
have
„ 102 + 88 + 124 + 80 nn , ,
Ec-H = — —— ---- —— = 99 kcal/mole
The atom-removal energy is known for few molecules, and con
sequently such a calculation can only be carried out in isolated cases.
If a molecule contains more than two different atoms, the concep
tion of the mean bond energy does not coincide with the conception
of the bond dissociation energy. If there are different types of bonds
in a molecule, each of them can be assigned, to a first approxima
tion, a definite value of E. This makes it possible to calculate the
energy of formation of the molecule from the atoms. Thus the energy
of formation of the pentane molecule from carbon and hydrogen
atoms is determined from the equation1
^c5Hi2 ~ 4£’C- c + 12£ c- h
Table 11.1 gives the energy of certain bonds. An examination
of these quantities reveals a regular change in accordance with the
periodic system of elements. Diminution of the bond energy in the
series C—X (X = F , Cl, Br, I) is due to weakening of the bond as the
internuclear distance of carbon—halogen increases. The great strength
of the C—F bond is one of the causes of the chemical inertness of
fluorine derivatives of hydrocarbons, among them the perfluoroal-
kanes, C„F2n+2. Augmentation of the bond energy in a series is
indicative of a strengthening of the bond and, as a consequence,
reduction of interatomic distances (see p. 129). Transition from a
1 Naturally this method yields only approximate results. Contrary to expe

rimental findings it gives the same atomic energies of formation for all the
isomers of a given compound.
Table 11.1
Lengths and Dissociation Energy of Chemical Bonds
Bond length, E,
Bond Compound kcal/mole
A
C -H Saturated hydrocarbons 1.095 98.7

C— F ch 3f 1.381 116.3
C -C l CC14, CHC13 1.767 75.8
C -B r Bromine-substituted saturated hydro 1.94 63.3
carbons
C -I Iodine-substituted saturated hydro 2.14 47.2
carbons
C -C Saturated hydrocarbons 1.54 79.3
CiîC Benzene 1.40 116.4
C= C Ethylene and its derivatives 1.34 140.5
C -C Acetylene hydrocarbons 1.20 196.7
C -=0 co2 1.160 191.1
0 —H h 2o 0.958 109.4
0 —H Alcohols 0.96 104.7
0 —0 h 2o 2 1.48 33.3
S —H h 2s. 1.346 86.8
S= 0 VS 02 1.432 125.9
N —H NH3, amines 1.008 92.0
NÔ NO 1.151 149.4
As —H AsH3 1.519 47.5
single bond to a double or triple bond between the same atoms brings
an increase in the bond energy but this is not proportional to the
increase in the multiplicity of the bond.
The regular change in E in bonds of the same type makes it possib
le to employ methods of comparative calculation for their estimation.
Thus it is possible to compare the values of the mean bond energy
in two series of single-type compounds, for example, in the series
R2 and HR, where R = Cl, Br and I. Fig. 11.7 gives an example of
such a comparison: the values of the mean bond energy of elements
of the main subgroup of Group VI are compared with carbon and
silicon, and by graphic extrapolation the unknown value of Z^c-Te
can be estimated as approx. 132 kcal/mole.
Let us now consider an example of the comparison of the values
of E with the values of another property in a series of compounds.
We have already seen that the bond energy diminishes as the length
of the bond increases. Let us assume that to a first approximation
this diminution of the energy of the given element’s bond with
a series of analogues is linear—for example, for the bond C—R,
136 P A R T III. STRUCTURE OF MOLECULES A N D CHEMI CAL BOND
where R = F, Cl, Br and I. The correctness of this assumption is

illustrated in Fig. 11.8. This example could be extended by comparing
the internuclear distances and the energies of the carbon—carbon
bond, depending on the multiplicity of the bond, and so on.
The values of E for organic compounds are more reliable than for
inorganic compounds, since the former are characterized by great
^c-r> kcaL
Fig. 11.8. Correlation diagram

of bond energies and lengths in
carbon halides
diversity of the molecules studied, along with a small number of

types of bonds; the latter, on the contrary, present a very great
number of bond types, with relatively scanty experimental material.
Without taking up methods of determining bond energies (by
measuring the energy changes of various processes, molecular spect
roscopy, etc.), it can be stated that if the energy change of a certain
process is known and also the energies of all the bonds except a given
bond, the latter can be calculated. Thus when hydrogen burns ac
cording to the reaction
2H2+ 0 2= 2H20
116 kcal are evolved. It can be assumed that in the process the
H—H and 0 —0 bonds are broken, and the free atoms combine to
Ch. 12. METHODS OF D E T E R M I N I N G MOLECULAR STRUCTURE 137
form H20 molecules, each of which contains two 0 —H bonds:
From the law of the conservation of energy it follows that

2 # h -h + Eo-o —4E0-H— — 146
whence, if the energy of any two bonds is known, the energy of the
third bond can be calculat ;d
j7 2£ h _ h + # o - o + 116
^ o -h = ----------- 4-----------
Substituting the values £ h -h = 104 and E 0 — 0 = 1 1 8 kcal/mole
in the above equation, we find £ o - h = 110 kcal/mole.
The energy of the formation of a compound from atoms is equal
in magnitude and opposite in sign to the sum of the bond energies.
It is to be understood that both the initial molecule and the products
of its dissociation are at absolute zero and have the properties of an
ideal gas (see p. 262), and, furthermore, that the dissociation products
are in the ground state. Nevertheless, the chemist should bear in
mind that many reactions take place between excited atoms at high
temperatures and pressures.
Although temperature and pressure have little effect on bond
energies (for which reason all values of Ehnd cited above are for
P = 1 atm and t = 25-°C)1, the transition to an excited state is
accompanied by a great energy change.
CHAPTER TWELVE
PHYSICAL METHODS OF DETERMINING MOLECULAR STRUCTURE

It was pointed out above that the Butlerov theory of chemical
structure established the fact that every organic molecule has a de
finite structure, and indicated chemical methods for determining
molecular structure. Chemical methods of studying structure have
also been developed for determining the structure of complex com
pounds which are an important class of inorganic substances (see
p. 235-236). The structures of an immense number of compounds
have been determined by chemical methods. This information, to
gether with the results of investigations into the properties of com-
1 The difference in the value of Etnd at T = 0 and T = 298°K does not, as
a rule, exceed 1 kcal/mole.
pounds and the principles underlying their changes, revealed through

the discovery and elaboration of the Periodic Law, have determined
the lines along which the science of chemistry has developed.
Chemical methods of studying molecular structure are still widely
employed, but along with them there are a number of physical methods
which make it possible to study features of molecular structure
which cannot be determined by chemical methods, such as the precise
values of interatomic distances and the angles between bonds, the
distribution of electric charges in a molecule, etc.
One of the methods of studying molecular structure extensively
used is electron-diffraction examination.
12.1. Electron-Diffraction Examination

The electron-diffraction examination method makes use of the
diffraction of electrons on molecules. Electrons like all other micro
particles possess wave properties. Therefore, when a beam of elect
rons characterized by a de Broglie wavelength X impinges on an
To powerful vacuum
pump
obstacle whose size is of the same order as X, a diffraction pattern

is produced corresponding to this wavelength.
Electron-diffraction is. examined by means of an instrument called
an electron-diffraction camera. A diagram showing its principle
of operation is given in Fig. 12.1.
The electron source in the electron-diffraction cameras is usually
a hot metal filament, since metals when strongly heated begin
Ch. 12. METHODS OF DE T ER MI NI NG MOLECULAR STRUCTURE 139
to emit electrons. The electrons escaping from the filament are

accelerated by a potential difference of tens of thousands of volts—
usually 30-60 thousand volts when studying molecular structure.
This gives electrons of great velocity—fast electrons. The value of X
for electrons accelerated by a potential difference, F, can be found
by substituting in de Broglie’s equation (3.18) the value of the
electron velocity, v, calculated from the ratio
eF = ^ l (12.1)
where e and me are the charge and mass of the electron, respectively.
Thus for V — 10,000 volts, X = 0.12 A.
A diaphragm is used to obtain a narrow beam from the stream
of accelerated electrons, the usual thickness in the electron-dif
fraction cameras being about 0.1 mm. To study molecular structure,
molecules of the test substance are introduced into the electron
beam. If the given substance is a gas or volatile liquid, it is placed
in a glass bulb having a delivery tube which ends in a narrow jet
tip (see Fig. 12.1). A stream of the test gas is supplied for an instant
(about 0.1 s) by turning a cock. If the substance has a high boiling
point, a small electric heater is mounted in the instrument to vapo
rize it. When the beam of electrons passes through the stream of
molecules from the test substance, electron-diffraction takes place,
and the diffraction pattern is recorded on a photographic plate
positioned a short distance (usually 10-25 cm) from the stream.
A very high vacuum must be maintained in the instrument when,
studying electron diffraction. Therefore, the camera is connected
to a powerful vacuum pump which constantly evacuates the gas
being introduced. This must be done because the electrons would
be quickly decelerated by the substance. Moreover, fast electrons
must be used. Slow electrons with energy of the order of 100 eV
are completely retarded if they collide with only five or six mole
cules of a substance.
The diffraction pattern obtained on the photographic plate—the
electron-diffraction photograph—consists of a central spot formed
by electrons which were not deflected, and rings of varying intensi
ties due to impingement on the plate of electrons scattered through
different angles, 0, to the original direction of the beam. The change
in intensity in the electron-diffraction photograph depending on
angle 0 is strictly definite and depends on the* molecular structure
of the test substance. Fig. 12.2 presents, by way of example, dif
fraction patterns of CC14 and CS2. Deciphering the photographs
makes it possible to determine the structure of a given molecule.
The electron-diffraction photograph is deciphered by using an
expression which determined the intensity I of a beam of electrons
scattered by molecules through an angle 0 to the original direction
140 PART I I I . S T R U C T U R E OF M O L E C U L E S A N D C H E M I C A L BON D
of the electron beam. It has the form1

n n
sin sr
^ ~ 2 I 2 m ^l^ m ( 12 . 2)
1 1
in which
4jt . 0
S~ ~ Sin 2 (12.3)
and r is the internuclear distance; ~ is the proportionality sign.
(a) (b)
Fig. 12.2. Electron diffraction patterns
<«> c c i 4; (b) CS2
When calculating with equation (12.2) summation of the values

sm sr
ZiZm sr
is performed for each of the atoms in the molecule. ZL is understood
to be the nuclear charge of the atom under consideration, and Zmr
the nuclear charge of the neighbouring atom, while r is the distance
between the., given atom and its neighbour. By the first neighbour
of the atom under consideration is meant the atom itself; in this
case Zi = Z m, r = 0 and (sin sr)/sr = 1, from which it follows that
the given member of the sum is simply Z2.
1 Derivatiou of the relationship (12.2) is given in Appendix V.
Ch. 12. METHODS OF D E T E R M I N I N G M O L E C U L A R STRUCTURE 141
We shall now consider some examples of the use of expression

<12. 2).
For diatomic molecules consisting of like atoms, for example,
Cl2, Br2, N2, etc., with charge Z, expression (12.2) acquires the form
I ~ Zz -\-Z2? ^ - ~ Z2{ l + ^ r 1 ) (12.4)
The expression obtained has the member (sin sr)/sr which includes
internuclear distance r; from (12.3) it can be seen that s has the dimen-
sin x
x
sion cm-1; therefore, sr is a dimensionless quantity. A graph of the

function
sin x
y x
(12.5)
is shown in Fig. 12.3. This is a periodic function with maxima and
minima, but these become less and less noticeable as x increases.
The first four maxima are found at values of x equal to 7.73, 14.06,
20.46 and 26.66. It is evident that multiplying the function (sin x)lx
by a constant quantity will change only the amplitude of the oscil
lations, without affecting the position of the niaxima. Nor will
their position be changed by adding some constant quantity to the
given function, this will only raise the entire curve with respect
to the x axis. Thus the intensity of electron scattering by diatomic
molecules described by expression (12.4) is a periodic function of the
quantity sr; this function has maxima at values of sr equal to 7.73,
14.06, 20.46, 26.66 and so on. It can be seen from Fig. 12.3 that the
first maximum is the sharpest.
142 PART III. S T R U C T U R E OF M O L E C U L E S A N D C H E M I C A L BON D
To decipher the electron-diffraction picture of a diatomic molecu

le one must find the value of angles 0 for which the intensity of the
stream of scattered electrons is maximum. These angles can be deter
mined by measuring the radii of the rings in the diffraction picture,
from which, knowing the distance from the molecule source to the
photographic plate, it is easy to calculate the angle sought. Substi
tuting the values of 0 in ratio (12.3) makes it possible to calculate
the values of s corresponding to the intensity maxima. On the other
hand, as was pointed out above, the values of the product sr at which
the intensity of electron scattering has a maximum arq known, from
which the value of the internuclear distance r can be determined.
The experimentally determined values of s for the first three maxi
ma in the chlorine electron-diffraction picture are 3.87, 6.93 and
10.29 A"1, from which calculation of r gives the following results
7.73 14.06 20.46
r 3.87
2.00 A; r ~ 6.93
2.03 A; r ~ 10.29 -=1.99 A
Thus the internuclear distance in the chlorine molecule can be consi

dered equal to 2.01 + 0.02 A.
When determining the internuclear distance in diatomic mole
cules it is sufficient to calculate the value of s corresponding to the
first intensity maximum in the diffraction picture, since this maxi
mum is the most intense and affords greater precision in calculations
than the following maxima.
When deciphering electron-diffraction pictures of more complicated molecu
les, the trial-and-error method is used. This consists in assigning a definite struc
ture to the molecule being studied, and then calculating by means of expression
(12.2) the dependence of the intensity of blackening in the picture on the value
of 5, comparing this dependence with experimental findings. If the maxima of
the theoretical and experimental curves coincide, it can be considered that the
molecule has the suggested structure. If the maxima do not coincide, other possi
ble structures must be tried, until the calculated and experimental curves coin
cide.
As an example, we shall now consider calculation of the curves of electron
scattering intensity for carbon tetrachloride, CC14, and benzene, CcH6. Let
us assume, in accordance with the teachings of organic chemistry, that the CC14
molecule is of tetrahedral structure (Fig. 12.4). It follows from an examination
of the model shown in Fig. 12.4 that if the distance C — Cl is designated as R,
the distance Cl — Cl will be 2 “1/2/3 R = 1.63 R. WTe now set up the function
of electron-scattering intensity for the given model. Let us first consider the
carbon atom. This atom’s first neighbour is itself, which gives the member Z£;
its other neighbours are four chlorine atoms, each of them at a distance R.
Thus in the sum there appears the component
sin sR
4Zc^ci sR
Next let us consider the chlorine atoms. Each of the four chlorine atoms has-
as its neighbours itself, one carbon atom at a distance /?, and three other chlori-
ne atoms at a distance of 1.63 R. This gives the following members in the sum
/^2 , /v ^
AZCI4 -4ZCZC1
sinsR + //x o v 2 sin 1.63sR
(4) 3ZCi
Adding all the members we obtain

r 7*2 I i | r \r j rj Sin S/? Af ) r7 ~ Sin 1 .63s/?
/ ~ Z C+ 4ZCl + 8ZCZC1 — + 12Zci
If R is taken to be 1.75 A , an intensity curve is obtained in which all the maxi

ma coincide with those found experimentally. Thus the electron-diffraction
Cl
Fig. 12.4. Diagram for calcula

ting the CC14 electron diffra
ction pattern
technique shows that the CC14 molecule is of tetrahedral configuration and that
the distances, C — Cl and Cl — Cl, are 1.75 and 1.63 x 1.75 = 2.85 A, res
pectively.
When calculating the intensity curve for benzene it must be borne in mind
that it is difficult to establish the position of the hydrogen atoms in the molecu-
Fig. 12.5. Diagram for cal

culating the C6H6 electron
diffraction pattern
le by the electron-diffraction technique. This is because the charge on the nucle

us of the hydrogen atom — the proton, equal to unity, is very small compared to
the charges on other atomic nuclei. For that reason the scattering of electrons
by protons is negligible and has little effect on intensity curves. Therefore, when
calculating the electron-diffraction pictures for benzene, it is possible, to a first
approximation, to take into account only the structure of the carbon skeleton.
Let us assume that the six carbon atoms of benzene lie in a single plane, thus
forming a regular hexagon with a distance R between adjacent atoms. In that
144 PART III. S T R U C T U R E OF M O L E C U L E S A N D C H E M I C A L BON D
case, as can be seen from Fig. 12.5, each carbon atom has, in addition to itself,
five neighbours — two at a distance R ; two others at a distance ^[/S R = 1.74 R
and one at a distance 2R. From this we find for the electron-scattering intensi
ty, the expression
/ 6Zc+12Z c S'sn/ + 12Zc sin 1.74 sR - . + 6 Zh sin 2sR
1.74sR 2sR
If R is taken to be 1.40 A the position of all the maxima on the theoretical and
experimental curves coincide. Thus the electron-diffraction technique confirms
the assumption that the ben
zene molecule has a cyclic stru
cture and makes it possible to
determine the dimensions of the
benzene ring. Electron-diffra
ction examination also shows
A r ^ - that all the bonds in the ben-
110.5° zene ring are of the same
length —something of extremely
great importance. For an expla
nation of this fact on the basis
//V /W W v /- of the modern theory of the
chemical bond see pp 192-194.
r A j-K -'s s Figure 12.6 shows the obser
ved and calculated electron-
Observed
»Intensity
' vA / v' \ A / - v- Fig. 12.6> Observed and calcu
lated electron-scattering inten
10 15 20 s , A m sity curves for CHF3
scattering intensity curves for CHF3. Calculations were made for different
values of the angle between the F — C — F bonds. From a comparison of the
curves the conclusion can be drawn that the value of the given angle in the
CHF3 molecule lies between 106 and 110.5°.
The disadvantage of the trial-and-error method is the necessity of assigning
some configuration to a molecule beforehand. In the case of complicated mole
cules where many structures seem possible it is difficult to single out a model
corresponding to the actual configuration of the molecule. There are other me
thods of deciphering electron-diffraction pictures which are free from this uncer
tainty but discussion of the same is beyond the scope of this book. It should be
noted, however, that a great amount of information of fundamental importance
for the science of chemistry has been obtained through the deciphering of elect
ron-diffraction pictures by the trial-and-error method.
Table 12.1 gives some results of electron-diffraction investigations

into molecular structure.
The electron-diffraction technique is not always applicable for
establishing the structure of molecules. As already noted it is dif
ficult to determine the position of hydrogen atoms by this metjiod.
Great difficulties are also encountered when employing the elect
ron-diffraction technique in the case of complicated molecules con
taining many different groups of atoms.
Table 12.1
Results of Electron-Diffraction Determination of Molecular Structure
Interatomic
Molecule Configuration of molecule
distance, A
Br2 B r -B r 2.28 Dumbbell

CI2 C l- C l 2.01 Same
C02 C -0 1.13 Linear
CS2 c-s 1.54 Same
so 2 S -0 1.43 Bent; angle O —S —O equal to 120±5°
CC14 C— Cl 1.75 Tetrahedron
p els P —Cl 2.00 Pyramid; angle Cl —P —Cl equal to
101± 2°
C,H, c —c 1.40 Flat ring
c 4h 12 c-c 1.52 Ring with zigzag arrangement of
atoms
c 2h , c-c 1.55
c 2h 4 c=c 1.34
c 2h 2 c = c 1.22
b f3 B —F 1.30 Flat; angle F —B —F equal to 120°
SiF4 Si —F 1.54 Tetrahedron
N(CH3)s N -C 1.47 Pyramid; angle C—N —C equal to
108°
P4 P -P 2.21 Tetrahedron
(HF)n F -F 2.25 Zigzag chain
CIS-C2H2CI2 C l- C l 3.22
C l- C l 4.27
12.2. Molecular Spectra

When discussing atomic spectra (see p. 57) it was shown that
spectral lines originate as the result of the transition of electrons
in the atom from one energy level to another. The existence of discre
te energy levels in the atom is due to the quantum-mechanical
character of electron motion. When it comes to molecules there
is possible, in addition to the movement of electrons, displacement
of the nuclei in respect to one another—the oscillation and rotation
of atoms round the centre of mass may appear. These motions are
also quantized, but because of the substantially greater mass of the
particles, the energy levels here are very close to one another, the
least distinct being the levels of molecular rotation.
It is usually the absorption spectra of molecules that are studied.
This is done by passing light through the test substance and finding
10 3aK . 15648
146 P A R T III. S T R U C T U R E OF M O L E C U L E S A N D C H E MI C AL BOND
by means of a spectrograph the wavelengths of the radiations absor

bed. When a molecule absorbs a quantum of radiation it passes from
one energy state to another. The only quanta absorbed are those
whose energy is equal to the energy of these transitions—thus the
absorption spectrum like the emission spectrum makes it possible
to judge of the energy levels in the molecule.
In accordance with the foregoing principles, there are three types
of molecular spectra—spectra of electron transitions, vibrational
spectra and rotational spectra. Fig. IL.2 (Appendix II) shows the
energies and radiation wavelengths corresponding to different chan
ges in the state of the molecule.
The lowest energies are those of rotational transitions; they cor
respond to radiation in the far infrared region. Rotational spectra
can be observed unhampered without the superimposition of changes
in other forms of motion—vibrational and electron transitions.
The energy of vibrational transitions is approximately ten times
that of rotational transitions, and the corresponding radiation lies
in the near infrared region. Changes in vibrational motion of the
molecule are always accompanied by changes in rotation, and for
that reason the vibrational spectrum, unlike the rotational spectrum,
cannot be observed in the ‘pure form7—these spectra are always
superimposed on one another, thus forming a vibrational-rotational
spectrum.
Transitions of electrons in molecules, as in atoms, correspond to
energies of several electron-volts; in this case the radiation is in the
visible or ultraviolet region. Transitions of electrons are accompanied
by changes in vibrational and rotational motion; this is reflected
in the spectrum, which in this case shows the aggregate of all forms
of energy changes in the molecules.
Study of molecular spectra provides much valuable imormation
about molecules, including their structure.
Examination of rotational spectra makes it possible to determine
molecular moments of inertia (the physical meaning of the concept
’moment of inertia’ is taken up in Appendix VI), knowing which,
definite conclusions can be drawn about the structure of the mole
cules concerned.
We shall now set up an equation describing the frequency of the
lines of the rotational spectrum of a diatomic molecule consisting
of atoms A and B. To do this we express the energy of rotational
motion through the molecule’s angular momentum M and moment
of inertia I (see Appendix VI).
Ero, = ^ - (12.6)
To find an expression defining the angular momentum M it is neces

sary to solve the SchrQdinger equation for a particle rotating uni-
formly at a constant distance from a centre; such a system is termed

a rigid rotator. The solution cannot be gone through here but it shows
that the angular momentum of a rigid rotator is defined by the
relationship
M = h V J ( J + 1) (12.7)
where J is the quantum number, which can have the values 0, 1,
2, . . . As can be seen we have a familiar expression—such a for
mula defines the angular momentum of the electron in the atom
[see equation (4.18)]. Substituting (12.7) in (12.6) we have
£ro< = - § - / ( / + 1 ) ( 1 2 .8 )
Equation (12.8) determines the energy levels of the rotating mole

cule.
We now find the energy difference AErot for two levels in which J
differs by one. Let the quantum number for one level be / , and for
the other / + 1. Then
AErot = — [(J + 1) ( / + 2) - / (J + 1)] = — (J + 1) (12.9)
The wave number of a spectral line corresponding to the transi
tion of a molecule from one rotational state to another is determined
by the relationship
~ 1 AErot
X he
from which, taking into account (12.9), we obtain
h
v 4j 2 / ( ^ + i )
i c
( 12 . 10)
Employing equation (12.10), the moment of inertia is found from

the wavelengths of the lines of the rotational spectrum, from the
moment of inertia the distance between the nuclei of atoms A and
B can be determined (see Appendix VI). Relationships making it
possible to determine interatomic distances can also be drawn up
for molecules consisting of more than two atoms.
When atoms in a molecule are displaced relative to the equilibrium positi
on, a restoring force / is set up. The assumption that it is proportional to the
displacement Ar, that is,
/ = fcAr (12.11)
corresponds to the condition of harmonic vibration. The proportionality coeffi
cient k is called the force constant. This quantity is determined on the basis of an
analysis of the vibrational spectrum from the equation
AEvib = H - / ± ( 12. 12)
10*
148 P A R T I I I . S T R U C T U R E OF M O L E C U L E S A N D C H E M I C A L BO N D
expressing the difference of two energy levels for a harmonic vibrator; m* is

the reduced mass (see Appendix VI). The value of k depends on the nature of
the atoms and on the multiplicity of the bond. Thus whereas &ci2 = 3.21 x
X 105 and /chci = 4.74 X 105 (single bond), ko2 = 11.3 X 105 (double bond)
and &n2 = 22.2 x 106 (triple bond) dynes/cm. In the series of analogues HF —
— HC1 — HBr — HI it diminishes from 8.65 x 105 (HF) to 2.89 x 105 (HI)
dynes/cm.
For large values of Ar, vibrations become nonharmonic; as the interatomic
distance increases the restoring force progressively diminishes and equation
<12.11) no longer holds true.
It is obvious that if the energy of radiation acting on a molecule
is increased, it will finally be possible to bring about its disintegra
tion—its dissociation. Further increase in the energy will only increa
se the translational velocity of the atoms formed through molecular
dissociation, and this is not quantized. Therefore, at a definite
wavelength the spectrum will become continuous. Just as it is pos
sible to calculate the energy of the detachment of the electron from
the atom—the energy of ionization (see p. 60)—from the short-
wavelength limit of the atomic spectrum, the short-wavelength
Table 12.2
Results of Spectral Study of Di- and Triatomic Molecules
for Substances in the Gaseous State
Molecule r, A Ebnd• eV
H2 0.74116 4.4763
-Li 2 2.6725 1.03
Na2 3.0786 0.73
CI2 1.988 2.475
HF 0.9175 5.8
HC1 1.2744 4.430
KC1 2.6666 4.97
Csl 3.315 3.37
Molecule Interatomic distance, A

Valence angle,
deg
(l) (3) (2) r 1-3 r 2-3
0 = 0 = 0 1.160 1.160 180

H —0 —H 0.958 0.958 104.5
0 = s = 0 1.432 1.432 122
0 3 1.278 1.278 117
0 = s
.-= c 1.160 1.560 180
II — C = N 1.059 1.157 180
Ch. 13. T Y P E S OF CHE MI CAL POND—IONIC A N D COVALENT BOND 149
limit of the electron spectrum of the molecule gives the energy of

dissociation of the molecule, from which it is possible to find the
energy of the bond.
Table 12.2 gives data on the structure of molecules and the energy
of bonds obtained through the study of spectra. A comparison with
the data in Table 12.1 shows that for Cl2, C02 and S 0 2 the results
of electron-diffraction and spectral determination of molecular
structure agree well.
Besides the electron-diffraction and spectral methods, the X-ray
diffraction analysis of crystals is of great importance in determining
molecular structure (see pp 273-277).
It should be noted that the configuration of the molecule in a crystal
may differ markedly from the configuration of the isolated mole
cule.
CHAPTER THIRTEEN
BASIC TYPES OF THE CHEMICAL BOND—IONIC

AND COVALENT BOND .
13.1. Electronegativity of the Elements

Let us assume that atoms A and B interact, producing a chemical
bond by the transfer of an electron from one atom to another. The
question then is which of these atoms draws an electron onto its
shell. Let us assume that an electron is transferred from A to B.
This involves the liberation of energy (EB—I A), where E& is the
electron affinity of B, and I A the ionization energy of A. In the
reverse process, the energy (EA—/ B) would be liberated. The direc
tion of the process is determined by the maximum liberation of
energy, since this stabilizes the system. Let us assume that the actual
electron transfer is from atom A to atom B. This means that
(Eb - I a) > ( E a - I b)
or
(^b + ^b) Z> (Aa + Ea)
The quantity 1/2 (/ + E) is termed electronegativity. Let it be
designed x. Then
a' = i ( / - f £ ) (13.1)
Thus the electron is transferred to the atom of the element which
has greater electronegativity. Electronegativity characterizes an atom's
power to add electrons, thus forming a chemical bond. The method
cited for calculating electronegativity was proposed by Mulliken
(USA). The disadvantage of this method is that the electron affinity
is known for only a few atoms.
150 P A R T III. STRUCTURE OF M O L E C U L E S A N D CHEMICAL BOND
Figure 13.1 shows the electronegativity values of various elements

according to Pauling, who has recommended another method for
their determination (see p. 232). In Pauling’s system the electro
negativity of fluorine is arbitrarily taken as 4. The somewhat unu
sual form of the diagram was dictated by the desire to give the graph
the appearance of the periodic system. As might be expected the
value of x is greatest for fluorine and least for caesium. Hydrogen

occupies an intermediate position—when it reacts with some ele
ments, for example, with F, it gives up an electron; when it reacts
with other elements, such as Rb, it acquires an electron.
Some 20 electronegativity scales have been proposed, different
properties being taken as a basis (internuclear distances, bond ener
gies, etc.). They give dissimilar values for x, but it is the differences
between these values that are of great importance. The relative
values of x are close. The qualitative coincidence of results in the
different scales indicates coincidence in the arrangement of elements
in the electronegativity series.
What is more important is to bear in mind that, when using ele
ctronegativity as a quantity characterizing an atom’s power to
attract valence electrons, it is impossible, strictly speaking, to
ascribe a constant electronegativity to an element. This depends
on the composition of the compound which contains the atom being
studied, on the elements whose atoms surround the atom under
Ch. 13. T Y P E S OF C H E M I C A L B O N D — I O N I C A N D C O V A L E N T B O N D 151
consideration. Thus the free chlorine atom and that in the mole
cules Cl2, NaGl, CC14 and PdCl2 have different properties. It follows
that what must be borne in mind, strictly speaking, is not the electro
negativity of an element in general but its electronegativity when
forming specific chemical bonds in specific surroundings and in
a specific valent state. But in spite of this, the conception of electro
negativity is useful in explaining many properties of chemical bonds.
13.2. Ionic and Covalent Bond

If atoms A and B have greatly differing electronegativity, the
transition of an electron during their interaction converts them onto
oppositely charged ions
A = A+ + e + I A (13.2)
B + e = ~ — Eb (13.3)
w here/A and E b are the ionization energy of atom A and the electron
affinity of atom B, respectively. The electrostatic attraction set
up between A and B causes the formation of a molecule
A+ + B" - A+B" + E (13.4)
where E is the energy change. As a result the ions in the molecule
must be at such a distance that the attraction balances the repulsion
of the like-charged electron shells of the ions and nuclei. It is in
this way that the ionic bond is formed (called also the heteropolar
or electrovalent bond). Consequently the ionic bond corresponds to
the equation
A + B = A+B" + I a - E b + E (13.5)
obtained by summing (13.2), (13.3) and (13.4).
When atoms of the main subgroups of the periodic system are
converted into ions their electron shells are converted into the stable
electron shells of the corresponding noble gases. Thus when KF is
formed, the K + ion acquires the electronic configuration of argon,
and the F" ion, that of Ne. These views were developed by Kossel
in 1916. As we shall see in Chapter Fifteen (p. 233), complete tran
sition of the electrons from one atom to another never takes place—
there is never a 100% ionic bond.
Chemical interaction is not limited to the formation of an ionic
bond. If, for example, it is assumed that in C1F, as in CsF, the valen
ce electron is drawn to the fluorine atom, calculation gives a nega
tive value for the Cl—F bond-dissociation energy. This would mean
that the C1F molecule cannot exist—something that is contrary to
fact. Although chlorine fluoride is very reactive, it is stable, the
experimental value of the bond energy being 2.6 eV. The obvious
conclusion is that the bond in this molecule is not ionic.
152 PART III. STRUCTURE OF M O L E C U L E S A N D C H E M I C A L B OND
Let us now consider a molecule consisting of atoms which have

the same electronegativity. H 2 can be taken as an example. In this
case the two atoms are equal in their power to attract an electron.
In 1916-1918 Lewis and Langmuir (USA) suggested that the chemical
bond is formed by a pair of electrons held jointly by two atoms.
Such a bond is termed covalent or homeopolar. If the valence line
is replaced by two dots representing the Lewis and Langmuir elect
ron pair, one can write H : H in place of H—H. Using this symbolism
for the outer layer (the valence electrons) of atoms, the structure
of the chlorine and nitrogen molecules can be represented as follows
:c i : ci:
Extending this system to molecules consisting of atoms of different

elements we obtain
H H H H
h :c: h h :'c: c i: c: : c
H H H ii
Obviously the number of such examples can easily be increased,

taking into consideration the enormous number of organic compounds
in which the covalent bond is typical. Here, as in the case of the
ionic bond, there is a tendency to form electron octets (or dublets).
Attention must be called to the fact that if molecules consisting
of like atoms are left out of consideration (and they are compara
tively few in number), all the other covalent molecules are characte
rized by a greater or less displacement of the electron pair to one
of the atoms, since they are composed of atoms which differ in their
electronegativity. Such a bond is polar covalent. CF4 and CH 3 CI are
examples of molecules with a polar bond. It is more convenient to
represent a polar covalent bond by an arrow rather than by a shift
of the relevant pair of dots: Cl —^F. The polarity of the 0 <- H
bond explains many properties of water, among them the electro
lytic dissociation of substances dissolved in it.
In the light of what has been said, the ionic bond can be conside
red the limiting case of the polar bond, where the electron pair
(or electron pairs) is practically completely displaced, i.e., becomes
a constituent part of the electron shell of one of the atoms.
The number of covalent molecules A2 is limited by the number
of elements in the periodic system, but there are few symmetrical
molecules An, where n > 2 ; there are practically no molecules with
a purely ionic bond (see p. 233). Therefore, the chemical bond in the
majority of known compounds (now more than three million) is
polar covalent.
Ch. 13. T Y P E S OF C H E M I C A L B O N D — I ONI C A N D C O V A L E N T B O N D 153
13.3. The Dipole Moment and Molecular Structure

Let us assume that we have found the ‘centres of gravity of the
negative and positive parts of a molecule. We must then discover
that all compounds can be divided into two groups. One group would
consist of compounds in whose molecules the two centres of gravity
coincide. These are called non-poiar molecules. They include all
covalent diatomic molecules of the A2 type, as well as tri- and poly
atomic molecules of high symmetry, such as C02, CS2, CC14, CbIIr,
and the like (see p. 145). The other group would include all compounds
whose molecules are electrically asymmetric, i.e., mblecules in which
the centres of gravity of the charges do not coincide. Such molecules
are said to be polar. They include compounds with an ionic type
of bond, such as CsF, and any AB compounds since their atoms have
unlike electronegativity, as well as AnBm compounds whose structure
is asymmetric.
A molecule’s polarity is characterized by the dipole moment
p = el (13.0)
where e is the charge of the electron and I is the distance between
the centres of gravity of the positive and negative charges. The
greater the polarity and the greater the displacement of the electron
pairs to one of the atoms, the greater the value of p. On the contrary,
if the electrical asymmetry of a molecule is slight, the value of p
is small.
Dipole moments are determined experimentally by measuring the
dielectric constant at different temperatures.
If a substance is placed in an external electric field produced by
a capacitor, the capacitance of the latter will be increased e times,
that is
~r~
c0 e (13-7)
where c0 and c are the capacitance in a vacuum and with the substan
ce, respectively, and e, the dielectric constant.
The increase in capacitance as a result of the reduction of the
strength of the electric field is caused not only by the presence of
the constant dipole moment p which is characteristic of the mole
cules of the given compound, but also by their deformation by the
field. In other words the electric field not only causes orientation
of the molecules of the polar compound in the direction of the field
but produces an induced dipole moment p ind through displacement
of the electrons and, to some extent, the nuclei.
If fields are not very strong the induced dipole moment can be
considered proportional to the field strength E
Hind = a E (13.8)
154 P A R T III. S T R U C T U R E OF M O L E C U L E S A N D C H E M I C A L B OND
Proportionality factor a is called the polarizability (or deformation

polarizability); the more the molecule is subject to deformation
(i.e., the weaker its electron shells), the greater the polarizability.
The induced dipole moment disappears as soon as the field is remo
ved—in accordance with (13.8), when E is zero, \iind is also zero.
It can be shown that for gases and for polar substances dissolved
in non-polar solvents, there is a relationship between e and p,
expressed by the equation
e -1 M 4 / . .V„u2 \
e+ 2 p 3 n A 0 [ a i 3RT ) (13.9)
where M is the molecular weight of the compound; p is the density;
N 0 is Avogadro’s number; R is the universal gas constant and T
is the absolute temperature. The first component in parentheses
corresponds to the deformation effect; the second, to the orientation
effect. Obviously the latter should be the greater, the more polar
the molecule, i.e., the greater p and the lower the temperature, since
heating increases the thermal motion of the molecules, thereby imped
ing their orientation1. In accordance with equation (13.9), the
orientation effect is predominant at low temperatures, and the
deformation effect, at high temperatures.
We' now substitute symbols for the members of (13.9)
e —1 M
e + 2 p ~lJ
(13.10)
4n N 0a A
3 ~ A
(13.11)
4jiA’q|i2 r
9R ^
(13.12)
1
T ~ X
(13.13)
Then equation (13.9) can be written as
y = A + Bx (13.14)
Therefore, if the reciprocals of the absolute temperatures are
£_ 1 M
laid off as abscissae, and the values of —r-z—
8+ 2 p
are used as ordinates,
a straight line is produced, and the dipole moment can easily be
found from its slope (B ). Actually, in accordance with equations
(13.12) and (13.14)
^ I ^ O T ^ 0-01282' 10’18 V** (13.15)1

1 a and therefore \iind do not depend on the temperature since the differen
ce between the normal and excited levels of the electrons in the molecules is
very great, i.e., much energy is required for altering the structure of the elec
tron shell, which corresponds to a very high temperature.
Ch. 13. T Y P E S OF C H E M I C A L B O N D — I O N I C A N D C O V A L E N T B O N D 155
In order to construct this straight line, the capacitance of a capa

citor with the test compound must be measured at least at two
temperatures, and the density of the compound at these tempera
tures must be known.
Fig. 13.2. Graph for determining the Fig. 13.3. Comparison of dipole mo
dipole moment (p = 0.01282 x 10"18 ments of hydrogen compounds of ele
V tanP) ments of Groups VII and V of the
periodic system
Fig. 13.4. Addition of dipole moments of molecular bonds

(a) HCN; (b ) S 0 2; (c) CH3C1
The method of calculation is illustrated in Fig. 13.2 (B = tan |J).

It can be seen from the graph that pnci > p-HBr > P-hi an(i that
ttH2 = 0- By using equation (13.9) it is also easy to determine a
from Fig. 13.2.
156 P A R T III. S T R U C T U R E OF M O L E C U L E S A N D C H E M I C A L BO N D
Let us consider the scale of \i. Since e = 4.802‘fO-10 and I is

comparable with the size of molecules (10“8 cm), \i is a quantity
of the order 10~18. This unit of dipole moments has been named
the debye (D) after Peter Debye, the Dutch physicist.
We begin as before with the way in which this property changes
in a group of similar compounds. For HC1, HBr and III the dipole
moments are 1.04, 0.79 and 0.38 D, respectively. The fall in the
value of [i in this series is explained by the diminution of the pola
rity of the bond due to diminution of the difference in electronegati
vity.
The parallelism in the distribution of \i values in two similar
series is illustrated in Fig. 13.3.
When considering the dipole moments of complex molecules it
is advisable to assign a definite value of \i to each bond, taking
into account not only the quantity but also the sign, depending on
the direction of displacement of the electrons, i.e., considering the
dipole moment of a bond as a vector. The contribution of unshared
electron pairs to the dipole moment must be taken into account
(see p. 199).
The dipole moment of a polyatomic molecule can be considered
equal to the vector sum of the dipole moments of all the bonds,
neglecting their mutual effect. The vector addition of the dipole
moments of bonds is shown in Fig. 13.4, it being understood that
in all cases the vector is directed from + to —.
From what has been said it follows that determination of the
dipole moment can reveal the character of the chemical bond (ionic,
polar or covalent) and indicates the geometrical structure of the
molecule. Thus in order to determine the structure of a compound,
|x is calculated for different models according to the rule for adding
vectors, and the model for which the calculated value is closest
to that found by experiment is taken to be the correct one.
We shall consider some simple examples. Of two possible structures
of ammonia (a and 6, Fig. 13.5), we choose b\ since measurements
show that the molecule is polar (see also p. 199). Say we have
synthesized dichlorobenzene, C6H 4C12. Which of the three isomers
have we obtained?
C l- _ / \ / \ _ Ql c i-A
II 1
C l- C l __II 1
C1 u
ortho- para- met a-
Let us assume that the dipole moment of the compound obtained

is found to be equal to the dipole moment of chlorobenzene, CeH5Cl.
Then by constructing the parallelograms of the dipole moments of
the bonds (Fig. 13.6), we see that it was the meta-isomer that was
Ch. 13. T Y P E S OF C H E M I C A L B O N D —I O N I C A N D C O V A L E N T BO N D 157
synthesized, since for para-dichlorobenzene p, = 0, and for ortho-

dichlorobenzene p = pc-ci v & .
By means of the dipole moment it is also possible to distinguish
cis and trans isomers. Thus in iraras-dichloroethylene, unlike cis-
dichloroethylene, the dipole moment is equal to zero.
Fig. 13.5. Dipole moments of two conceivable structural variants of the NH3
molecule (a) planar; (b) pyramidal
Fig. 13.6, Dipole moments of dichlorobenzene isomers (arrows are scaled to

the value of pG__ G1)
If there are polyatomic substituents in a molecule, their nature

must be taken into account. As an example take the case of nitro
benzene and toluene
/ ^ j c —no2 jP^jC —ch 3
\J \ /
The N 02 and CH3 groups make contributions to the molecular dipole
moment differing not only in magnitude but also in sign.1
1 When measuring \i for orô-dichlorobenzene, we would find that the

experimental value was lower than that calculated as the repulsion of the
closely situated negatively charged chlorine atoms would make itself
felt.
15S P A R T III. S T R U C T U R E OF M O L E C U L E S A N D C H E M I C A L BOND
13.4. Effective Charges

When an atom forms a chemical bond, its electron density changes.
This change can be assessed by assigning to the atom a certain ef
fective charge 8 in units of the electron charge. Effective charges
express the asymmetry of the electron cloud and are conventional
in character, since the electron cloud is delocalized and cannot be
divided between the nuclei.
For atoms forming a covalent molecule, 6 = 0, while in the ionic
molecule the effective charge of the atoms is equal to the charge
of the ion. For all other (polar) molecules it has intermediate values.
Under the influence of an external field 6 increases up to a value
corresponding to the ionic bond. This increase is the greater, the
greater the polarizability, which in turn increases with the electro
negativity.
The value of 6 is determined from optical absorption spectra,
X-ray spectra, nuclear resonance, etc.
The effective charge can be considered a measure of the polarity
of the covalent bond. Thus the calculation for HC1 on the basis of
X-ray absorption spectra gave the following values
H + o . 2 C 1 - o .2
This result can be interpreted thus: in the HC1 molecule the zone of
maximum overlapping of the bonding electron clouds is shifted
toward the more electronegative chlorine atom, which results in
the hydrogen atom being polarized positively (6 h = 0.2) and the
chlorine atom being polarized negatively (6Ci.= —0.2). It can also
be said that the bond in the HC1 molecule is approximately 20%
ionic, and is thus close to covalent. On the other hand, the NaGl
molecule, for which the values Na+0-8Cl"0-8 are found, is a compound
in which the chemical bond is close to ionic. This is borne out by
the following figures: phci = 1-0D; p-Naci — 10D.
The values of 6 for analogous atoms in compounds of the same type
change regularly. In the HHal series we find: 6p = 0.45; 6ci = 0.2;
6Br = 0.12; 6i = 0.05. Evidently in such series of molecules there
is a quantitative parallelism between the values of 6 and the values
of p, (a).
Table 13.1 gives the values of the effective charges of the atoms
of compounds as found from X-ray absorption spectra. Although
these values of 6 are approximate they provide a basis for definite
conclusions: it is noteworthy that there is no compound in which
the effective charge of an atom is greater than 2; in compounds of
the same atom, its effective charge falls as the degree of oxidation
increases (Cr+2C12—Cr+3C13—K 2Cr+60 4), i.e., the greater the formal
valence, the greater the proportion of the covalent bond, which
is due to diminution of the polarity of the bonds as their number
increases.
Ch. 14. QUANTUM-MECHANICAL EXPLANATION OF C O V A L E N T B OND 159
Table 13.1
Effective Charges of the Atoms in Compounds
Compound Atom 6 Compound Atom 6
HC1 Cl - 0.2 GeBr4 Ge + 1.1

SO2CI2 Cl - 0.1 Br - 0 .3
C0H4CI2 Cl 0 ZnBr2 Zn + 0 .5
KCJO3 Cl + 0 .7 Br - 0 .2 5
LiC104 Cl + 0.8 IBr Br - 0.1
II2S s - 0.2 GeH4 Ge + 0.2
S 02 s - 0.1 Ge0 2 Ge + 1.0
SC12 s + 0.2 K2Cr20 7 Cr + 0.1
CaS04 s + 0 .4 K2Cr04 Cr + 0.2
KMn04 Mn + 0 .3 CrCl3 Cr + 1.3
CaT i0 3 Ca + 1.5 CrCl2 Cr + 1 .9
CHAPTER FOURTEEN
QUANTUM-MECHANICAL EXPLANATION OF THE COVALENT BOND

It was only after the laws governing the motion of microparticles
became known and quantum mechanics was formulated that it was
possible to establish the physical causes of a bond between atoms.
In 1927, a year after the publication of Schrodinger’s article in which
he proposed the equation which now bears his name, the work of
Heitler and London (Germany) appeared which contained the quan
tum-mechanical calculation of the hydrogen molecule. This marked
the beginning of the use of quantum mechanics for the solution of
chemical problems.
It should be noted that the exact solution of Schrodinger’s equation
for concrete problems encountered in atomic and molecular theory
involves extremely great mathematical difficulties which it has
been possible to overcome in only a few cases1.
Up to now the exact solution has only been found for single-elect
ron systems—the hydrogen atom and hydrogen-like ions, as well
1 Here and below, what is meant by the exact solution of Schrodinger’s

equation is the exact mathematical solution, i.e., finding analytical expressions
for of) and E, which when substituted in Schrodinger’s equation for the problem
considered will give identity. An approximate solution is not necessarily less
exact in the practical sense. For example, as we shall see below, the energy of
the bond in the hydrogen molecule can be calculated by approximate methods
with as great precision as that achieved experimentally.
1G0 P A R T III. S T R U C T U R E OF M O L E C U L E S A N D C H E M I C A L BOND
as the ionized hydrogen molecule, II*. For other atoms and mole
cules only approximate solutions of Schrodinger’s equation are pos
sible at present. These solutions are of great importance for chemistry
since they explain the nature and properties of chemical bonds.
For that reason it would be advantageous to take up some of the
mathematical techniques employed for obtaining approximate solu
tions of Schrodinger’s equation before considering the quantum-
mechanical interpretation of the chemical bond.
14.1. Solution of the Schrodinger Equation

Using Approximate Functions
The Schrodinger equation for a simple imaginary model—the
movement of a particle in a potential well—as well as its solution
have been considered in Chapter Three (see pp 35-45). In the poten
tial well problem it was possible to find a function and an expres
sion for energy E which satisfied the Schrodinger equation for the
given case. The solution was not difficult because the potential
energy of the particle U could be considered equal to zero. The prob
lem then consisted in finding a function whose second derivative
was expressed by the same function with the opposite sign. This
condition is satisfied by the function of the sine.
As we shall see below when considering the hydrogen molecule,
the potential energy of the electrons is defined by a six-membered
expression. For other molecules the potential energy of the electrons
is described by still more complicated relationships. In such cases
it proves impossible to find a function which would satisfy the
Schrodinger equation, and consequently a function and a value
of E are sought, which are close to the unknown and E that would
be a solution of the equation.
For the motion of a single particle the Schrodinger equation has
the form
This can be written in a more compact form which is convenient

for finding approximate solutions. We introduce symbols making
it possible to give the equation such a form that for a certain function
of the potential energy U (x, y , z) the value of E can be found by
means of a function which can be considered a sufficiently close
approximation to the unknown function \|) which would be a solu
tion of the Schrodinger equation.
We use the symbol
v dx2 ' dy% ' dz2

(14.2)
Ch. 14. QUANTUM-MECHANICAL EXPLANATION OF C O V A L E N T B O N D 161
The symbol V2 (nabla square) is called the Laplacian operator1.

It is an abbreviated conventional expression for denoting the sum
mation of the second derivatives of a function with respect to coor
dinates. When this symbol is introduced the Schrodinger equation
becomes
= ^ (14-3)
it being understood that U is a function of the coordinates x , y and z.
For still greater simplification we introduce the Hamiltonian
operator
hi
H — 8ji2m V2+ E/ (14.4)
When the Hamiltonian is introduced, the Schrodinger equation
assumes a very simple form
H \|) = (14.5)
It should be emphasized that in (14.5) it is only the notation that
has been changed. Equations (14.5) and (14.1) are the same.
We now multiply both sides of (14.5) by function i|). We then
obtain
= ill'll) (14.6)
or
'll?//'i|) = Ety2 (14.7)
from which
t?_y n y
^ ~~ i|>2 (14.8)
If an expression were known for function \J), which is a solution
of the Schrodinger equation, equation (14.8) would give a value
of E for any point in space if the values of and ty2 were substi
tuted2.
Obviously if an approximate function a|) is used, substituting
the values of this function will give approximate values of E. Since
the function is approximate, employing some of its values can give
a value of E greatly differing from the true value. Uncertainty in
the choice of values of and ty2 can be obviated if these functions
are integrated over all space (from the value — oo to + o o for each
coordinate). The equation for calculating the energy will then take
1 A n operator is a symbol of mathematical operations which transforms one

function into another. For example, in the expression tan a, logi0 a and dy/dx ,
(tan], [log10] and [did.r] are operators.
2 The expression tyHty cannot be replaced by or i/rf2, or V2\|) by \fV2,
just as it is impossible to replace
y dyldx by dy2!dx
• 11 3aK. 15648
the form
^ tyHtydv
(14.9)
^ ty2 dv
where dv is the volume element. If the function \|p is normalized
(for the operation of normalization see p. 43), the integral in the
denominator is equal to unity and equation (14.9) takes the form
E = J tyHty dv (14.10)
Equations (14.9) and (14.10) can likewise be employed for calcu
lating the energy in systems containing several electrons. In that
case the Hamiltonian is written in the following form:
H — s s rS v i+ t' (14.11)
where V? is the Laplacian operator containing the coordinates of
electron i, summation being carried out to cover all the electrons.
Using equation (14.9), the energy of a system (atom or molecule),
its basic characteristic, can be calculated approximately, providing
that a function is found which is sufficiently close to the system’s
correct wave function. The choice of the best form of the approxi
mate function is made by the variational method.
The variational method is based on the fact that the smaller the
value of E obtained through equation (14.9) or (14.10), using the selected
function 'll), the closer it is to the value of the energy of the system in the
ground state, and the closer the selected wave function■is to the correct
function. This can be rigorously proved. Without going into this
proof, the principle of the variational method can be explained by
the following reasoning. As we know, the wave function describes
the distribution of the density of the electron cloud. The ground state
of the molecule, the state of the lowest energy, corresponds to a cer
tain distribution of the electron density, which is expressed by the
correct wave function which is unknown. The approximate wave
function corresponds to some other distribution of the electron den
sity for which the energy will be greater. The closer the selected
function is to the correct function, the lower the energy calculated
with this function will be, and the closer it will be to the actual
value of E for the ground state of the system.
The solution of quantum-mechanical problems in molecular theory
comes down to testing various functions with equation (14.9), coor
dinating these functions with the physical pattern of electron motion
in the molecule. The function which gives the minimum value of E
can be considered the best for describing the state of the system.
When using the variational methods, the approximate function
is usually taken as the sum of the products of independent functions
Cli. 14. QUANTUM-MECHANICAL EXPLANATION OF C O V A L E N T BON D 103
<plf <P2> 93 > • • • and the coefficients c1# c2, c3, . . ., that is
^ = cl9l + ^2^2 + ^3^3 + • • • rf" ^7i<Pn (14.12)
Functions q^, cp2, . . . must meet the same requirements as the
wave functions; they must be finite, continuous and single-valued,
and must become zero at an infinitely great distance from the mole
cule (see p. 36). It goes without saying that the functions must
correspond to the problem being considered, namely they must
depend on the charge of the nucleus and the coordinates. It is
desirable that they should take into account all the features of the
system as precisely as possible, in particular the mutual repulsion
of the electrons.
Obviously, when function (14.12) is substituted in equation (14.9)
the value of E depends on the values of the coefficients cu c2, c3, . . . .
In accordance with the principle of the variational method these
coefficients must be so selected that the value of E is minimum.
This is conveniently done by considering the coefficients as variables
on which the value of E depends. In that case the condition of the
minimum E is expressed by a system of equations
dE
-4^-
oci
= 0; —
dc2
= 0; dcn
= 0 (14.13)
Solution of this system of equations makes it possible to find such

values of clt c2, c3, . . ., at which the value of the system’s energy
is minimum. In this case function represented by the sum (14.12)
will be as close as possible to the correct wave function which is
a solution of the Schrodinger equation.
We can see how this is done by considering the example of a wave
function containing two coefficients ct and c2
^ = Ci<Pi + ^2<P2 (14.14)
and then extending the result to a function containing any number
of coefficients.
Substituting the expression (14.14) into equation (14.9) and taking
into account that the Hamiltonian of the sum can be represented
as the sum of the Hamiltonians of the components we have
c\ (pi/Tcp! d v - \ - c i c 2 J <PiJET(p2 d v - \ - c i c 2 J y 2H y > i d v - { - c \ J (p2H ( p 2 dv
(14.15)
ci i W i d v + 2cic2 cpi<p2 d v + c\ £ cpfcft;
For the sake of brevity it is convenient to denote the integrals in

equation (14.15) with letters. We denote the integrals containing
the Hamiltonian with the letter H , and those not containing the
Hamiltonian, with the letter S . The subscripts show what functions
stand under the integral sign. Thus H n (to be read H one—one)
11*
164 P A R T III. STR U CTU RE OF M O L E C U L E S A N D C H E M I C A L B ON D
corresponds to the integral j qq/fipj dv. Similarly
H i2 = j (piHip2dv
21 = j q>2H (fi dv
H 22 = j <jp2ffcp2 dv
S n = j rpi <Pi dv — j <p*dv
S 12 - - $21 = ^ <Pi(P2 dv
$ 2 2 = j cp2(p2 dv = ^ <pl do

lt can be demonstrated that
Hi 2 = H 21 (14.16)
Using this notation equation (14.15) is written in a more compact
form
£ _ cjg)i + 2Clc2//<2+ CIH22
Cl ^ l l + 2 C jC 2 iS 12 — c 2 $ 22
In accordance with conditions (14.13) the expression for E must

be differentiated with respect to c±and c2 and the derivatives obtained
equated to zero. For this purpose the formula expressing the deriva
tive of a fraction should be used
( u \' vu' — uv'
T) = ^
Here, u and v represent the expressions in the numerator and deno
minator of (14.17). In this case
V2
-= 0
from which
W -(^)V -0
or
u’ — Ev' = 0
Differentiating with respect to clT we obtain

2c2H i2 — E (2c1*Sr11 -f- 2c2*î2) = 0
or
(Hn — ESn) Cî2 ^ S l2) c2 = 0
Ch. 14. QUANTUM-MECHANICAL EXPLANATION OF C O V A L E N T BON D 165
Differentiating with respect to c2 and taking account of (14.16)

gives an analogous expression
(#21 — “f" (#22 — E S 22) = 0 C o
Thus in order to determine the necessary coefficients ci and c2

we must solve the system of equations
(7 /n — E S {x)ci + (#12 — E S i2)c2= 0 1
( # 21 - E S 2i) Ci + (7722 - E S 22) c2= 0 j {UAH)
These equations are called secular equations, a term borrowed from
celestial mechanics. Using similar systems astronomers determine
changes in the movement of planets taking place over the ages.
On inspection of the above system of equations we see that the first
index in the symbols for the integrals coincides with the number of
the equation and the second, with the number of the term in the
equation. In the general case, when the wave function isgiven by
expression (14.12) containing n coefficients, the system of secular
equations becomes
(H n -E S ^ C t+ W u -E S ^ C i + .. . + (IIln—E S in)cn =0 '
(.H21- E S 2l) Cl+ (H 22 - E S 22)c2+ .. . + (//2„—E S 2n) cn -0 y
(Hn1—E Sni) cl-\-(IIn2—E Sn2) C2 + . . . + —E Snn) cn = 0 ; (lin n

(14.19)
The above equations can be written in a shorter form
2 (II u ~ ESu) cj = 0
where i and j are the numbers of the equations and of their terms,
respectively, passing through all the values from 1 to n.
The solutions of such systems of equations differ from zero when
the determinant composed of the coefficients of the unknowns in
the system is equal to zero (see Appendix X), i.e.,
77u - E S n 7712 - E S i2 .. . H in - E S in
#21 — #^21 # 2 2 ,— E S 22 .. . H 2n — E S 2n _ n
I I 711 — ESni II7 i 2 — E$n2 • • • H un — ^ n n
This is called a secular determinant. The given condition can be

written more compactly in the general form
\ Hi j - E S i j \ = 0
Having solved the secular determinant we find the expression for
energy 7?, and substituting it into the system of secular equations
determine coefficients cit c2 . . . .
166 P A R T III. S T R U C T U R E OF M O L E C U L E S A N D C H E M I C A L BON D
If the sum (14.12) contains a large number of terms, the form of

functions cp4, (p2, . . . has comparatively little effect on the result.
If coefficients q , ct , . . . are properly selected, a sufficiently close
approximation of the correct wave function can be obtained. If there
is an infinitely large number of terms in equation (14.12) with any
functions corresponding to the problem under consideration, and if
appropriate coefficients are found, the value of E calculated from
equation (14.9) will coincide with the actual value. Naturally, if
there is a large number of terms in equation (14.12), the calculation
becomes very difficult.
On the contrary, if a sum is taken which consists of a small number
of terms, the choice of functions making up the sum is of great impor
tance. Obviously, a satisfactory result can only be obtained if the
function taken reflects the actual condition of the system, and if
there is a small number of terms, this can only be achieved if there
is a good choice of functions.
14.2. Potential Energy Curves for Molecules

Before taking up the results of the quantum-mechanical treatment
of molecules we must examine the relationship between the potential
energy of the molecule and the interatomic distance.
Let us consider two atoms whose nuclei are at a distance r from
one another, and ascertain how the potential energy of such a system
changes when there are changes in r. It is convenient to consider
zero potential energy to be that of a state in which the atoms are
at an infinitely great distance apart and do not interact. If atoms
are capable of uniting to form a molecule, the attractive force begins
to act as the distance between them decreases, whereupon the poten
tial energy of the system falls. This continues up to a certain distan
ce r 0. As r continues to decrease, the potential energy begins to rise
due to action of the repulsive force which is of considerable magnitude
at small interatomic distances. It follows that the relationship
between the potential energy and r is expressed by a curve having
a minimum. The potential energy curve for the hydrogen molecule
is shown in Fig. 14.1.
We now consider the potential energy curve for two atoms which
cannot combine to form a molecule. In that case the repulsive force
(for repulsion of the electron shells) predominates at all values of r,
and the potential energy curve has the shape shown in Fig. 14.2;
as r decreases, the curve continuosly rises1.
The above potential energy curves for molecules are a formal
description of the sum of energy changes taking place in atoms as the
1 A weak attractive force also acts between atoms which do not form mole
cules. This is due to the so-called dispersion interaction (see p. 265), but it is
very slight and can be neglected here.
Ch. 14. QUANTUM-MECHANICAL EXPLANATION OF C O V A L E N T B ON D 1G7
internuclear distance decreases. Deducting the slight zero-point

oscillation energy of the nuclei (see below) which appears as the atoms
approach one another, the change in the energy of the system E
expressed by the potential energy curve (the depth of the potential
curve) is the sum of changes in the total energy of the electrons and
the potential energy of the nuclei in the molecule. Therefore, the
quantum-mechanical calculation of the energy in a molecule with
•equation (14.9) for different values of r makes it possible to find
£,eV
Fig. 14.1. Potential energy curve for Fig. 14.2. Potential energy curve for
the hydrogen molecule two atoms which do not form a che
mical bond
the theoretical potential energy curve of the molecule. The correct

ness of the calculation can be judged by the degree to which the
theoretical and experimental curves coincide.
If a molecule is in an unexcited state its energy is minimum.
Accordingly, the atomic nuclei are at the equilibrium distance r 0.
It is. also evident that the minimum in the potential energy curve
corresponds to the bond-dissociation energy Ebnd taken with the
opposite sign, since according to the definition (see p. 133) this
quantity represents the energy required to separate the atoms making
up the molecule, and displace them beyond the space within which
they interact (theoretically, until they are at an infinitely great
distance apart).
The parameters of the potential energy curve can be determined
experimentally. The internuclear distance is found by the methods
discussed in Chapter Twelve. The value of Ebnd can be calculated from
spectroscopic data (see p. 146) or by thermochemical methods which
are taken up in courses of inorganic chemistry.
168' P A R T III. S T R U C T U R E OF M O L E C U L E S A N D C H E M I C A L BOND
According to the results of spectral investigations of the hydrogen

molecule, r0 = 0.74116 A and Ebnd = 4.4763 eV (see Table 12.2).
It has been pointed out that even in the unexcited state every mole
cule has a certain nuclear zero-point oscillation energy (for a discus
sion of zero energy see p. 91). For the hydrogen molecule which con
tains the lightest nuclei this energy is comparatively high, it amounts
to 0.2703 eV. When the molecule is ruptured this energy is liberated
(the hydrogen atom, unlike the molecule, has only one nucleus, and
oscillation is therefore impossible). Thus in order to find the depth
of the potential curve on formation of the bond, 0.2703 eV must
be subtracted from the bond energy taken with the minus sign.
This gives the value E — —4.7466 eV.
There are also experimental methods for determining the other
parameters of the potential energy curves of molecules.
14.3. Results of Quantum-Mechanical Treatment

of the Hydrogen Molecule by Heitler and London
In the hydrogen molecule there are two electrons moving in the
field of two nuclei. If the distances between the particles are desig-
Eieciron f
Fig. 14.3. Distances between

particles in the hydrogen mo
lecule
nated as in Fig. 14.3, the expression for the potential energy is writ
ten in the form
e2 e2 e2 e2 e2
U (14.20)
Rab r i2 r a\ r a2 ?b2 rbl
Heitler and London based the wave function for the electrons in
the hydrogen molecule on the wave function of the electron in the-
hydrogen atom in the normal, ls-state. This wave function is deter
mined by the relationship
y it = -± = e-r (14.21>
yn
(see Table 4.1), in which r is the distance of the electron from the*
nucleus expressed in atomic units.
Ch. 14. QUANTUM-MECHANICAL EXPLANATION OF C O V A L E N T B O N D 169‘
It is evident that if the atoms are a great distance apart, tho

movement of their electrons will not undergo significant changes-
and the wave function of the system can be expressed by the product
of the wave functions of two hydrogen atoms
♦ = *«(l)iM 2) (14.22)
This form of the general wave function depends on the fact that ty2'
expresses the probability of the electron being found in the volume
element under consideration, while the probability of finding elect
rons simultaneously in the relevant volume elements is determined
by the product of the probabilities, since, according to the law of
probability, the probability of two events taking place simultaneous
ly is equal to the product of their probabilities. The quantity r in
expression (14.21) is equal to ral for function i|)a(l), and to n>2 for
function 'i|^(2).
Since electrons are absolutely indistinguishable, it can be consi
dered, conversely, that electron 1 moves about nucleus 6, while
electron 2 moves about nucleus a. Therefore, similarly to (14.22)
we can write
(14.23)
Because of the indistinguishability of the electrons, expressions
(14.22) and (14.23) are equivalent, both, however, are poor approxi
mations of the correct form of the wave function in the hydrogen
molecule because the movement of electrons in a molecule is greatly
different from their movement in free atoms.
Heitler and London assumed that an expression wThich took into
account the possibility of electron movement expressed by both
relationships would be a sufficiently close approximation of the*
correct wave function in the hydrogen molecule. Thus a wave func
tion for the electrons in the hydrogen molecule was ‘constructed’
by linearly combining functions (14.22) and (14.23)
= CjiJv (1) % (2) + c2\|>b (1) ij;a (2) (14.24)
It is evident from relationship (14.21) that the wave function
proposed by Heitler and London for the electrons in the hydrogen
molecule takes account pf their interaction with the nuclei (the-
value of falls as r increases), but neglects the mutual repulsion of
the electrons.
Using function (14.24) the energy of the electrons wras calculated
for different values of the distance between the nuclei R ab-
It can be shown by the variational method1 that in this case two
1 Heitler and London carried out their calculation by the so-called per
turbation technique, but the variational method discussed previously gives the-
same results. The latter method is simpler and was employed by other investi
gators in later, more exact quantum-mechanical calculations of the hydrogen
molecule.
170 P A R T I I I . S T R U C T U R E OF M O L E C U L E S A N D CHEMICAL BOND
solutions of the system of equations (14.13) are possible

ci — c2
and
c2 = —ci
Therefore, two forms of wave function (14.24) are possible
1% (1) tfo, (2) + tfe (1) l>a (2)] (14.25)
and
= CA [ta (1) % (2) — (1) (2)] (14.26)
In these equations the coefficients cs and cA are normalization
factors and are so selected that the summarized probability of the
electrons being found in space is equal to unity (see p. 43).
Equation (14.25) shows that if the nuclei or electrons are inter
changed (i.e., if indices (1) and (2) or a and b are interchanged),
function remains the same. It is therefore said to be symmetric
with respect to the coordinates of both nuclei and electrons. On the
contrary, if there is such an interchange in (14.26) the sign will
be reversed. Accordingly, function i|>A is said to be antisymmetric.
How are symmetric and antisymmetric functions to be interpreted
physically? Recall Pauli’s exclusion principle. According to that
principle there cannot be two electrons in an atomic or a molecular
system in which all four quantum numbers are identical (see p. 55).
The quantum numbers determine the form of the wave function
characterizing the state of the electron. Since the symmetric function
remains the same when the electrons are interchanged it might seem
that these electrons are in the same state, which is contrary to the
Pauli principle. However, the wave functions of the hydrogen atom
(14.21) from which function (14.24) is derived neglect the spin of
the electron. Therefore, the electrons in the molecule whose state is
expressed by the symmetric \])-function must have different spin quantum
numbers, i.e. these electrons have oppositely directed or antiparallel
spins.
Conversely, the antisymmetric function corresponds to a state in
which the electrons have undirectional or parallel spins.
When wave functions (14.25) and (14.26) are substituted into equation (14.9)
and the mathematical operations indicated in the latter carried out, expressi
ons for the energy are obtained which can be written in the general form as
follows:
For the symmetric function
Es = T + j t
For the antisymmetric function
J —K
Ea — (14.28)
1-S2
The quantities in the above equations denoted by the letters / , K and S are
determined by integrals whose value depends on the distance between the atomic
nuclei. In general form these integrals may be written
/ = j 'ifilT'il)! dv
K— \ dv
Here
(!) tb (2)
t n = to 't’**t*1)
S = j t a (1) tb (!) dv= j t o (2) tb (2) dv
The integral denoted by the letter / is termed the Coulomb integral since
it characterizes the electrostatic interaction of electrons with nuclei, as well as
electron and nuclear interaction. Integ
ral K is the exchange integral 1. It de E} eV
termines the reduction in the energy
of the system due to the movement of
each electron about both nuclei (this
movement may conventionally be cal
led the exchange of electrons). The
physical meaning of this principle will
be discussed below. The exchange inte
gral has a negative sign; it makes the
principal contribution to the energy of
the chemical bond. Integral S is the
overlap integral. It shows to what
Fig. 14.4. Comparison of hydrogen

molecule calculation with experimen
tal findings
1 —experimental curve; 2 —calculation by
means of symmetrical wave function
<14.25): 3 —calculation with antisymmetri-
cal function (14.26)
extent the wave functions of the electrons of the hydrogen atoms overlap.
The integral varies from 1 at Rab = 0 to 0 at Rah = oo; at Rab = r0 it is
equal to 0.75.
The results of calculation of the electron energy in the H 2 molecule
are shown in Fig. 14.4. Two curves are obtained corresponding
to expressions (14.27) and (14.28), respectively. In the case of the
symmetric wave function the curve has the form characteristic of
J IK
1 Relationship (14.27) is sometimes given the form Es = 2 E j j -\- --~l~ c »
1 o"
wherein the first term (2E h ) expresses the energy of the two hydrogen atoms in
the normal state, and the second term, the energy changes taking place as the
atoms approach one another. When relationship (14.27) is written in this form
the Coulomb and exchange integrals acquire a somewhat different appearance.
172 PART III. S T R U C T U R E OF M O L E C U L E S A N D C H E M I C A L BO N D
the molecule, it shows the formation of the chemical bond. The

equilibrium interatomic distance, r0, as calculated by Heitler and
London is 0.869 A, and the depth of the potential curve comes to
3.14 eV (72.3 kcal/mole). As noted previously the most precise
experiment gives r0 = 0.74116 A and E = 4.7466 eV. Taking
account of the very approximate character of the wave function
employed, composed of the unaltered wave functions of the atoms
and allowing but indirectly for the mutual repulsion of the
electrons, such a coincidence can be considered quite satisfactory.
Thus Heitler and London’s calculation provided a quantitative
explanation of the chemical bond on the basis of quantum mechanics.
It showed that if the electrons of the hydrogen atoms have antiparal
lel spins there is a substantial diminution of the energy of the system
as the atoms approach one another, and a chemical bond is formed.
The formation of the chemical bond is due to the fact that if the
electrons have antiparallel spins their movement about both nuclei
becomes possible, something that at times is termed, not too aptly,
the ‘exchange of electrons’. The possibility of the electrons’ movement
about both nuclei brings about a considerable increase in the den
sity of the electron cloud in the space between the nuclei. A region
with a high density of the negative charge appears between the posi
tively charged nuclei which draws them together. The attraction
reduces the potential energy of the electrons and consequently the
potential energy of the system, and a chemical bond results1. It
follows that the formation of the chemical bond is explained by the
lowering of the potential energy of the electrons, resulting from the
increased density of the electron cloud in the internuclear space.
Figure 14.5 shows the distribution of electron density in a system
consisting of two nuclei and two electrons as calculated with wave
functions (14.25) and (14.26) (recall that the density of the electron
cloud is determined by the square of the wave function). Areas with
a higher density of the electron cloud are made darker in Fig. 14.5.
The wave functions of electrons with antiparallel spins are added
together, and therefore the density of the electron cloud between
the nuclei increases. In this case it is said that the electron clouds,
or to be more exact, the wave functions, overlap.
Overlapping of the electron clouds cannot be regarded simply
as the superimposition of the electron cloud of one isolated atom
on the electron cloud of another isolated atom. Since the wave func
tions are added, the electron density between the atoms determined
by the quantity is greater than the sum of the densities of the
electron clouds of isolated atoms for the same distances from the
1 Precise calculations show that there is a slight increase in the kinetic

energy of the electrons when the bond is formed but it is small in comparison
to the fall in the potential energy.
nuclei. When a chemical bond is formed the electron clouds stretch

out toward one another as it were.
On the contrary, in the case of the antisymmetric wave function
which is characteristic of electrons with parallel spins, the density
of the electron cloud between the atoms falls to zero because the
electrons are forced out of the space between the nuclei and a chemi
cal bond is not formed.
The quantum-mechanical calculation of the hydrogen molecule was first
carried out by Heitler and London who used the approximate function (14.24).
The calculation was later repeated by other investigators using more complicate
(a) (6)
Fig. 14.5. Electron clouds of hydrogen atoms with different relative orientation
of electron spins
(a) antiparallel spins—atoms combine to form a molecule; ( b ) parallel spins—atoms are
repulsed
expressions for the wave function which took into consideration the deformation
of the electron shells, the mutual repulsion of the electrons, etc. In 1935 James
and Coolidge (USA) employed a 13-termed expression for the approximate wave
function, and obtained the values of r0 and E which were very close to those
found experimentally (see Table 14.1). In 1960 Kolos and Roothan (USA) used
a still more complicate expression consisting of 50 terms, and differences between
the results obtained and the experimental values were infinitesimal.
Table 14.1
Quantum-Mechanical Calculations of the Hydrogen Molecule
Number of
terms in wave
Investigators function **o» A E, eV
equation
Heitler and London (1927) 2 0.869 3.14

James and Coolidge (1935) 13 0.740 4.72
Kolos and Roothan (1960) 50 0.74127 4.7467
Experimental values 0.74116 4.7466+

±0.0007
PerformiDg the calculations in this case was only possible thanks to the use of
an electronic computer.
It follows that in spite of it being impossible to find the exact solution
of the Schrodinger wave equation for the hydrogen molecule, the use of appro
ximate methods makes it possible to calculate the system with a very high
degree of accuracy.
Heitler and London also carried out the quantum-mechanical

calculation of the interaction energy of the hydrogen molecule and
a third atom of hydrogen. The calculation showed that the third
atom would not be attracted, which means that formation of the H3
molecule is impossible1. This was theoretical substantiation of a most
important quality of the covalent bond, saturability.
Without going into the calculation, the result can be explained
on the basis of what was said above about the calculation of the 1I2
molecule. The addition of a third atom to the H2 molecule is impos
sible because the condition for overlapping of the electron clouds and
formation of the chemical bond, namely the presence of electrons
with antiparallel spins, is not fulfilled. The spin of the third electron
would inevitably coincide in direction with the spin of one of the
electrons in the H2 molecule and a repulsive force would act between
them, just as a repulsive force appears on the approach of two hydro
gen atoms with parallel spins.
14.4. Valence of the Elements on the Basis

of the Heitler and London Theory
The possibility of interaction between the He atom and the H
atom can be treated in the same way as the possibility of interaction
of the H2 molecule with a third H atom.
The electronic structure of the helium atom in the normal state
is expressed by the formula Is2. This means that the helium atom
has two electrons in which n = 1, I = 0, and m — 0. According
to Pauli’s exclusion principle these electrons must have antiparallel
spins. It is evident that the electron of the hydrogen atom would
have a spin coinciding in direction with one of the electrons of the
helium atom. Consequently, there can be no common electron cloud
joining the He and H atoms, and no chemical bond can be formed.
Heilter and London likewise treated the interaction of two He
atoms. Here, too, the formation of a chemical bond is impossible
because each of the electrons of an He atom would have a spin which
coincided in direction with the spin of one of the electrons in another
He atom.
1 Although the addition of a third atom of hydrogen to the hydrogen molecu

le is impossible, the hydrogen ion H + deprived of its electrons can be added, the
ionized molecule HJ exists.
The electronic configuration of the He atom -can be represented

by the diagram
He [Tfl
which shows that two electrons are contained in a single quantum
cell (see p. 57). Two electrons with opposite spins occupying the
same quantum cell are said to be paired. Using this terminology it
can be said that according to the Heitler—London theory hydrogen
can form an H2 molecule because it contains an unpaired electron,
but helium cannot form an He2 molecule because the electrons in
the He atom are paired.
We next consider the interaction of two Li atoms. The electronic
configuration of the Li atom (l5225) is represented by the diagram
This atom contains one unpaired 25-electron. Therefore, one can

expect the formation of an Li2 molecule by the pairing of the lone
5-electrons, in the same way as the H2 molecule is formed. Actually,
the Li2 molecule exists. The bond energy of the Li2 molecule (1.13 eV)
is roughly one-fourth that of the H 2 molecule (4.48 eV). This is
because of the first electron shell around the Li nucleus, the Li—Li
bond, is considerably longer than the H —H bond (2.67 A as compared
to 0.74 A). Moreover, the two pairs of first-shell electrons strongly
screen the charges of the nuclei and repulse one another. All this
causes a substantial weakening of the bond.
Extending this principle to other systems it can be said that
a chemical bond is formed when two atoms having unpaired electrons
come into contact. Overlapping of the electron clouds of the unpaired
electrons, or more exactly their wave functions, then becomes
possible, and a zone of high electron density appears between the
atoms, resulting in the formation of a chemical bond. It follows that
if an atom has n unpaired electrons it can form chemical bonds with n
other atoms, each of which has a single unpaired electron. Therefore,
according to the Heitler—London theory, the valence of an element
is equal to the number of unpaired electrons in its atom. Thus Heitler
and London’s quantum-mechanical calculations provided theoretical
substantiation for Lewis’s assumption that the chemical bond is
formed by a pair of electrons.
On the basis of the foregoing discussion we now consider the valen
ce of the elements of the second period of the periodic system.
Lithium. As we have seen, lithium has one unpaired electron and
its valence is therefore one.
170 PART I I I . STRUCTURE OF M O L E C U L E S A N D CHEMICAL BOND
Beryllium. The beryllium atom has the electronic configuration

ls22s2; distribution of the electrons in quantum cells is represented
by the diagram
£
1
Thus in the normal state the beryllium atom has no unpaired elect
rons and its valence is therefore zero. However, imparting a certain
amount of energy to the beryllium atom (62 kcal/g-at) brings it
into an excited state
Be ( I s 2 2 s 2) 6 2 k ca l ^ B e*(lsa2s2p)
P P
s
In this state there are two unpaired electrons, i.e., the beryllium atom
now has a valence of two. The energy expended in bringing the atom
into an excited state is more than compensated by the energy liberated
on formation of the chemical bond (recall that the energy pf the
single bond is of the order of 100 kcal (see p. 134)).
Boron. The electronic configuration of the unexcited atom is
ls22s22p. Distribution of the electrons in the quantum cells is repre
sented by the diagram
P
T
The presence of an unpaired 2/?-electron in the normal state would
indicate a valence of one. Univalence, however, is not characteristic
of boron, since it is converted into the excited state by a relatively
small amount of energy.
B (is22s22 p ) ^ 7' 3HCa-L - B # ( I s 2 2 s 2 p 2 )
n 3 — ■ T
s
m rn
n
In this state it has a valence of three.
Carbon. The electronic configuration of the carbon atom, ls22s22p2,

corresponds to a distribution of the electrons in the cells in which,
in accordance with Hund’s rule, there are two unpaired electrons.
Bivalence is not characteristic of carbon1 as it is comparatively easily
converted into the excited state, in which it has a valence of four.
C(ls22s 22p2) 96kcaL - C * ( l s 22 s 2p 3 )
2 nn rmr
1 u
As in the case of Be and B, the excitation energy of the carbon atom
is compensated by the formation of a large number of chemical
bonds.
Nitrogen. The electronic configuration of the nitrogen atom cor
responds to the following diagram of the electrons in the quantum
cells
P
S
2
N (1 s22 s2 2p3)
1
In accordance with Hund’s rule, the nitrogen atom has three lone
p-electrons, and the valence is therefore three. It should be noted
that nitrogen does not exhibit a valence of five. This would require
the transfer of electrons to a new shell (the third), which would
require such a great expenditure of energy that it could not be com
pensated by any chemical bond with some other atom. For that
reason nitrogen, unlike other elements of Group V, does not form
such compounds as NG15, NBr5, etc. The configuration of the nitric
acid molecule in which the degree of oxidation of nitrogen is + 5 ,
will be considered later (see p. 199); it will only be noted here that
nittogen in H N 03 and N20 5 is not pentavalent.
Oxygen. The electronic configuration of the oxygen atom and the
distribution of the electrons in the quantum cells are as follows:
P
s
O (is2 2s22p4)2
nrrn
Accordingly, oxygen has a valence of two.
1 As we shall see later, carbon in the CO molecule is not bivalent (see p. 197).
12 3aK . 15648
Fluorine. The electronic configuration of the atom and the arrange

ment of the electrons in the quantum cells
£
TTTTTTT
s
F ( l s ’ 2 s J 2 p 5; 2
1
it
show that there is only one unpaired electron; therefore, fluorine*
is monovalent.
Neon. The electronic configuration of the atom and the arrangement
of the electrons in the quantum cells
s
N e ( t s 2 2 s 2 2 p 6) timiii
m
are such that there are no unpaired electrons. Neon, like helium,
does not form molecules with other atoms, and its valence is zero.
A very great amount of energy would be required to excite the Ne*
atom since the electrons would have to be forced to a new electron
shell.
The foregoing consideration of the valence of the elements of thfr
second period of the periodic system makes it clear why there is such
a great difference between these elements and all the other elements
of the periodic system, a distinction which was pointed out earlier
(see p. 84). It is particularly striking in three elements—nitrogen,
oxygen and fluorine. Besides the peculiar features of these elements
due to the small radius of their atoms and ions, there are also dif
ferences arising from the fact that the external electrons are in the*
second shell in which there are only four quantum cells. For that
reason these elements do not have the high valences of their analogues.
Heitler and London’s concept of the formation of the chemical
bond proved to be very fruitful and was the basis for the explanation*
and approximate calculation of the bond in more complicate mole
cules. Their ideas were developed into the theory of the chemical
bond which has come to be called the valence bond method or the
electron pair method. Slater and Pauling (USA) contributed largely
to the formulation and development of the valence bond method.
The basic principles of the valence bond method are as follows:
1. The single chemical bond is formed by two electrons with
opposite spins, belonging to different atoms. The wave functions
of the two electrons overlap and a zone of high electronic density
is produced between the atoms; this lowers the system’s potential
energy and a chemical bond is formed.
Ch. 14. Q U A N T U M - M E C H A N I C A L EXPLANATION OF C O V A L E N T BO N D 179
2. The bond lies in the direction in which the possibility of the

overlapping of the wave functions of the electrons forming the bond
is the greatest.
3. Of the two orbitals of any atom, the strongest bond is formed
by the orbital on which the orbital of another atom is more strongly
superimposed.
Proceeding from these principles, the method of valence bonds
provides theoretical substantiation of the orientation of the chemical
bond.
14.5. Explanation of the Orientation of Valence

The atoms of elements of the second and following periods can be
considered as consisting of a core containing the inner shells and of
the outer, valence electrons. Approximate expressions for the wave
functions of the valence electrons in different atoms are now known,
and are extensively used in quantum-mechanical calculations of
molecules. However, for a qualitative, graphic examination it is
more convenient to simplify and consider the wave functions of the
2s-, 2p- and 3d-electrons, etc., the same in form for all the atoms,
as for the hydrogen atom. This simplification will be consistently
employed henceforth.
Data concerning the wave functions of the electron in the hydrogen
atom were previously cited (see pp. 47-49). We shall now supplement
them with a graphic representation.
Figure 14.6 shows the curves of the radial components of the wave
functions R (r) for the Is-, 2s- and 2/?-states of the electron (see Tab
le 4.1). The diagrams under the curves give a clear idea of the depen
dence of the wave function on the direction in space, determined by
its angular component 0 (0) O (cp). These diagrams show the depen
dence of on the angles 0 and (p when r is fixed. They are constructed
in such a way that the radius vector connecting any point on their
surface with the origin of the coordinates is proportional to the value
of \|) at a point lying in the given direction at a definite distance,
for example 1 A, from the nucleus.
It can be seen that the wave functions of the s-electrons have a
spherical symmetry; the sign of the wave function of the 2s-electrons
reverses—at short distance from the nucleus it is negative; at long
distances, positive. The function for the 2p-electrons has a cylin
drical symmetry (in case of 2pz-electrons, the axis of symmet
ry being the z axis); it consists of two halves having different
signs.
We shall now consider examples to illustrate the general princip
les concerning the orientation of the chemical bond.
We shall first discuss the chemical bond in the water molecule.
The. H20 molecule is formed from an atom of oxygen and two atoms
12*
Fig. 14.6. Wave functions of Is-, 2s- and 2pz-electrons (*+* and * are signs of
wave function)
Values R (r) correspond to charge on nucleus Z — 1
Fig. 14.7. Wave functions of hydrogen and oxygen atoms (diagrammatic rep
resentation}
Ch. 14. QUANTUM-MECHANICAL EXPLANATION OF C O V A L E N T BO N D 181
of hydrogen. The oxygen atom has two unpaired p-electrons (see

p. 177), which occupy two orbitals at an angle of 90° to one another.
The hydrogen atoms have ls-electrons. The wave functions of the
unpaired electrons in the oxygen and hydrogen atoms are shown
in Fig. 14.7. It is evident that on the approach of a hydrogen atom
having an electron whose spin is directed opposite to the spin of one
of the unpaired p-electrons of an oxygen atom, these electrons will
form a common electron cloud linking the oxygen and hydrogen atoms.
The angle between the bonds should be close to 90°, i.e. the angle
between the p-electron clouds.
The deviation of the actual angle between the bonds in the H20
molecule (104.5°) from the value of 90° which might be expected
on the basis of the diagram can be explained by two causes:
1. The 0 —H bond is of the polar covalent type, that is the elect
rons are drawn more strongly to the oxygen atom (see the electro
negativity graph, Fig. 13.1). As a result the hydrogen atoms have
a certain positive charge, and the repulsion of these charges tends
to increase the angle between the bonds.
2. Electrons of the two 0 —H bonds in the H20 molecule have
like-directed spins, which causes a, repulsive force to act between
them. (There is a similar effect during the interaction of two He
atoms.)
A diagrammatic representation of the overlapping of the electron
wave functions in the H 20 molecule is shown in Fig. 14.8.
The effect of the above factors is less in hydrogen sulphide (H2S),
an analogue of water. In this compound the bond is less polar (see
the electronegativity graph), and the distance between the atoms
greater. In H2S the angle between the bonds is 92°, and in H2Se,
91° (see p. 129).
The configuration of the ammonia molecule can be found in a si
milar way. The nitrogen atom has three unpaired /^-electrons whose
orbitals are arranged in three mutually perpendicular directions.
It is evident that in accordance with the requirements of the method
of electron pairs, the three N—H bonds must be situated at angles
of about 90° to one another. The NH3 molecule should have the shape
of a pyramid with the nitrogen atom at the vertex (Fig. 14.9). The
value of the angle between the bonds in the NH3 molecule, as found
experimentally, is 107.3°. The causes of the discrepance between the
experimental value and that to be expected from the diagram are
the same as in the case of the H 20 molecule, and as in the preceding
example the effect of side factors diminishes as the size of the atom
increases. In the compounds PH 3, AsH3 and SbH3 the angles between
the bonds are 93.3, 91.8 and 91.3°, respectively.
The conclusion can be drawn on the basis of the foregoing discus
sion that, neglecting the effect of secondary factors, the bonds formed
by p-orbitals are situated at an angle of 90° to one another.
182 P A R T 111. STRUCTURE OF M O L E C U L E S A N D C H E MI C AL BOND
The picture is more complicate when it comes to the formation

of bonds by the carbon atom. As already noted, this atom has four
unpaired electrons when in the excited state—one 5-electron and
three p-electrons.
By analogy one might expect the carbon atom to have three
bonds directed at an angle of 90° to one another (the p-electrons),
and one bond formed by the 5-electron, which could be directed at
any angle since the 5-orbital has a spherical symmetry.
The bonds formed by the p-electrons should be stronger, since the
p-orbitals extend further from the nucleus than the 5-orbital, and
z
Fig. 14.8. Overlapping of electron Fig. 14.9. Diagram of overlapping of

wave functions when H20 mole- electron clouds when NH3 molecule is
cule is formed formed
should overlap to a greater extent the orbitals of other atoms forming

bonds with carbon. But as we know, all the bonds of the carbon
atom are equivalent and are directed toward the vertices of a tetra
hedron, the angle between them being 109.5°.
A theoretical explanation of this fact was suggested by Slater
and Pauling. They demonstrated that in the interpretation and
calculation of the chemical bond, several orbitals which do not
differ greatly in energy could be replaced by the same number of
like orbitals called hybrids. The wave function of the hybrid orbital
is formed by the wave functions of the electrons involved multiplied
by certain coefficients. Thus, as regards the formation of four bonds
by the carbon atom, the hybrid wave functions of the carbon elect
rons are expressed by relationships in the form
K b = 4 I + ^ 1 + c|>p1 y, + % z1 (14.29)
The values of the coefficients a, fc, c and d are found from the norma
lization and some other mathematical requirements which the wave
Ch . 14 QUANTUM-MECHANICAL EXPLANATION O F . C O V A L E N T B O N D 183
functions must meet. The coefficients may be either positive or

negative. The operation of finding hybrid orbitals is similar to replac
ing a vector by the sum of its projections on the axes of the coordi
nates.
From Fig. 14.6 it can be seen that the 2s wave function is negative
at small distances from the nucleus, and positive at long distances.
However, the negative part is close to the nucleus and corresponds
to only a small part of the ele
ctron cloud; practically speaking
it takes no part in the overlap
ping of the orbitals. The 2p-or-
bital is positive to one side of
the origin of the coordinates,
and negative to the other. When
the 2s-orbital is added to the
2p-orbital, the positive part of
the 2p-orbital is augmented,
Fig. 14.10. Surface of hybrid wave

function
and the negative part diminished. On the contrary, when the

2p-orbital is subtracted from the 25-orbital, the negative part
increases, and the positive part decreases. Therefore, the hybrid
wave functions have a small value on one side of the nucleus, and
a large value on the other. The surface representing the hybrid fun
ction is shown in Fig. 14.10.
The four hybrid orbitals of the carbon atom are arranged at an
angle of 109.5° to one another, they are directed toward the vertices
of a tetrahedron, the carbon atom being at its centre. The shape of
the electron clouds for the hybrid orbitals of the carbon atom is
shown in Fig. 14.11. From Figs 14:10 and 14.11 it can be seen that
the hybrid orbital is extended strongly to one side of the nucleus.
This causes a much greater overlapping of these orbitals with the
orbitals of electrons belonging to other atoms than the overlapping
of the orbitals of s- and p-electrons. In accordance with the third
principle of the method of valence bonds, this produces a stronger
bond, and for that reason hybridization causes the formation of more
stable molecules. Hybridization is also furthered by the fact that
the electrons in many-electron atoms are mutually repellent and tend
to move in such a way as to be as far distant from one another as
possible. Hybrid orbitals correspond to this tendency better than
non-hybrid orbitals.
184 P A R T I I I . S T R U C T U R E OF M O L E C U L E S A N D C H E M I C A L B ON D
R e g r o u p in g of th e e le c tr o n c lo u d s of th e ca rb o n a to m to form h y b
rid o r b ita ls is sh o w n d ia g r a m m a tic a lly in F ig . 1 4 .1 2 .
I t fo llo w s th a t w h en ca rb o n c o m p o u n d s are fo rm ed , th e d iffe r e n t
o r b ita ls o f th e v a le n c e e le c tr o n s of th e ca rb o n a to m — on e 5 -o rb ita l
an d th ree p - o r b it a ls — are tra n sfo rm ed in to fo u r e q u iv a le n t h y b r id
sp 3-o r b ita ls . T h is e x p la in s th e e q u iv a le n c e of th e fo u r b o n d s of t h e
ca rb o n a to m in th e c o m p o u n d s C H 4, CC14, C (C H 3) 4, e t c ., and th e ir
b e in g o r ie n te d a t e q u a l (te tr a h e d r a l) a n g le s to on e a n o th er.
H y b r id iz a t io n of th e o r b ita ls of th e v a le n c e e le c tr o n s is n o t lim it e d
to carb on co m p o u n d s. The necessity of employing the hybridization
Fig. 14.11. Spatial arran- Fig. 14.12. Diagram showing redistribution of

gement of hybrid electron electron density when hybrid orbitals are formed
clouds of carbon atom
concept arises whenever several bonds are formed by electrons belonging

to different subshells which do not differ greatly in energy (a substantial
difference in energy prevents hybridization).
W e n o w c o n sid e r e x a m p le s of d iffe r e n t ty p e s of h y b r id iz a tio n
in v o lv in g s - an d p -o r b ita ls (th e e x p r e ssio n s for th e ir w a v e fu n c tio n s
are g iv e n in A p p e n d ix V I I ).
H y b r id iz a tio n of one s- an d on e p -o r b ita l ^ - h y b r i d i z a t i o n ) ta k e s
p la c e d u r in g th e fo r m a tio n of b e r y lliu m , z in c , c a d m iu m and m ercu ry
h a lid e s . In th e n o rm a l s ta te th e a to m s of th e se e le m e n ts h a v e tw o
p aired 5 -electro n s in th e ir o u te r s h e ll. W h en e x c it e d , on e o f th e
5 -electro n s is p ro m o ted to b eco m e a p -e le c tr o n , r e s u ltin g in tw o u n
p aired e le c tr o n s — on e of th e m an 5 -electro n , th e o th e r a p -e le c tr o n .
W h e n a c h e m ic a l b on d is fo rm ed , th e se u n lik e o r b ita ls are tra n sfo rm ed
in to e q u iv a le n t h y b r id 5 p -o r b ita ls o r ie n te d a t an a n g le of 180°
to on e a n o th e r b eca u se th e tw o b on d s are o p p o site in d ir e c tio n
(F ig . 14.-13). E x p e r im e n ta l d e te r m in a tio n of th e c o n fig u r a tio n of
th e m o le c u le s B e X 2, Z n X 2, C d X 2 and H g X 2, w h ere X is a h a lo g e n ,
h as d e m o n str a te d th a t th e y are lin e a r and th a t th e tw o m e ta l-h a lo g e n
b o n d s are of e q u a l le n g th .
Ch. J4. QUANTUM-MECHANICAL EXPLANATION OF C O V A L E N T B O N D 185
Hybridization of one s- and two p-orbitals (sp2-hybridization)

takes place when boron compounds are formed. As stated above
(see p. 176), the excited boron atom has three unpaired electrons, one
s-electron and two p-electrons. From their orbitals there are formed
three equivalent sp2-hybrid orbitals located in a single plane at an
angle of 120° to one another (Fig. 14.14). Experimental investiga
tions have demonstrated that the molecules of such boron compounds
Fig. 14.13. Arrangement of ele

ctron clouds following sp-hybri
dization
as BX3 (X—-halogen), B(CH3)3 (trimethylboron), and B(OH)?

(boric acid) do have a planar configuration, all three bonds being
of the same length and arranged at an angle of 120° to one another.
This shows the advantage of quantum-mechanical conceptions

over the old structural theory. From the standpoint of the old theory
there, is no difference between the bonds in the compounds
Cl Cl
I I
B and N
/ \ / \
Cl Cl Cl Cl
Actually the former compound is flat while the latter is in the form
of a pyramid with the nitrogen atom at the vertex. This results, for
one thing, in a difference in polarity (1bci3 = 0, p,Nci3 ¥= 0- From
what has been said it can be seen that the present-day theory of the
chemical bond gives a precise explanation of this fact and enables
one to foresee other similar principles. Thus quantum chemistry,
besides making it possible to calculate the properties of molecules,
creates a new system of concepts; it provides a new chemical langua-
ge which supplements and develops the principles of A, Butlerov’s

theory of chemical structure.
Hybridization of one s- and three p-orbitals (sp3~hybridization), as
noted above, explains the valence of the carbon atom. The formation
of 5p3-hybrid bonds is likewise characteristic of the carbon analogues,
silicon and germanium. Their valences also have a tetrahedral orien
tation.
Since hybrid orbitals provide greater concentration of the electron
clouds between nuclei and, consequently, a stronger bond, one may
wonder why such hybridization does not occur in H20 and NH3.
Actually the orientation of the bonds in these compounds can also
be explained by 5p 3-hybridization. Such an approach is even more
•exact than that set forth on pp. 179 and 181. It should not be forgot
ten, however, that both approaches are approximate. When the H 20
molecule is formed the outer shell of the oxygen atom can assume
the configuration ^ 3, ^ 4* where ^31 and ^4 are 5p8-
hybrid wave functions, the upper index showing the number of
•electrons occupying the given orbital. Thus two of the four hybrid
orbitals are occupied by unpaired electrons and can form chemical
bonds. The angle between these bonds should be 109.5° which is
•closer to the experimental value, 104.5°, than the 90° given in the
•diagram on p. 180. But whereas on pp. 179 and 181 it was necessary
to account for the deviation of the theoretical value from the expe
rimental in the H 20 molecule, it is now necessary to explain why
the angles between the bonds in the analogues of water, H 2S, H 2Se
and H 2Te, differ markedly from 109.5°. This is explained by a num
ber of factors, in particular by the fact that in compounds containing
large atoms the bond is weaker and the gain in energy resulting
from the formation of the bond by hybrid orbitals would not compen
sate for a certain increase in the energy of the 5-electrons that would
be caused by their transition to sp3 hybrid orbitals. This hampers
hybridization. Moreover, recent precise calculations have demonstra
ted that when R —H bonds are formed, the 25-orbitals of oxygen
and nitrogen overlap to a greater extent with the ls-orbitals of hydro
gen than do the 2p-orbitals. When it comes to the analogues of oxygen,
on the contrary, the p-orbitals overlap to a greater extent. This
brings about the greater contribution of the 5-state (hybridization)
in the formation of the chemical bond in the H 20 molecule than in
its analogues. Therefore, the valence angles in H 2S, H 2Se and H 2Te
are close to 90°.
Similarly, the structure of the NH 3 molecule is explained by sp3-
hybridization. The electronic configuration of this molecule can
be represented by the diagram
:n : h
H
Ch. 14. QUANTUM-MECHANICAL EXPLANATION OF C O V A L E N T BOND 187
The valence electrons of the nitrogen atom occupy four sps hybrid
orbitals. One electron is found in each of three orbitals and the
latter form bonds with hydrogen atoms. The fourth orbital is oc
cupied by two electrons which do not form chemical bonds. Investi
gations have demonstrated that ammonia’s dipole moment is chiefly
due to this unshared pair of electrons.
The unshared pair contribution to the dipole moment is shown by
comparison of the values of p, for NH3 and NF3, molecules which
have a similar configuration. Since the N—F bond is more polar
than the N—H bond (see the electronegativity graph, Fig. 13.1)
it might be expected that jj,n f 3 would be greater than |Xn h 3* Actual
ly the reverse is the case: pnh3 = 1-48 Z), and pnf3 = 0.24 D.
This is explained by the fact that the direction of the dipole moments
of the N—H and N—F bonds is different; in the NH3 molecule the
nitrogen atom is charged negatively, and in the NF3 molecule, posi
tively. In the NH3 molecule the total moment of the bonds and the
moment of the unshared pair of electrons have the same direction
and are added; in the NF3 molecule these moments have opposite
directions and are subtracted. Consequently NF3 has a small dipole
moment.
From what has been said it follows that the H —N—H valence
angles in the NH3 molecule should be equal to 109.5° which is close
to the value found experimentally, 107.3°. The deviation from this
value in the NH3 analogues, PH 3, AsH3 and SbH3, is due to the
same factors as in the case of H 20 analogues.
Hybridization is not limited to the cases considered, other types
are possible, among them hybridization involving d-orbitals (see
p. 239).
14.6. Single, Double and Triple Bonds
Discussion of this problem can conveniently begin with the bonds
in the N2 molecule.
The electronic configuration of the nitrogen atom is Is2, 2s2, 2p3
with three orbitals oriented perpendicular to one another along
the x , y and z axes. Assume that two nitrogen atoms approach one
another, moving in the direction of the x axis. When they are suf
ficiently close, two 2px-orbitals overlap, forming a common electron
cloud along the axis connecting the atomic nuclei. The bond formed
by an electron cloud having maximum density on the line connecting
the centres of atoms is called a a bond.
We shall now consider the other unpaired electrons of the nitrogen
atoms. The diagram in Fig. 14.15 presents the surfaces of the wave
functions of the 2p z nitrogen electrons. It can be seen that the wave
functions overlap but the overlapping is different than that which
produces the o bond. There are now two areas of overlapping, one
on each side of the line connecting the atomic nuclei, the plane of
188 P A R T III. S T R U C 1 U R E OF M O L E C U L E S A N D C H E M I C A L BO N D
symmetry being that which passes through the coordinates x and z.

The bond formed by electrons whose orbitals have maximum overlapping
on both sides of the line connecting the centres of the atoms is called
a jx bond.
It is evident that the two 2py-electrons of the nitrogen atoms form
a second n bond located about the plane passing through the coordi
nates x and y.
Thus there are three chemical bonds in the nitrogen molecule but
these bonds are not the same—one of them is a a bond, the other
two are n bonds. The three lines used to designate the bonds in the
nitrogen molecule in accordance with the old structural theory are
not equivalent.
We next examine the chemical bonds in some carbon compounds.
Figure 14.16 is a diagram representing the structure of the ethane
molecule, C2H6. In this compound the four bonds of the carbon
atoms are formed by hybrid 6-p3-orbitals, which are arranged at an
angle of 109.5° to one another. All the bonds are single, a bonds.
The electron cloud of the a bond located along the axis connecting
the centres of the carbon atoms has a cylindrical symmetry in respect
to the axis. Revolution of one of the atoms round this axis does not
change the distribution of the electron density in the a bond; conse
quently this can be done without breaking or deforming the bond.
This explains the possibility of the revolution of the atoms round
the C—C bond, thus preventing cis—trans isomerism in ethane
derivatives and other organic compounds having a single bond bet
ween the carbon atoms—something that chemists have long known.
We now take up the bond in the ethylene molecule, C2H 4. Consi
deration of the various possibilities of bond formation in this mole
cule shows that the greatest overlapping of the orbitals takes place,
and consequently the system with the least potential energy is
formed, when one s-orbital and two p-orbitals of the carbon atoms
form three sp2-hybrid orbitals, while the third p-orbital remains
a simple p-orbital. A diagram of the bonds forming in this case is

shown in Fig. 14.17.
As we already know, the electron clouds in the case of ^ - h y b r i
dization are found in a single plane at an angle of 120° to one another
(see p. 185). In the ethylene molecule the hybrid orbitals form three a
bonds, one C—G bond and two C—H bonds which lie in the same
plane at angles of 120° to one another. Experimental investigation
has demonstrated that the C2H 4 molecule does have a planar confi
guration. It is evident that the p-orbitals of the carbon atoms re
maining non-hybrid form a bond. Because of the mutual repulsion
jt
of the electrons of different bonds (see p. 181), the bond is in

jt
a plane perpendicular to that in which the cr bonds lie (see Fig. 14.17).
When the bonds are so arranged, the molecule has the least poten
tial energy, i.e., this state is the most stable.
Thus the two carbon—carbon bonds in the ethylene molecule are
not equivalent—one of them is a a bond, and the other, a bond. jt
This explains the peculiarities of the double bond in organic com

pounds. Overlapping of the orbitals is less in the bond than in
jt
the cr bond, and the zones of high electron density lie further from
the nuclei. For that reason the tc bond is weaker than the a bond,
and because of the lower strength of the bond, the energy of the
jt
C=C double bond is less than twice the energy of the C—C single
bond (see Table 11.1); formation of two single a bonds from a double
. bond results in a gain of energy, which accounts for the unsaturated
character of organic compounds having a double bond.
Unlike the cr bond, the bond has no cylindrical symmetry with
jt
respect to the axis connecting the centres of the atoms. Therefore,

rotating one of the atoms round this axis changes the configuration
of the electron clouds. From Fig. 14.17 it is readily seen that the jt
bond is broken if the atom is rotated through 90°, while the o bond
remains unchanged. Since considerable energy is required to break
the jt bond, free rotation round the C—C bond in the C2H 4 mole
cule is impossible. This gives rise to cis—trans isomerism in ethylene-
derivatives. On the other hand, if the molecule is subjected to the-
action of considerable energy, for instance, if the substance is heated
the it bond may be ruptured and one of the carbon atoms may rotate^
through 180° about the a bond, after which the jt bond may formL
again; as a result a cis isomer is transformed into a trans isomer
Fig. 14.18. Diagram showing formation of chemical bonds in C2H2 molecule
Such transformations are well known. For example, when maleic

acid which has a cis configuration is heated, it is converted into-
fumaric acid, the trans isomer
HOOCv yCOOH Hv y COOH
)c = c ( — > =<
W \h HOOC/ \h
Maleic acid Fumaric acid
s/^-Hybridizatipn which determines the orientation of the chemical
bond in ethylene also occurs in other molecules in which the carbon
atom is joined to three other atoms or groups, for example, in the*
compounds
O
\
c l\
xc=o
o
sc
H - < °
1
Cl/ \ h X )H
Carbonyl chloride Formaldehyde Formic acid
1 (phosgene)
the bonds of the carbon atoms are in a single plane, and the angle*
between them is close to 120°.
Figure 14.18 is a diagrammatic representation of the arrangement
of the bonds in the acetylene molecule, HC==CH. In this case, only
two electrons of the carbon atom form hybrid orbitals through
^p-hybridization. The two sp-hybrid orbitals are arranged at an
angle of 180° to one another, forming a cr bond between the carbon
atoms, and also forming the C—H bonds. The C2H2 molecule has
a linear configuration. The remaining two non-hybrid p-orbitals
of the carbon atom are arranged at an angle of 90° to one another.
They form two n bonds whose electron clouds lie about two mutually
perpendicular planes.
Since the valence electrons of the carbon atom which are not
involved in the formation of the triple bond are oriented at an angle
of 180° to one another, such compounds, unlike carbon compounds
containing a double bond, cannot exhibit cis—trans isomerism.
Fig. 14.19. Diagram showing formation of chemical bonds in C02 molecule
It should not be thought, however, that cis—trans isomerism is

impossible in principle for compounds of the general formula A2B2.
Thus the recently prepared difluorodiazine, N2F2, exists in two isd-
meric forms
N= Nv /N = n /
\p Y/
Cis form Trans form
In this case the molecule is not linear, while the nitrogen atoms are
joined by a double bond. Free rotation about such a bond is impos
sible. These factors bring about cis—trans isomerism.
The C—H bonds in acetylene formed by sp-hybrid orbitals are
different in properties from the C—H bonds in saturated hydro
carbons formed from sp3-orbitals. Thus, for example, the hydrogen
in acetylene is rather easily replaced by metal, an instance being
the precipitation of copper acetylide, Cu2C2, when acetylene is.
passed into solutions containing Cu+.
sp-Hybridization likewise occurs in the molecule of carbon dioxide
C02 (Fig. 14.19). Two sp-hybrid orbitals of the carbon atom form
two a bonds with the oxygen atoms, while the remaining, non-hyb
rid carbon orbitals form mutually perpendicular Jt bonds with the
two p-orbitals of the oxygen atoms. This explains the linear struc
ture of the C02 molecule.
A particular form of the chemical bond is found in the benzene

molecule
CH
^ \
HC CH
HC CH
\ /
CH
In this molecule each carbon atom is joined to two other carbon
atoms and one hydrogen atom. As in the formation of ethylene and
the other molecules considered above in which the carbon atoms are
Fig. 14.20. Diagram showing formation of chemical bonds in C6H6 molecule

(for the sake of simplicity only three p-orbitals are shown)
joined to three other atoms, sp2-hybridization takes place in benzene.

Three hybrid orbitals form three a bonds, two with carbon atoms,
and one with hydrogen, which are located in a single plane at angles
of 120° to one another. This explains the flat configuration of the
C6H6 molecule which has the form of a regular hexagon. The orbitals
of the unhybridized p-electrons in C6H6, as in C2H 4, are arranged
perpendicular to the plane of the molecule. The diagram in Fig. 14.20
represents the surfaces of the wave functions of the p-electrons of the
carbon atoms in the benzene molecule. It is evident that each p-orbi-
tal overlaps other p-orbitals on two sides. We know that overlapping
of the orbitals of two atoms makes it possible for the electrons to be
about both atoms. Since all the p-orbitals in the C6H6 molecule over
lap one another, each p-electron can be about any of the carbon atoms.
The ji bonds in the benzene molecule bind all the carbon atoms in
the same way, and it is impossible to say to which atoms each of the
three pairs of electrons forming the n bonds belongs. In the C6H6
molecule the n bonds are delocalized. The p-electrons in the C6H 6
molecule move along the ring of carbon atoms without meeting
Ch. 14. QUANTUM-MECHANICAL EXPLANATION OF C O V A L E N T ^BOND 193
resistance in the way, this takes place in some metals at very low
temperatures, giving rise to the phenomenon of superconductivity.
The possibility of the electrons moving about all the carbon atoms
in the C6H6 molecule lowers their kinetic energy, and consequently
strengthens the bonds. This explains the chemical properties of
benzene, particularly its diminished tendency to react by addition,
as compared to ethylene and other unsaturated hydrocarbons.
Delocalization of the electrons also explains the properties of
benzene derivatives. When one of the hydrogen atoms in the C6H6
molecule is replaced by some other atom or group, the latter strongly
influences the probability of a second substituent occupying one
of the possible positions—ortho-, meta- or para-. The mutual action
of several functional groups in aromatic compounds is also great.
These facts are explained by the extension of the perturbation of
the electron cloud about one of the carbon atoms to the entire ben
zene ring. Thanks to the delocalization of the valence electrons,
free radicals such as triphenyl-methyl are comparatively stable
(see p. 116).
Since the n bonds in the C6H6 molecule are delocalized, it is evi
dent that the structural formulae with double bonds employed in
elementary discussions do not illustrate the actual electronic confi
guration of this molecule. The true structure of the C6H6 molecule
is intermediate between the two variants represented by the follow^
ing structural formulae:
CH CH
Since the three pairs of n-electrons shown by lines to indicate a

double bond are distributed evenly among all the carbon atoms,
the following representation of the C6H6 molecule corresponds more
closely to the actual configuration
CH
CH
Here the dotted line indicates the delocalized jt bonds. It is note

worthy that on the basis of an analysis of the behaviour of benzene
in different reactions, the German chemist Thiele came to the con-
13 3aK . 15648
194 P A R T III. S T R U C T U R E OF M O L E C U L E S A N D CHEMICAL BOND
elusion that the double bonds in the C6H6 molecule were not loca
lized and suggested the above formula as being^the best representa
tion of the molecule’s structure. That was in 1899, long before the
development of the quantum-mecha
nical theory of the chemical bond.
The conclusion that the bonds in
the C6H6 molecule are intermediate in
character between single and double
bonds is borne out by the fact that
their length (1.40 A) lies between the
lengths of the single and the double
bond (1.54+0.02 A and 1.32±0.02 A,
respectively) (Fig. 14.21).
The formation of delocalized elect
ron pairs is characteristic not only of
Fig. 14.21. Relationship between bond length

and bond multiplicity
MultipUcLty of bond I —ethane; II— ethylene; I I I — acetylene: IV —
benzene
the benzene molecule but also of many other organic molecules in

which #a carbon chain contains several carbon atoms near one
another, each of which is joined to three other atoms. Examples
of such molecules are
CH2 = CH ~CH = GH2
/GH3
CH2= CH — (7
\ h 2
CH2 = CH — CH= CH - CH = CH2
In these molecules, represented here in the traditional form, doub

le bonds alternate with single bonds, forming what is called a system
of conjugated double bonds. In such a system, as in the C6H6 molecu
le, the clouds of unhybridized p-electrons of all the carbon atoms
overlap and the electrons can move freely along the chain of carbon
atoms.
As was pointed out earlier (see p. 39), the movement of Ji-electrons
in a system of conjugated bonds is similar to the movement of par
ticles in a one-dimensional potential well. In many cases the spectra
of compounds containing conjugated double bonds can be calculated
with sufficient precision with the aid of this simple quantum-mecha
nical model. Examples of such calculations are given in Appendix IX.
,h. 14. QUANTUM-MECHANICAL EXPLANATION OF C O V A L E N T BOND 195
It was noted above (p. 193) that neither the first nor the second structural
ormula (valence diagram) of benzene reflects the properties of the latter, i.e.,
loes not correspond to the actual structure of its molecule. Let wave functions
and correspond to these structures. What was said above signifies that
mither nort|?2 (each of which represents localized bonds) describes the benzene
nolecule which is characterized by the presence of unlocalized n bonds. A linear
’ombination of xpi and would be a better approximation
0|) = + c2^2 (14.30)
which assumes equality of the length of all bonds. Each of the components of
the mixed wave function corresponds to the ultimate (unperturbed) structu
re (1) or (2).
Since the ultimate structures (1) and (2) differ only in the arrangement of the
single and double bonds, it follows that q = c2; function will correspond to
the smaller value of energy, i.e., we approach the results of the correct solution
of the Schrodinger equation.
The result is still more exact if five valence diagrams are introduced into the
calculation, adding to (1) and (2) the following three structural formulae sug
gested by Dewar:
(D 3
In that case the mixed wave function will have the form
'I? = q% + c 21]?2 + c 3yp 3+ c4\|>4 + c5\p5 (14.31)
where \|)3, and t|?5 are wave functions for diagrams with diagonal bonds. Obvio
usly, c3 = c4 = c5, i.e., there are only two coefficients in (14.31). True, the
energy of the electron states corresponding to structures (3), (4) and (5) is higher
than for structures*(1) and (2) since one of the n bonds in the Dewar structures
is weaker than the others. Therefore, their contribution to the value of \|) will be
less than that of the first two structures. This means that when calculating to
a first approximation, one can limit oneself to and a|?2 while \|?3, \|?4 and
can be neglected.
The method of calculating the chemical bond in molecules considered above
in its application to benzene is called the m eth o d of va len ce d i a g r a m s u p e r p o s i t i o n
{of re so n a n ce th e o ry ). The wave functions employed have the form
(14.32)
in which in each of the components corresponds to a certain arrangement of
the bonds in the molecule. The less the energy of the structure to which each
*i|?l entering (14.32) corresponds, the greater its coefficient q (or as it is said,
the greater its weight).
It goes without saying that the method of superimposing valence diagrams,
which employs different variants of the representation of the wave function of
electrons in the molecule (the C6H0 molecule, for example), which is less exact
in (14.30) and more exact in (14.31), is only a mathematical technique. The
true distribution of electron density in a molecule is absolutely definite, unique
and unchanging in a given energy state. Therefore, it would be incorrect to
think that benzene consists of a mixture of molecules in five different states,
or that the molecular structure determining the properties of this compound is
the superposition (resonance) of five actually existing structures. The super
position of valence diagrams cannot be considered a physical phenomenon. It
is a method of quantum-mechanical treatment of the state of electrons whose
13*
196 P A R T I I I . S T R U C T U R E OF M O L E C U L E S A N D C H E M I C A L B O N D
movement is not localized about a definite pair of electrons. This technique is

only used in the method of electron pairs ana has no place in the method of mo
lecular orbitals, another extensively applied quantum-mechanical theory which
we shall consider later.
It would, therefore, be a mistake to gather the idea from the method of
superimposing valence diagrams that these structures actually exist, continu
ally changing from one to another, a mistake to think that the superposition of
structures stabilizes a molecule and accounts for its properties and even its
existence, and that certain ultimate structures are responsible for certain pro
perties of the molecule.
For complicate molecules, carrying out calculations when valence diagrams
are superimposed involves great mathematical and computing difficulties.
If n is the number of unlocalized electrons in a molecule, the number of inde
pendent valence diagrams which must be taken into account in the calculation
will be
n\
t '(t + 0 !
Accordingly, for naphthalene
\/\/-
it will be 42, while for anthracene
it will come to 429. In such cases calculations are very difficult and cannot be
performed without a large number of assumptions, which greatly lowers the
reliability of results.
During the ‘forties and ‘fifties when the resonance theory was the vogue,
many chemists, acting without any quantum-mechanical substantiation, chose
from a collection of valence diagrams those which it seemed to them were more
in line with the properties of the compound in question, maintaining that its
chemical behaviour was determined by the given structure. Naturally, it was
necessary to use certain valence diagrams to explain some reactions of a com
pound, and other diagrams to explain other reactions. This often led to con
fusion and misunderstanding. ,
The sharp increase in the difficulty of calculating by the method of valence
diagrams as the number of atoms in the molecule increased was one of the reasons
prompting rapid development of other quantum-mechanical interpretations of
the chemical bond.
14.7. The Donor-Acceptor Bond
We shall now examine the chemical bond in the carbon monoxide
molecule CO. The distribution of electrons in the quantum cells
in the excited carbon atom
n
Ch. U. QUANTUM-MECHANICAL EXPLANATION OF C O V A L E N T BON D 197
and in the oxygen atom

O TTTTTT1
EU
ti
is such that the formation of two chemical bonds is possible since

there are two unpaired electrons in the oxygen atom. However, when
an electron is transferred from oxygen to carbon there will be»three
unpaired electrons in each of the C“ and 0 + ions formed
t t t 0 m i l
TT TT
I I 11
These ions have the same electronic configuration as the nitrogen

atom (see p. 177). When the C" and 0 + ions combine, a triple bond
appears, similar to the bond in the N2 molecule. It is evident that
the triple bond is stronger than the double bond; its formation leads
to a state having lower potential energy. Therefore it can be assumed
that this bond is formed in the CO molecule. Actually, as is readily
seen from Table 14.2, the physical properties of carbon monoxide
and nitrogen are very close. This bears out the above assumption
regarding the character of the bond in the CO molecule.
Table 14.2
The Properties of Carbon Monoxide and Nitrogen
Property CO n2
Intemuclear distance, A 1.13 1.09

Ionization energy, eV 14.1 15.6
Bond-dissociation energy, kcal/mole 256 225
Melting point, °K 66 63
Boiling point, °K 83 78
Density in liquid state, g/cm3 0.793 0.796
A somewhat different course of reasoning which leads to the same

result is also possible. The unexcited carbon atom has two unpaired
C
m
TT
Li
198 P A R T I I I . 8 T R U C T U R E OF M O L E C U L E S A N D C H E M I C A L BO N D
electrons which can form two shared electron pairs with the two un^
paired electrons of the oxygen atom. But the two paired p-electrons
in the oxygen atom can form a third chemical bond since there is
an unfilled quantum cell in the carbon atom which can receive this
pair of electrons. A chemical bond formed by a pair of electrons pre
viously belonging to one of the atoms is called a donor-acceptor bond.
The terms semipolar and coordinative are also used for designating
this type of bond. The atom contributing to the electron pair is
called the donor, and the atom to which the pair is transferred is
called the acceptor. Shifting of the electron pair makes the bond
polar, which explains the origin of the term ‘semipolar’.
In formulae the donor-acceptor bond is denoted by the signs -f-
and — after the relevant atoms, which shows that an electron pair
is shifted to one of the atoms, or by an arrow which also indicates
the shift of an electron pair
C --0 + C Ô
It should be noted that the above scheme of the chemical bond
in the GO molecule is only a first approximation. The transition of
an electron pair from the oxygen atom to be shared with the carbon
atom should make the molecule highly polar. Nevertheless the
dipole moment of carbon monoxide is very-small, only 0.12 D.
On the basis of the above scheme this can be attributed to a certain
shift to the oxygen atom of the electron pair forming the bond.
Below (p. 210) is given a more exact description of the CO molecule
which also leads to the conclusion that the bond in this molecule
is triple.
We shall now consider several more molecules containing, a donor-
acceptor bond.
The NH3 molecule has the following electronic configuration:
H
h : n:
H
Three electron pairs form N—H bonds, while the fourth pair of outer
electrons belongs to the nitrogen atom alone. It can form a bond
with a hydrogen ion, resulting in the formation of an ammonium ion
H
H:N: + H + - H
H H -*
It follows that nitrogen in the NH* ion is tetravalent. It should
be emphasized that all four bonds in the ammonium ion are equiva
lent because of the electron density being evenly distributed among
them.
j
At this point it would be well to call attention to the fact that
because of the free electron pair the dipole moment of ammonia
is greater than the calculated value if only the shifting of the elect
ron pairs of the N—H bonds is taken into account (this applies
in an even greater degree to the H20 molecule in which the oxygen
atom has two unshared electron pairs). Ignoring the unshared pairs
may even lead to an incorrect determination of the direction of the
vector (i.
The ammonia molecule can also combine with other particles that
can accept an electron pair, for example
H :f : II :f :
h *.n : + b : f ; - -> h :n ;*bV f
In the compound
H F
| i
1
H —N - B—F
i i
1 1
H F
both nitrogen and boron are telravalent.

Tetravalent nitrogen is also contained in nitric acid, the formula
of which can be written
ii-o -rr
The transfer of one of the nitrogen electrons to oxygen results in

the appearance of four unpaired electrons in the nitrogen atom, which
can form four chemical bonds. Since the oxygen atoms joined only
lo nitrogen are the same, there is the same probability of an elect
ron passing to either of them. For that reason the formula
X-0
H— O— NK
indicating that the fourth bond is divided evenly between the two
oxygen atoms, represents the configuration of H N 03 more exactly
than the preceding one. The H N 03 molecule has the structure shown
in Fig. 14.22. It can be seen that the molecular structure corresponds
to the foregoing formula: the NjlliO bonds are equal in length and
shorter than the N—0 bond.
In the isolated (vapour) state, the nitric anhydride molecule,

N20 5, has the structure
% N— O— N
o f
similar to the structure of HNOJ.

In (N 03)” ion the unlocalized jt bonds are uniformly distributed
among all the oxygen atoms
'N'
0
A donor-acceptor bond is likewise formed in H2S 0 4 and H3P 0 4.
The electronic configuration of these molecules can be written
H :0 -H
1 1
h
■
:o1
1 1. ••
: o - p —> 0 :
•• | ••
:o—H
h - o: ” 1
:o
But the sulphur and phosphorus atoms, unlike nitrogen, have free d-
orbitals in the outer shell, which are filled to a certain degree,
with the unshared electron pairs of the oxygen atoms. Thus the
sulphur and phosphorus bonds with oxygen are intermediate between
single and double. Their electronic configuration is better expressed1
1 Investigations have demonstrated that crystalline nitrogen pentoxide is

an ionic compound having the structure (N 02)+ (N 03)~ which is nitronium nit
rate. In that case both nitrogen atoms in N20 5 are also tetravalent.
by the following formulae:
H—° \ H~ ° \
*';sf
H— O ^ -0 —H H- 0^ \>
The effective charges on the oxygen atoms and the contribution

of these atoms to the formation of jt bonds are different depending
on whether they are joined to hydrogen or not. This difference is
shown by a dotted or a dashed line. In the (P 0 4)3“ and (S04)2" ions
having a tetrahedral structure, the n bonds are equivalent; it can
be expressed by structural formulae
This also applies to other oxygen-containing acids of elements of

the third and following periods. So far as nitrogen is concerned, it
cannot be pentavalent. It is evident that the impossibility of exhi
biting covalence equal to the number of the group is also characte
ristic of the elements following nitrogen—oxygen and fluorine—
because they have no d cells in their outer electron layer.
We see that the conventional structural representation of mole
cules, though very useful in giving an idea of the spatial sequence
of atomic coupling, is often very inaccurate in portraying electron
configuration. In many cases it is impossible in general to denote the
features of electronic configuration by means of valence lines. It is
urgently necessary to devise convenient and graphic methods of
representing the chemical bond which will give more information
about electronic configuration than the usual structural formulae.
14.8. The Bond in Electron-Deficient Molecules

There are molecules having fewer electrons than necessary for
the formation of two-electron bonds. As an example we shall consi
der the diborane molecule B2H6. It would seem that its structure
should be similar to that of ethane, but unlike the latter, diborane
has only twelve valence electrons. Experimental findings show that
the hydrogen atoms in the B2H6 molecule are not equivalent: four
of them are easily replaced (e.g., by the CH3 group), but replacement
of the other two groups involves breakup of the molecule, for examp
le, into two B(CH3)3 molecules. The non-equivalence of the hydro
gen atoms in B2H6 is borne out by study of its nuclear magnetic
202 P A R T I I I . S T R U C T U R E OF M O L E C U L E S A N D CHEMICAL BOND
resonance. For that reason there are grounds for ascribing to diborane
the following structure:
H .
H
H Ê H
In this structure there are four two-electron terminal B—H bonds;

the remaining four electrons unite the BH2 radicals by means of
Fig. 14.23. Structure of dibo

rane
hydrogen bridges lying in a plane perpendicular to the plane in which

the boron atoms are located, and at a greater distance than the other
hydrogen atoms. Thus a distorted tetrahedron is formed round
each boron atom (Fig. 14.23).
Each hydrogen bridge atom forms with two boron atoms a common
two-electron three-centre B—H—B bond. As regards energy this bond
is more advantageous (by 14 kcal) than the usual two-centre bond,
and is formed by the overlapping of two sp3-orbitals of the boron
atoms and one 5-orbital of the hydrogen atom
rB B
It is in this way that ‘banana’ bonds are formed:

Ch. 14. QUANTUM-MECHANICAL EXPLANATION OF COVALENT BOND 203
Electron-deficient compounds are electron acceptors. Therefore

when B2H6 reacts with potassium., acceptance of electrons from the
latter produces potassium boronate K 2B2H6, and as a result all
the bonds become two-electron ones.
Other examples of electron-deficient molecules are A12(GH3)2 and
Be2(CH3)4.
14.9. Molecular Orbital Method
•
In addition to the difficulties encountered in carrying out calcu
lations when employing the valence bond method discussed above,
there are likewise, in a number of cases, difficulties of a fundamen
tal character. Thus, investigations show that in some molecules
it is not electron pairs that are involved in the formation of the
chemical bond but individual electrons.
The possibility of the formation of a chemical bond by means of
a single electron rather than an electron pair is seen most clearly
in the case of the ionized hydrogen molecule H^. This particle was
discovered at the end of the 19th century by J. Thomson. It is formed
when hydrogen molecules are bombarded with electrons. Spectro
scopic studies show that the internuclear distance in this particle
is 1.06 A, and its bond energy 2.65 eV. Since there is only one elect-
ton in H+ it is evident that in this molecule we have a one-electron
bond.
Many polyatomic particles have unpaired electrons. Particular
mention should be made of free radicals (see p. 115)., Free radicals
are highly reactive particles containing unpaired electrons. Unpaired
electrons are likewise found in some ordinary molecules, among
them NO, N 02 and C102 which contain an odd number of electrons,
and also in the oxygen molecule 0 2. The latter is of particular inte
rest for the theory of the chemical bond.
The oxygen atom has two unpaired electrons
O Ti m n
n
n
Therefore, on the basis of the valence bond method it could be
expected that when two atoms combine, two electron pairs are
formed and there are no unpaired electrons in the 0 2 molecule.
However, study of the magnetic properties of oxygen (see Appen
dix VIII) shows that there are two unpaired electrons in the 0 2
molecule.
From the standpoint of the valence bond method it is not clear
what part unpaired electrons play in the formation of the bond in
the specified molecules. A number of investigators attempted to
204 P A R T I I I . S T R U C T U R E OF M O L E C U L E S A N D CH E MI C AL BOND
improve the valence bond method and make it more suitable for
interpreting these facts. Nevertheless, another approach to the expla
nation and calculation of the covalent bond, called the molecular
orbital method (MO), has proved more fruitful. Formulation and deve
lopment of this method was to a great extent due to the work of
R. Mulliken (USA). It is apparently the best approach to the quan
tum-mechanical interpretation of the chemical bond we now have.
Whereas in the Heitler and London method a wave function (14.24)
was set up describing the movement of both electrons in the I i2
molecule, the molecular orbital method proceeds from the wave
functions of the individual electrons, the wave functions of the 1st,
2nd, . . n-th electrons in the molecule being found, i.e.,
It is thus considered that each electron in the molecule is found

in a certain molecular orbital described by an appropriate wave
function. Each orbital corresponds to a definite energy. In a single
orbital there can be two electrons with opposite spins. Wave function
characterizing all the electrons in the molecule which are conside
red, can be found by multiplying the wave functions of the indivi
dual electrons
ijj = aM 2 . . . t|3„ (14.33)
Approximate expressions are found for the above one-electron
wave functions. As in the valence bond method, the variational
principle is employed in selecting expressions which are closest to
the correct expressions.
There are several variants of the molecular orbital method. In
a method extensively used today, molecular one-electron wave
functions are taken as linear combinations of the wave functions
of the electrons in the atoms of which the molecule is composed.
Since this form of the molecular orbital method is based on the linear
combination of atomic orbitals it is known as the MO LCAO.
To make clear the peculiarities of one-electron molecular wave
functions it would be well to examine the principle of the quantum-
mechanical calculation of the simplest system, the ionized hydrogen
molecule H*.
Distances between particles in the H+ molecule are shown in
Fig. 14.24. The potential energy of the particles in the system is
expressed by the relationship
p2 p2 p2
C/ = - ------- ---—
Xab ra rb
In accordance with what has been set forth above, the wave
function of the electron in H* can be expressed by the relationship
t = crfa + c2^ b (14.34)
where tj?a and are the wave functions of the electron in the unexcited
hydrogen atom determined by equation (14.21). The value of the
coefficients in (14.34) can be found by the variational method. Two
results are obtained,
Consequently, two wave functions describing the movement of an

electron in a system containing two nuclei are possible
1|>S = C s (l|>a + % ) ( 1 4 .3 5 )
and
= c A (i|5a — i|>6) ( 1 4 .3 6 )
The first is symmetric; the second, antisymmetric.

When the calculation of H* is carried out with the symmetric
function, we obtain equilibrium distance r0 = 1 . 3 A, and the depth
Fig. 14.24. Distances between

particles in the ionized Hf mo
lecule
of the potential curve equal to 1.77 eV. According to experimental

findings r0 = 1.06 A and the depth of the potential curve 2.79 eV, i.e.,
the calculation accords qualitatively with the experimental values.
When the calculation is performed with the antisymmetric function
(14.36) we find that in this case the molecule is not formed.
We shall now consider the physical meaning of the symmetric
and antisymmetric wave functions, which in this case is somewhat
different than in the calculation of the H2 molecule by the Heitler—
London method, since here we are concerned with only one electron.
In accordance with (14.21) and the diagram in Fig. 14.24, the
expression for the antisymmetric wave function (14.36) has the form
S ’a = ( e ~ r° — e ~ Tb)
V ji
It is evident from the foregoing that when ra = rb, the wave function
is equal to zero. In that case the value of a|)2 which characterizes the
probability of finding the electron also becomes zero. The set of
points for which ra = rb forms the molecule’s plane of symmetry.
Therefore, if the electron movement is described by the antisymmetric
function, the probability of finding the electron in the molecule’s
plane of symmetry is equal to zero. In that case the electron cloud
206 PART III. STRUCTURE OF MOLECULES AND CHEMICAL BOND
will not be concentrated between the nuclei and the latter will be
mutually repulsed. Therefore, if the electron movement is described
by the antisymmetric function, the molecule will not be formed.
Thus we can draw the general conclusion that an orbital described
by an antisymmetric wave function will not bring about the for
mation of a chemical bond; on the contrary it will make a molecule
unstable. Such an orbital is said to be antibinding.
As regards the symmetrical function, on the other hand, it can
be seen from (14.35) that when ra = rh the electron density between
the nuclei is not equal to zero, its value exceeds the sum of the elect
ron densities of the isolated hydrogen atoms at the same distance
from the nucleus, since when (14.35) is squared, the member 2cs'i|)a,i])&
appears in addition to the members cstyl and csfyb. Thus a substan
tial part of the electron cloud is concentrated in the space between
the nuclei, causing them to be drawn together and a chemical bond
to be formed. An orbital described by a symmetrical function can
accordingly be termed binding.
The foregoing conceptions form the basis of the LCAO variant.
In this method electrons are added, one at a time, to the system of
atomic nuclei ‘fixed’ in their equilibrium positions. The interaction
of the electrons with*one another is neglected in this case. When an
electron is transferred from an atomic orbital to a molecular binding
orbital its energy diminishes. Such an orbital’stabilizes the system.
Conversely, an antibinding orbital corresponds to a higher energy
since it is more advantageous for the electron to be in the atom than
in the molecule. The problem of the molecule stability comes down
to the energy balance of all the binding and antibinding electrons
it contains. As a rough approximation it can be considered that an
antibinding electron neutralizes the effect of a binding electron.
If a parallel is drawn with the valence bond method, it can condi
tionally be considered that the formation of a single bond depends
on the presence in the molecule of two binding electrons whose effect
is not counteracted by antibinding electrons.
We shall first consider qualitatively the results obtained with
the molecular orbital method when applied to diatomic molecules
formed from atoms of elements of the first and second periods, and
then examine in greater detail the variant of the MO LCAO known
as the HiXckel method, which is extensively employed in organic
chemistry.
14.10. Molecular Orbitals in Diatomic Molecules

Each electron in a molecule is characterized by a set of quantum
numbers, as is the electron in the atom. The electron’s energy in
the atom depends on the values of the quantum numbers n and I.
The magnetic quantum number m, which determines the magnitude
of the projection of the orbital momentum on any axis and characte

rizes the position of the orbital in space, has no effect on the elect
ron’s energy. This is explained by the fact that in the atom there
is no preferred direction for the orientation of the orbital—all posi
tions of the orbital are energetically equivalent. In the molecule
the situation is different. In the diatomic molecule there is one
direction which stands out from among all the others, this is the
direction of the line connecting the atomic nuclei. For that reason
the energy of the electron in a molecular orbital depends on the
position of the orbital.
The position of the molecular orbital in space is characterized
by the quantum number X, which determines the magnitude of the
projection of the orbital momentum on the line connecting the atomic
nuclei in the molecUle. It follows that in its physical meaning, the
molecular quantum number X is analogous to the atomic quantum
number m. Like m, quantum number X has the values 0, ± 1 , ± 2, ...,
denoted by the letters a, Jt, 8, ... in the same way as the values of
I are denoted by the letters s, p, d, ... . Accordingly one speaks of
the a-, jt- and 8-states of the electron in the molecule.
According to the Pauli exclusion principle there can be two elect
rons with different spins in a o-orbital. There can be four electrons
in the jt- and 6-orbitals; unlike the o-orbital for which X can only
be equal to zero, two values of X are possible here—positive and
negative; for each of them there can be two values of spin.
Making use of the conceptions discussed, we shall now consider
the formation of diatomic molecules from atoms of elements of the
first period.
Formation of the molecule H2 can be represented as follows:
H[ls] + H+-* H 2+ [als]
which shows that a binding o-orbital is formed from a Is atomic
orbital. Similarly we can represent the formation of the H2 molecule
2H[ls] + H2[(ols)2]
Two electrons in the hydrogen molecule occupy two binding orbitals
formed from Is atomic orbitals. As was pointed out above, it can
conventionally be considered that two binding electrons correspond
to a chemical bond.
We now examine the structure of two unusual molecules He2
and He2. In the first molecule the three ls-electrons of the helium
atoms are to be assigned to molecular orbitals. Evidently two of
them will fill the binding ols-orbital, while the third electron will
go into the antibinding cr*ls-orbital (antibinding orbitals are custo
marily indicated with an asterisk). Thus the He* molecule will
have the electronic configuration
He* [(ols)2 (a*ls)]
208 P A R T III. S T R U C T U R E OF MO L E C U L E S A N D C H E M I C A L BOND
In it there are two binding electrons and one antibinding electron,

and according to the above rule such a molecule should be stable.
Actually the He* molecule does exist though less stable than the H 2
molecule since the bond energy in He+ (70 kcal/mole) is less than
in H2 (104 kcal/mole).
If an He2 molecule is to be formed, the four ls-electrons of the
atoms must be in molecular orbitals. It is evident that two of them
would occupy a binding orbital, the other two, an antibinding orbital
He2 [(als)2 (a*l$)2]
But as was stated above an antibinding electron neutralizes the
effect of a binding orbital (actually the antibinding electron is even
somewhat stronger than the binding electron), and therefore such
a molecule cannot exist. Thus the molecular orbital method, like
the valence bond method, demonstrates the impossibility of the
formation of the He2 molecule.
We next consider molecules formed from the atoms of elements
of the second period. In these molecules it can be considered that
the electrons of the first electron shell (the A-shell) of the atoms
are not involved in the formation of the chemical bond but consti
tute the core, denoted by the letter K in representations of the mole
cular structure. It must also be borne in mind that since the p-elect-
rons in the atom can have a quantum number m equal either to
0 or to ± 1 , they can fill both a- and Ji-orbitals.
Studies of molecular spectra have demonstrated that the orbitals
of the molecules in question are arranged in the order of increasing
energy as follows:
a ls < a*ls < o2s < o*2s < o2p < n2p < n*2p < a*2p
zo <Cyo* <C xo <C um << vtc* < uo*
There is little difference in the energy of electrons in the o2p-
and jt2p-orbitals, and in some molecules (B2, C2, N2, 0 2) the relation
ship between them is the reverse of that given because the o2p
level is higher than the n2p level. The above sequence determines
the order in which the molecular orbitals are filled: when molecules
are formed the electrons arrange themselves in orbitals with the lowest
energy. In the line under the symbols of the orbitals are other deno
tations suggested by Mulliken (the symbols of the molecular orbitals
are preceded by the letters of the Latin alphabet in the reverse order).
This makes the notation more compact. Both systems are used in
the literature.
In accordance with the foregoing, formation of the Li2 molecule
can be written
2Li [K2s] Li2 [KK (o2s)2]
This molecule has two binding electrons.
Ch. 14. QUANTUM-MECHANICAL E X P I A N A T I O N OF C O V A L E N T B O N D 209
The Be2 molecule should have the following electronic configuration

Be2 [KK (o2s)2 (o*2s)2}
in which there are four electrons in molecular ôrbitals, two from
each atom. In such a molecule the number of binding and antibind
ing electrons would be equal, and therefore it should be unstable.
Like the He2 molecule, the Be2 molecule has not been discovered.
In the B2 molecule there are six electrons in molecular orbitals,
the electronic configuration being
B2 [KK (o2s)2 (o*2s)2 (n2p)2]
In this molecule there are two electrons on the n2p level which can
accommodate four electrons. In accordance with Hund’s rule they
should have parallel spins. Actually, experimental investigations
show that the molecule has two unpaired electrons.
In the carbon molecule, C2, there are eight electrons in molecular
orbitals. The molecule has the configuration
C2 [KK (o2s)2 (a*2s)2 (jt2p)4]
There are two antibinding electrons in the C2 molecule, and six
binding electrons—an excess of four. Consequently it can be consi
dered that the bond is double.
When the nitrogen molecule is formed ten electrons must be ac
commodated in molecular orbitals. In accordance with the sequence
of filling the orbitals given above, the configuration of the N2 mole
cule should be
N2 [KK (o2s)2 (a*25)2 (jt2p)4 (o2p)2]
or in the alternative notation
N2 [KK (zo)2(yo*)2 (u?n)4 (xo)2]
It follows that there are eight binding electrons in the nitrogen
molecule, and two antibinding electrons, i.e., an excess of six bind
ing electrons, that is the N2 molecule has a triple bond. The forma
tion of the molecular orbitals in the nitrogen molecule is illustrated
in Fig. 14.25. For simplicity, only the formation of molecular orbitals
from 2p atomic orbitals is shown.
In the 0 2 molecule 12 electrons must be distributed among mole
cular orbitals. The molecule has the following configuration:
0 2 [KK (zo)2 (z/a*)2 (u;jc)4 (xo)2 (un*)2]
In this molecule there are only two electrons in the vn* antibinding
orbital, although four are possible. In accordance with Hund’s
rule they should have parallel spins. This is not at variance with
Pauli’s exclusion principle, since one electron will have X = 1,
and the other, Ji = —1. Thus the molecular orbital method explains
the presence of two unpaired electrons in the molecule, which ac-
14 3aK. 15648
counts for oxygen’s paramagnetism (see'Appendix VIII). The excess

of binding electrons in the 0 2 molecule comes to four.
The fluorine molecule has the electronic configuration
F2 [KK (zo)2 (yo*)2 (xo)2(wri)4(vn*)4]
Here there is an excess of two binding electrons, and the bond is
single.
It is evident that in the molecule
Ne2 [KK (zo)2 (yo*)2 (xo)2 (urn)4 (im*)4 (ua*)2]
there would be an equal number of binding and antibinding elect
rons. Therefore, like the He2 molecule, it is not formed.
Fig. 14.25. Diagram of formation of molecular orbitals in N2 molecule (only the

2p-electrons of the N atoms are shown)
It is interesting to examine the change in bond energy and bond

length in the molecules discussed above. The data are given in
Table 14.3. We see that an increase in the excess of binding electrons
increases the strength of the bond. As we pass from Li2 to N2 the
interatomic distance diminishes. This is due to the reduction in
the size of the atoms because of the charge on the nucleus and to the
increase in the strength of the bond. From N2 to F2 the bond length
increases due to weakening of the bond. This explains the regulari
ties in the change in the covalent radii of atoms (see p. 92).
We shall now consider some diatomic molecules formed from
different atoms.
In the CO molecule there are 10 valence electrons in the orbitals,
and the electronic configuration is the same as in the N2 molecule
CO [KK (zo)2 (yo*)2 (urn)4 (xo)2]
Ch. 14. QUANTUM-M ECHANICAL EXPLANATION OF C O V A L E N T B O N D 211
Table 14.3
Characteristics of Diatomic Molecules
Molecule Li2 b2 c2 n2 o2 •
Lx cess of binding electrons 2 2 4 6 4 2
Kbndy kcal/mole 25 69 150 225 118 36
Interatomic distance, A 2.67 1.59 1.31 1.10 1.21 1.42
The fact that the N2 and CO molecules have the same electronic
configuration, thus demonstrated by the molecular orbital method,
logically explains the similarity in properties of the two substances
(see p. 197).
The excess of binding electrons in the CO molecule is 6, and accor
dingly the bond can be considered triple, thus we have arrived in
a different way at the conclusion drawn on p. 197.
In the NO molecule there are 11 electrons in the orbitals, which
gives the configuration
NO [KK (zo)2 (yo*)2 (iot)4 (xo)2 (im*)]
It can be seen that there is an excess of 5 binding electrons. In the
ionized NO+ molecule there is an excess of 6 binding electrons
NO+ [KK (zo)2 (yo*)2 (wtc)4 (xo)2]
Therefore, the NO+ molecule should be more stable than the NO
molecule. Actually, whereas the bond energy in NO is 149 kcal/mole,
it is 251 kcal/mole in NO +. Nevertheless, ECo+ = 192 kcal/mole
is less than E co = 256 kcal/mole.
Thus even this small number of examples shows how effective
the molecular orbital method is in interpreting and foretelling the
properties of molecules.
14.11. Hiickel Method

During recent years the molecular orbital method has been employed
for calculating the characteristics of a very large number of mole
cules. It has been extensively applied even in fields of science where
a few score years ago the very idea of the possibility of using quantum
mechanics seemed sheer fantasy, such as the theory of organic reac
tions, biochemistry, molecular biology. The variant of the M
O
14*
LCAO suggested by E. Hiickel has found particularly wide applica

tion in the above fields.
The Hiickel method is employed to find the energy and wave
functions of electrons forming delocalized bonds. An enormous
j i
number of molecules containing such bonds are known, and they

play a very important part in many organic* reactions and biological
processes.
It was pointed out above (p. 194) that the movement of the elect
ron in a system of delocalized ji bonds can be examined by means
of a one-dimensional potential well model. However, this does not
always give correct results, since the model is very crude. Moreover,
it only determines the energy level of electrons, and provides no
information about the distribution of electronic density in mole
cules or the strength of the bond between atoms. For that reason
such examinations find only limited application. The Hiickel method
is incomparably more effective.
Three conceptions have taken shape through the use of this method,
which are of great importance in modern organic chemistry; they
are ideas about the bond order, jx-electron density and the free valence
index. We shall now characterize these conceptions in the most
general terms and show their applicability; a strict formulation
will be given below after summarizing the essence of the method
discussed.
It was previously stated that chemical bonds in molecules with
delocalized electrons could be regarded as intermediate between
single and double. The conception of the bond order characterizes
this idea quantitatively. If the bond order is equal to one, the bond
should be considered single, and if equal to two, double, but there
can be intermediate values. The higher the bond order, the stronger
it is, provided other conditions are the same. In the structural for
mula of butadiene
1.894 1.447 1.894
H,C CH CH CH,
the figures over the bonds indicate their order1. These values were
found by using the Hiickel method. It is evident that the bond order
is higher at the ends of the chain.
The ji electron density characterizes the probability of a deloca
lized ji-electron being found about the atom under consideration.
The higher this value, the more negatively charged is the atom.
This determines the direction of transformation of a molecule under
the action of charged particles. Thus, for example, in the nitro
benzene molecule
1 In such formulae where figures characterizing the bond are given, all
bends are usually denoted with a single line.
Ch. 14. QUANTUM-MECHANICAL EXPLANATION OF COYALENT BOND 2\Z
N 02
I
0 .79/ \ o . 79
0.95^ Jo.95
(h oi
the ji electron density is indicated with figures beside the atoms.
It can be seen that these values are highest for atoms meta to the
nitro group. Consequently, when nitrated (acted upon by NO+
ions), the second nitro group will for the most part occupy the meta
position. Thus the quantum-mechanical calculation explains why
it is mostly meta-dinitrobenzene that is obtained through the
nitration of nitrobenzene. The free valence index characterizes
the ability of an atom in a molecule to react with neutral atoms
and free radicals. This characteristic is denoted by an arrow pointing
to the value. Thus in butadiene the free valence indices will be
H H H H
xc r. o /
H I i ! w
0.838 0.391 0.391 0.838
From this we can conclude that when butadiene is reacted with
neutral atoms (e.g. on brominatibn) the latter will be added to the
end carbon atoms, and this is found to be the case. The values of
the free valence indices for naphthalene
0.452
0.104
0.404
show bromination of naphthalene produces a-bromonaphthalene

Br
L / \ /
which fully accords with experimental findings.

We shall first consider the application of the Hiickel method for
the simplest case, the ethylene molecule CH2=C H 2-
The structure of this molecule (carbon skeleton) can be represented
by a diagram showing the carbon atoms and the bond between them
There are two electrons here which form a jt bond, and it is to them
that the calculation by the Hiickel method relates. The wave func
tion for each of these electrons, in accordance with the requirements
of the molecular orbital method, is fornied from the wave functions
of the p-electrons of the carbon atoms <pi and <p2
^= cm + c2(p2
For our discussion there is no need to know the formulae express
ing cpi and <p2, so we shall only note that in cases where they are
required, the approximate expressions for the wave functions of
electrons as obtained by Slater are usually taken.
As was pointed out previously (p. 165), in order to find coeffi
cients q and c 2 a system of secular equations must be set up and
the secular determinant equated to zero.
The secular equations in our problem have the form
(# 1 1 — E ^ l l ) ci + (# 1 2 — # $ 12) c2 = 0 1
(# 2 1 - E S 2l) c, + ( H 22 - E S 22) c2 = 0 j (}
The integrals in these relationships have the following expres

sions and names:
£11 = j <Pi<Pi dv = j cpj dv = 1 and S 22 =
= j 92^2 dv= j (p2dv = 1 — normalization integrals

— exchange (or reso
# 2 1 = # 1 2 = j <PiH ^ 2 dv
nance) integrals
S 12 = S2i — ] 9 1 CP2 d v —overlap integrals

Hn = t tptH tpidv and H 22— \ cp2/f(p2di; — Coulomb integrals
The expressions for the Coulomb and exchange integrals contain

the Hamiltonian, but it does not enter the normalization or overlap
integrals. The Coulomb integral contains two like functions, and the
exchange integral, different functions. A similar difference is found
in the normalization and overlap integrals.
The following assumptions are made in the Hiickel method: (1)
exchange integrals for wave functions not relating to neighbouring
carbon atoms are considered equal to zero; (2) exchange integrals
for any neighbouring carbon atoms are alike; all Coulomb integrals
are also alike; (3) all overlap integrals are considered equal to zero.
It is clear that these assumptions are very crude but they greatly
simplify calculations. The last assumption seems the least substan
tiated since calculations show that the overlap integral containing
the wave functions of the /^-electrons of neighbouring carbon atoms

is approximately equal to 0.25. Nevertheless, more rigid treatment
of the problem, including overlap integrals, has demonstrated that
the error introduced by assumption (3) is not very great. Nor should
it be thought that the overlap integral must not be equated to zero
since the chemical bond originates as the result of the overlapping
of atomic orbitals. It must be borne in mind that this overlapping
also brings about the appearance of the exchange integral whose
contribution to the bond energy is substantially greater than that
of the overlap integral. Therefore the latter can be neglected when
calculating to a first approximation.
In the simple problem under consideration assumptions (1) and
(2) are not made, since the two carbon atoms in the ethylene mole
cule are neighbours. As a result of assumption (3) and the fact that
the normalization integrals are equal to unity, the system of secular
equations acquires a simpler form
(Hii — E) ct + I fl2c2 - 0 |
If 2 ici + (H 22 — E) c2 = 0 J (14.38)
The Coulomb integrals are customarily denoted by the letter a,

and the exchange integrals, by the letter (J1. In that case (14.38)
has the form:
(cx — E) Ci P<?2 — 0
PCt -f (a — E) c2 = 0
We divide both equations by P and substitute x from the equation
a —E (14.39)
=x
“~F~
The system of secular equations is written in the final form
xci -j- c2 — 0 1
ci + xc2 = 0J (14.40)
Equating the secular determinant to zero, we have

x 1
= x2— 1 = 0
1 x
from which x = ± 1 . Substituting x = —1 and x = 1 in (14.39),
we obtain two expressions for the energy of the electrons
Ei = a + p (14.41)
1 In the Hiickel method the values of a and P are found from experimental
data, P being calculated from a comparison of the bond energy in different
compounds. The numerical values of a and p are rarely used; ordinarily when
explaining or foretelling the properties of compounds, relationships are emplo
yed in the general form containing the parameters a and p.
and
E2 = a — p (14.42)
We have obtained two relationships expressing the energy of the
electrons forming the jx bond in the C2H 4 molecule. Exchange integ
ral p is always negative, and consequently energy is less than E z.
Equation (14.41) determines the energy of the binding orbital;
equation (14.42), the energy of the antibinding orbital. In the unex
cited C2H 4 molecule, the two electrons under
consideration will be in the state having the
lowest energy, i.e., they will be in the bind
ing orbital. This is shown diagrammatically
o r -/
__ in Fig. 14.26.
We shall now find the values of the coeffi
cients in the wave function expression. For
the binding orbital x = —1. Substituting
oc ---------------- this value of x in (14.40), we have
Ci = c2 or q/c 2 = 1 (14.43)
<x+j3
-N -
Fig. 14.26. Energy levels of ji-electrons in ethylene
We have obtained an expression determining the relationship

of the coefficients. To determine their absolute values we must
make use of the normalization equation
j i|>2 dv = j (cm + c2<p2)2 dv= I (14.44)
Opening the parentheses, we obtain
c\ j dv + 2ciC2 j <Pi<P2 dv + c] j cpgdy=l
Since the atomic wave functions <Pi and <p2 haye been normalized,
the first and last integrals are equal to unity. The second integral—
the overlap integral—is considered equal to zero in the Hiickel
method, from which we obtain
c\ + c\ = 1 (14.45)
Taking account of (14.43) and (14.45), we have
1
Cj = C2 =
V2
Thus the expression for the wave function of the binding orbital is
^ = -y=-(<Pi + <P2) (14.46)
( h. 14. QUANTUM-MECHANICAL E X P L A N A T I O N OF C O V A L E N T B O N D 217
Carrying out similar operations with x — 1, we find
Ci=y w and C 2=" 7 r

from which we have for the antibinding orbital
%= —<P2) (14.47)
We have found expressions for the energy and wave functions of tho
ji-electrons in the C2H 4 molecule, i.e., we have solved the given
quantum-mechanical problem.
We shall now consider a more complicated example, the state*
of the n-electrons in the free allyl radical
CH2= C H - C H 2—
1 2 3
There are equal grounds for representing the structure of this par
ticle by the formula
—CH2—CH =CH a
1 2 3
This shows that the n bonds in this structure are not localized.
Three electrons not involved in the formation of a bonds are to bo
assigned to ji molecular orbitals. (In the Hiickel method the a bonds
are not taken into account, it being considered that they do not
interact in any way with the n bonds.)
The structure of the carbon skeleton is represented by the diagram
1 2 3
Q ' ■■■— o - o
and the wave function of the electron, by the relationship

= Cl<Pi + C2SP2 + ^3^3
where <pt, (p2 and <p3 are the wave functions of the p-electrons in the
corresponding carbon atoms.
The system of secular equations has the form
(Ha ESa) — C i + E S i2) c2-\-(Hi3 E S i3) c 3= 0
( # i 2 — —
( H 2i E S 2i) Ci~\-(H
— E S 22) c2-\ - (H23 E S 23) c 3= 0
2 2 — —
(H E S 3i) Ci~\-(H
3 i — E S 32) c2+ ( H 33 E S 33) c 3= 0
3 2 — —
With the assumptions given above and using the brief notation, it
has the form
xci + c2 = 0 t
Ci + xc2 ■+ c3 = 0 > (14.48)
C2 + x c 3 = 0 J
The secular determinant has the solution

x 10
x 1 1 1 = x(^2—1) — x = x* — 2x =
1 x 1 =x M -1
0 a: = x (x2— 2) = 0
0 1x
From this we find three values of x
Xi = 0; x 2 = — ] / 2 ; x 3 = ] ^ 2
Substituting each of these values in relationship (14.39), we obtain

three expressions for the energy
Ef = a; E 2 = a + 1^20; E 3 = a — ]/2|J
We now examine the expressions obtained.
The first relationship does not contain the exchange integral |5.
The energy of the electron in the corresponding orbital is practically
the same as in isolated atom—the Coulomb integral a expresses
the energy of electrons in the
E absence of a chemical bond. This
orbital is termed nonbinding. Accor
<x-2j3 ding to the Hiickel method, an
electron in such an orbital has no
influence on the strength of the
a~ J3 bond. This is, of course, over
simplification; actually, this elec
tron does make a certain contribu
oc — 1----------------- H - - tion to the energy of the bond, but
it is not great, and can be negle
cted in a rough evaluation.
cx+J3
■H" -H-, "b"

c 3h- C3H, c3h; Fig. 14.27. Energy levels of rc-electrons
<x+2J3 in C3 H 5 radical and in ions formed
from it
The second expression gives the lowest value of E (the exchange

integral is negative). This energy corresponds to a binding orbital.
The third relationship gives the greatest energy, such an electron
occupies an antibinding orbital.
Figure 14.27 gives a diagram of the energy levels in particles
having the formula G3H5. In the neutral radical C3H5 there are three
electrons in the Hiickel molecular orbitals—two occupy a binding
orbital, and one, a nonbinding orbital. In the C3H^ ion there are
no electrons in the nonbinding orbital, but in the C3H" ion there
are two electrons in this orbital.
We now find the coefficients in the wave function expressions.

For their calculation we must use the normalization equation; it
is similar to (14.44) and gives the expression
c\ + c\ + cl = 1 (14.49)
analogous to (14.45).
For the nonbinding orbital x = 0. Substituting this value in the
system of secular equations and taking account of (14.49), we find
Ci = — ; c2 = 0; c3 = - y = -
The wave function is written
It can be seen that the electron cloud of the nonbinding orbital

is concentrated about end atoms 1 and 3, wave function <p2 is absent
from the expression for the molecular orbital. Therefore, this elect
ron has no appreciable influence on the strength of the bond.
For the binding orbital x = —"J/^2. Using this value in the same
way, we obtain
1 1
Ci = c3— ~y ;
The wave function is determined by the expression
^2 = 4 “<Pl + <P2+ * 4 (Ps

The examples discussed give an idea of the Hiickel method. The
calculation procedure is standard. Solution of the secular determinant
gives the values of parameter x from which are found the expres
sions for energy and the values of the coefficients in the wave
function equations. The delocalized Jt-electrons are then assigned
to the orbitals with the lowest energy.
The highest power of x in the equation obtained by opening the
secular determinant is equal to the number of atoms in the mole
cule under consideration (for ethylene, a quadratic equation is
obtained; for allyl, a cubic equation). Therefore, when performing
calculations for polyatomic molecules, algebraic equations of a high
power must be solved. Various mathematical techniques are known
which make it possible to find the roots of such equations with a
high degree of accuracy.
Computational difficulties are substantially less in the molecular
orbital method than in the valence bond method. Whereasr when exa
mining delocalized n bonds by the MO method, the power of the al
gebraic equation obtained is equal to the number of atoms in the
molecule, it is equal to the number of valence diagrams used, when
employing the valence bond method. We have seen (p. 195) that
even for fairly simple molecules the number of valence diagrams
is high.
We shall now consider calculation of n electron densities, bond
orders and free valence indices.
Electron density q, created by an electron on atom r under consi
deration, is determined by the square of the coefficients cT with
which the orbital of atom r enters the expression for the molecular
wave function. The total electron density is taken as the sum of the
electron densities created by each electron. It can be expressed by
the equation
(14.50)
j
where qr is the electron density on atom r; cjr, the coefficient for the
wave function of atom r in the expression for the / molecular orbital
occupied by n electrons. Thus for atom 1 in ethylene
The same value of q is obtained for atom 2.

In allyl
Here, too, the electronic density on all atoms is equal to unity.

Bond order p is characterized by the contribution of neighbouring
atoms (they can be denoted r and s) to the overlapping of the orbitals.
It is determined by the product of the coefficients for the correspond
ing atomic wave functions, the sum of these values for all the elect
rons being taken
Prs = 2 rijCjrCjs (14.51)
Thus for ethylene we obtain
Pi2==2( ~ y T ' ^ y f ) =zi

For allyl
-^ - = 0 .7 0 7
The electron in the nonbinding orbital makes no contribution to

the bond order, since for it c2 = 0.
The above calculation gives the order of the n bond; to find the
complete bond order, unity must be added, that is the order of
Ch. 15. IONIC BOND 221
the a bond. Thus the bond orders in ethylene and allyl will be
2 .0 0 1 .7 0 7 i T o?
CH , ===== C H 2; CH2 = CH 111— CH2
Free valence index F is determined by the relationships

F = N maiX- N r (14.52)
N T=2>Prs
where N r is the sum of the bond orders of all the n bonds in which
atom r is involved, and N max, the maximum value of the bond
possible for the carbon atom; it is realized in the radical trimethyle-
nemethane
3
CH2
Cl
2 / \ 4
- h 2c c h 2-
In this particle, the order of the n bond of the C atom is equal to 1^3 =
= 1.732 which is the maximum value of N possible. Using relation
ship (14.52) and the values of the bojid orders of the jt bonds already
obtained, we find the free valence indices in ethylene and allyl
0 .7 3 2 0 .7 3 2 1 .0 2 5 0 .3 1 8 1 .0 2 5
t t t t t
h 2c --------— c h 2 c h 2 — — CH — c h 2 -
CH APTER F IF T E E N
THE IO N IC BOND
15.1. Energy of the Ionic Bond

Ionic molecules are something chemists encounter incomparably
less often than particles in which the atoms are joined by a covalent
bond. As we shall see later (p. 278) there are no individual molecules
in ionic crystals. In most solutions of ionic compounds their mole
cules are also not found, since when dissolved in polar solvents,
such as water, alcohols, etc., ionic compounds are completely disso
ciated, while in nonpolar solvents, CC14, G6H6, etc., they are insoluble.
Ionic molecules can be detected in the vapours of ionic compounds.
As we know, such substances must be heated to a high temperature
if they are to be vaporized. It should be noted that the vapours of
ionic compounds contain not only molecules but other particles as
well—associates of several molecules, and also simple and complex
ions. In the vapour of sodium chloride, for example, are found, in
addition to NaCl molecules, such associates as (NaCl)2 and (NaCl)3,
and the ions Na2Gl+ and NaGl".
Since the attraction of ions obeys Coulomb’s law, it is compara

tively easy to calculate the energy of the bond for ionic molecules.
If ions are regarded as undefor-
med charged spheres, the attra
ction between them is determi
ned by Coulomb’s law
fallr — (15.1)
where fattr is the force with
which ions at a distance r and
bearing charges et and e2 are
attracted to one another. The
energy of the interionic bond
which is liberated when oppo
sitely charged ions approach
Fig. 15.1. Potential energy curves for

the ionic molecule
one another from infinity to a distance r, will be equal to
Uattr = j fattr d r j — - dr = — — (15.2)
The relationship Uattr = / (r) is shown by the lower dotted curve

in Fig. 15.1.
At very short distances the repulsive force due to the interaction
of the electron shells also makes itself felt. In 1918 Born and Lande
demonstrated that to a first approximation the repulsive force can
be considered inversely proportional to the distance between the
ions to the power n:
UreP= — (15.3)
The relationship Urep = f (r) is shown by the upper dotted curve

in Fig. 15.1.
Quantity n is called the Born repulsion factor. It is considerably
greater than unity since the repulsive force diminishes rapidly as
the distance increases. Factor n depends on the nature of the ion;
for ions with the configuration of He, Ne, A, Kr and Xe it is equal
to 5, 7, 9, 10 and 12, respectively. It can be determined by the
compressibility of crystals (see p. 289) and on the basis of the optical
properties of a substance. Since the force is the first derivative of
Ch. 15. I O N I C B O N D 223
the energy with respect to distance, we obtain from (15.3)

, dUrep nB
trev - ~ d 7 ~ rn+l (15.4)

As r changes, there are corresponding changes in f attT and f rep-
At the equilibrium distance between the ions, r0, these forces are
equal. Therefore, in accordance with (15.1) and (15.4), and on condi
tion that = e2, we obtain
e* nB
from which
n
(15.5)
Combining (15.5) with (15.3) when r = r0 gives
(15.6)
In this way quantity B can be removed from equation (15.3).

Since the interaction energy
U — U a ttr “ I- Urep
it is evident that in accordance with (15.2) and (15.6) for r = r0
we obtain the equation
c/0 = (15.7)
which gives the interaction energy of two ions at the distance r0
from one another. It is known as the Born equation. The curve of the
relationship U = f (r) is also shown in Fig. 15.1, the minimum on
it corresponding to the equilibrium distance r 0 and interaction
energy U0 in the equilibrium state. From the character of the curve
it is evident that equation (15.7) could have been obtained from the
conditions of the potential energy minimum
The potential curves for ions fall comparatively slowly to the

zero value which is due to the action of electrostatic attraction
extending to a much greater distance than the action of the covalent
bond.
For molecules consisting of two polyvalent ions equation (15.7)
has the form
(15.8)
224 PART I I I . S T R U C T U R E OF M O L E C U L E S AND C H E M I C A L BON D
where zx and z2 are the ionic charges. Equation (15.8) is used much
less frequently than (15.7) since the bond in such molecules is almost
always far from ionic; consequently equation (15.8) must be regarded
as only a very rough approximation.
Two circumstances following from the character of equation (15.7)
must be borne in mind: (1) the value of U0 is only slightly sensitive
to fluctuations in n\ thus if we use n = 11 instead of n = 9 we
change the value of U0 by only ^2 % (10/11 — 8/9 = 2/99); (2)
approximation (15.3) has practically no effect on results, since the
repulsive energy is only ~ 1 0 % of t/0*
For the formation of an ionic molecule from atoms of monovalent
elements, which can be represented by the general equation
A + B = A +B~
the relationship for the bond energy becomes
- £ ab = - ( 1 - 7 ) + « b - / a (15.9)
in which EAB is the energy of formation of the gaseous molecule AB
from free gaseous atoms which is equal in value to the bond energy;
/ A, the ionization energy of atom A; and EB, the electron affinity
of atom B.
As an example we shall now use equation (15.9) for estimating
the value of EAB for the gaseous molecule KC1 (r0 = 2.67 A;
£ci = 3.81eV; /k = 4.34eV). Assuming n = 9, we obtain in
accordance with (15.9)1.
—£ kci = 2.67 x l £ * x 1.8 X 10-12' ( 1 — T ) + 3 *81 “ 4 ’34 =

= 4.78 + 3.81 - 4.34 « 4.25 eV = 97.8 kcal/mole
The result obtained is close to the value of the bond energy of the
KC1 molecule obtained in experiments (101.2 kcal/mole).
It must be emphasized that even for atoms of the alkali metals
and halogens, whose conversion into the cation (A+) and anion (B“)
involves the loss and gain of only one ion, respectively, it is impos
sible to speak of an ideal, ‘100 per cent’, ionic bond. Actually, becau
se of the wave character of the electrons the probability of an elect
ron being found around atom A in molecule AB, although very small,
is not equal to zero, i.e., in this case complete separation of the char
ges, characteristic of free ions A+ and B“, is impossible. This is borne
out by the following example: if the properties of the free ions Na +
and Cl“ persisted in the gaseous molecule NaCl, the value of p,Naci
for an interatomic distance of ~2.5 A would be 12 D; actually,
P-NaCl < 1 2 D.
1 The factor 1.6 x 10~12 has been introduced into the denominator to con
vert ergs into electron-volts.
Ch. 15. I O N I C ' B O N D 225
15.2. Ionic Polarization

Deviation from the pure ionic bond in any compound can be
regarded from another standpoint—it can be considered that the
ions in the molecule act upon one another. This fact was not taken
into account in equation (15.9), which is one of the reasons of the
divergence from experiment.
The influence.on one another of closely situated oppositely charged
ions . causes their mutual polarization. As the ions approach one
8, . Sz
Fig. 15.2. Mutual polarization

of ions
another, the electrons are displaced with respect to the nucleus,

resulting in an induced dipole moment, \kind (Fig. 15.2). The more
the ions are polarized, the greater the deviation from equation (15.9).
Since this process strengthens the bond between the ionj, the calcu
lation according to (15.9) gives low results.
When estimating polarizability coefficient a (see p. 154), it must
be taken into account that a is measured in units of length cubed.
Actually the induced dipole moment is determined by the product
of the charge by the length of the dipole; at the same time it is expres
sed by the product aE, where the strength of the field E , in accordan
ce with Coulombls law, is equal to {Fie) = e/r2. From this it follows
that polarizability has the above dimension, i.e., that it is a measure
of the space occupied by the particles. When calculating to a first
approximation it can be assumed that
a « r8 (15.10)
i.e., « 10~24 cm3. This is borne out by the figures given in
Table 15.1.
Polarization is a bilateral process; in it the polarizability of the
ions and their polarizing action are combined.
The polarizability of ions depends on the type of electronic confi
guration, their charge and size. Since it is the outer electron subshell
that is least strongly bound to the nucleus, for the sake of simpli
city it can be assumed to a first approximation that the polarization
of an ion involves only the deformation of that subshell, i.e., the
displacement of the outer electron shells of the two ions relative
to their nuclei. If the charges are alike and the radii approximately
equal, minimum polarization is found in ions having the configura-
15 3 a n . 15648
226 P A R T I I I . S T R U C T U R E OF M O L E C U L E S A N D C H E M I C A L B OND
Table 15.1
Polarizability a (A3) and Cubes of Radii r 3(A3) of Ions
Ion a 7*3 Ion a r3 Ion a r3 Ion a r,i
Li+ 0.029 0 .32! Be2+ 0.008 0.043 F" 0.96 2.35 0 2“ 2.74 2.30
li
Na+ 0.187 0.92 Mg2+ 0.103 0.29 c i- 3.57 5.92 S2- 8.94 5.3
1
K* 0.888 2.35 Ca2+ 0.552 0.97 Br~ 4.99 7.5 Se2‘ 11.4 7.0
Rb* 1.49 3.18 1 Sr2+ 1.02 1.40 I- 7.57 10.6 j Te2~ 16.1 9.4
Cs+ 2.57 4.67 Ba2+ 1.86 2.40
tion of a noble gas, and maximum polarization in ions with 18 outer

electrons; a has an intermediate value in the ions of the transition
elements with an unfilled d-subshell. The high polarizability of
ions of the non-noble gas type is explained by the large number
of electrons in their outer shells.
In the subgroups (like electronic configurations and ionic charges),
polarizability rises as the atomic number increases (Table 15.1).
This is explained by the fact that the increase in the number of
electron shells in the ion-analogues causes the outer electron shell
to be further removed from the nucleus and at the same time increases
the screening of the nucleus by the inner electron shells, which brings
about greater deformation of the ions.
If an element forms ions having different charges, the polarizabi
lity will be smaller, the greater the charge, since increase in the
latter diminishes the radius of the ion and strengthens the bond of
the electrons with the nucleus.
In a group of isoelectronic ions with a noble-gas configuration
the polarizability increases as the positive charge diminishes, as for
example, in the group Mg2+—Na+—Ne°—F “—O2” (see Table 15.1).
In such series the number of electron shells is the same, and the
increase in polarizability is due to the diminution of the nuclear
charge.
The polarizing action of ions also depends on their type, charge
and radius. It is greater, the greater the charge, the less the radius
and the more stable the electron shell. The greatest polarizing action
is exerted by ions that are themselves little polarized. For that rea
son, if an element forms ions having different charges, their polariz-
Ch. 15. I O NI C B O N D 227
ing strength sharply rises as the charge increases because their radius
decreases at the same time. Conversely, polyatomic (complex) ions
of large size, which as a rule are greatly deformed, usually have
mi insignificant polarizing
action.
The intensity of the po
larizing action of ions is
noteworthy. Employing
Coulomb’s law we shall
estimate the strength of the
electric field at a distance
r = 10"7 cm (10 A) from
an ion
e- 4.80 X 10-10 _
r2 ~ 10-7 x 10-7 ~
= 4.80 x 104 esu/cm2
or 4.80 X 104 X 300 =
-=14.4 X 106 V/cm, which
is an enormous value.
The above regularities in
changes in polarizability
can be made quantitative
by using methods of com
parative calculation. This
is evident from the exam
ples shown in Fig. 15.3. In
Fig. 15.3a are compared
values of a in groups of nob
le gas atoms and isoelectro-
Fig. 15.3. Comparison of pola

rizabilities
(a) atoms of noble gases and ions of
alkali metals; (b) two series of
isoelectronic anions
nic single-charge cations of the lithium subgroup; in Fig. 15.36 the

polarizability of pairs of particles selected by a different principle
is compared. These examples of methods of comparative calculation
like others, including those previously discussed (see Fig. 5.6, 5.7,
7.2, 8.1, 11.7,’l l .8 and 13.3), provide further evidence of the regular
change in the most diverse properties in Mendeleev’s periodic system
of the elements.
Since anions are characterized by large size and a small charge,
while their electronic configuration corresponds, as a rule, to that
15*
228 P A R T I I I . S T R U C T U R E OF M O L E C U L E S A N D C H E MI C A L B OND
of a noble gas, their polarizing action on cations is usually small,

and can often be neglected, that is, it can be considered that polari
zation is unilateral in character. If, however, a cation is easily de
formed, the resulting dipole heightens its polarizing action on the
anion, and the latter, in turn, exerts an additional action on the
cation, and so on. This gives rise to a supplementary polarization effect
which is the greater, the more the cation and anion are polarized.
The supplementary polarization effect, and with it, the total
polarizing action are especially great for 18-electron cations, such
as Zn2+, Cd2+ and Hg2+.1
The increased constriction of ions resulting from their deformation
causes the length of the dipole to be less than the internuclear distan
ce; for instance, the length of the dipole in the KC1 molecule is 1.67 A,
whereas the internuclear distance is 2.67 A. This difference is parti
cularly great in hydrogen-containing compounds. If the size of the
hydrogen ion is neglected the distance between the nuclei of hydrogen
and halogen, in the case of a purely ionic bond, should be equal to rx-.
But d n -x < rx- for all halogens. Thus rGi- = 1.81 A, but dH—ci =
= 1.27 A . This means that unlike other cations, the proton penetra
tes the electron subshell of the anion, there exerting a strong polariz
ing action which causes a sharp reduction in the polarity of hyd
rogen compounds as compared to analogous compounds of other
cations. Due to the polarization effect the HC1 dipole has a length
of only 0.22 A instead of 1.27 A . Finally, the proton penetration
into the anion causes a diminution of the latter’s deformability.
The polarization of ions constitutes a certain degree of electron
displacement, which is of great importance, since by causing a
reduction in interatomic distances and, as a consequence, a reduction
of the dipole moment, it converts an ionic bond into a polar covalent
bond. As the deformability of the anion increases, there may be
complete transfer of the electrons from the anion to the, cation,
i.e., a covalent bond is formed. The latter differs from the ionic bond
in a number of properties, among them, direction. On the contrary,
the less the polarization of the ion (an anion, for example), the closer
a compound of the given atom will be to the ionic type. Since pola
rization sharply rises as the charge of the ions increases, it is evident
that among compounds of the type A2+B2“ or A+B2", and still more
so A3+B3” or AgB3“, there can be none with a purely ionic type of
bond, and this applies even to noble-gas configurations.
1 In addition to the Coulomb and polarization interaction the so-called

dispersion forces must also be taken into account (see p. 265). At this point we
shall only state that these forces (a) act between like-charged ions, in this way
weakening their Coulomb repulsion, and (b) increase in proportion to the product
pf the polarizability coefficients a of the interacting particles, and are therefore
particularly great for 18-electron cations.
Ch. 15. I O NI C BON D 229
Polarization conceptions are important because they provide

corrections to the Kossel diagram, thus making it possible more
precisely to describe the properties and peculiarities of the most
diverse compounds.
15.3. Effect of Polarization on Properties of Substances

Polarizability and the polarizing action explain many individual
features of substances.
The polarization of the ions in a molecule, or in other words, the
presence of a certain proportion of covalent bond, enhances its
stability. In that case the member Ep which takes account of the
polarization must be added to equation (15.9) for calculating th
energy of the bond. The equation then has the form
_ £ a b = - ^ ( 1 - - ^ ) + £ p + £ b - / a ( 1 5 . 1 1 )
It should not be thought, however, that molecules with a substan

tial proportion of covalent bond are always more stable than ionic
compounds. A distinction must be made between stability as regards
dissociation into ions and dissociation into atoms.
Equation (15.11) represents the energy of the formation of a mole
cule from atoms; this quantity greatly depends on the atomic ioni
zation energy / A, which can vary over a wide range. It is evident
from equation (15.11) that an increase in / a lowers the bond-dis
sociation energy. The energy of the dissociation of the molecule
into ions is determined by the first two members of equation (15.11).
What has been said can be illustrated by comparing two compounds
HgCl2 and CaCl2. The HgCl2 molecule which contains the strongly
polarizing ion Hg2+ dissociates into ions to a much lesser degree than
CaCl2; in ^N aqueous solution mercuric chloride is practically undis
sociated, whereas calcium chloride dissociates completely. However,
the ionization energy of the mercury atom (Ii = 10.4 eV; / 2 =
= 18.7 eV) is considerably greater than the ionization energy of
the calcium atom (Ii = 6.1 eV; / 2 = 11.9 eV). Consequently, the
HgCl2 molecule is much less stable as regards dissociation into atoms
than the CaCl2 molecule.
Polarization lessens the distance between ions and increases the
energy of the bond between them. Therefore, although rAg+ is com
mensurable with rNa+ and rK+, the polarizing action of Ag+ is much
greater than that of Na+ and K +. This is one of the reasons why the
solubility of AgCl is much lower than that of NaCl and KC1. In
a series of compounds of a given cation with anion-analogues, for
example, with Cl", Br" and I", the degree of dissociation in solutions
diminishes, as a rule, here the increase in the polarizability of the
anion makes itself felt.
The mutual polarization of ions facilitates the disintegration of

crystals, i.e., lowers the melting point, this effect being the gre
ater, the more the crystal lattice is deformed as a result of polariza
tion. Thus, although the radii of the cations in RbF and T1F coin
cide, the T l+ ion polarizes more and exerts a considerably greater
polarizing action on the F" ion than the R b+ ion; among other
things, this affects the melting points of the above salts: (m.p.) Rbr =
— 780, but (m.p.) t i f = 327°C.
The presence of maxima on the melting point-molecular weight
curves of the halides of the alkali metals is understandable if the
decrease in the polarizing action in the group L i+ — Na+ —K+ —
Rb+ — Cs+ and the increase in polarizability of the group F~ —
Cl Br“ — 1“ are taken into account.
Raising the temperature usually promotes polarization, since
heating increases the swing of ionic oscillations, thus bringing them
closer together, it can draw some ions to others, i.e., change the
structure of the substance, which means that a polymorphic trans
formation takes place (see p. 271). The possibility that heating will
draw the electron (electrons) completely from the anion to the cation
is not to be ruled out. This will cause thermal dissociation of the
substance. The greater the polarization (polarizing action), the
lower the dissociation point will be. For that reason the d.p. for a
given cation falls from MCI to MI, and for a given anion from NaX
to LiX. Or another example: whereas a high temperature is required
for the dissociation of Cal2, the reaction Aul 3 = Aul + I2 proceeds
at a low temperature. The dissociation of Cul 2 should take place at
a still lower temperature and therefore this compound does not exist
in ordinary conditions.
Lowering of the dissociation point as the polarizing force increases
c a n be illustrated with more complicated examples in which the
mechanism of thermal dissociation is different. Thus,
CdC03 (rcd2+ = 0.97o A) decomposes at 360°C, while
CaC03 (rCa2+ = 0.99 A) decomposes at 850°C. In this case the pro
cess proceeds according to the equation
MC03= M 0 + C02
since one of the oxygen atoms is polarized not only by the carbon
but by the metal as well (contrapolarization). The difference in the
polarization of the bonds M — 0 and 0 — C diminishes as the
temperature rises, which finally leads to dissociation. This contra-
polarization diminishes in a series of ions of the same type (e.g.,
Ca24 — Sr2+ — Ba2+), which explains the rise in the dissociation
points of the corresponding carbonates. Contrapolarization can be
so great that a compound is only stable at very low temperatures.
Chemists know very well that free oxygen-containing acids are,
as a rule, less stable than most of their salts. H 2C0 3 and H 2S 0 3 are
Ch. 15. I O NI C BO N D 231
examples. This is explained, firstly, by the strong polarizing action

of H f and, secondly, by the different character of contrapolariza-
tion. A proton penetrates the oxygen anion, lowering its charge and
diminishing deformability; for that, reason HG03" and H 50“ are
less stable than C03” and S 03”, respectively. A second proton makes
the particle still less stable; therefore H 2C0 3 and H 2S 0 3 easily lose
water. This is one of the reasons why oxygen-containing acids are
stronger oxidizing agents than their salts.
The deformability of the electron subshell also affects optical pro
perties. Absorption of certain rays is associated with excitation of
the outer electrons. The more polarizable a particle is, the less the
energy of electron transitions. If a particle is stable, excitation
requires much energy, corresponding to ultra-violet radiation. If
an atom (or ion) is easily polarized, the excitation energy is low,
corresponding to the visible region of the spectrum, and the substance
is coloured. For that reason, along with substances whose colouring
is due to the colour of an ion (or ions) they contain, there are colo
ured compounds formed from colourless ions. In such cases the
colour is the result of ionic interaction. The greater the polariza
tion' and polarizing action of the ions, the greater the probability
that a substance will be coloured. It is likewise evident that as
these effects are heightened, the colour should deepen. These prin
ciples are confirmed by a multitude of examples. We shall limit
ourselves to a few of them, leaving it to the reader to explain each
case. P b l 2 is coloured but Cal 2 is colourless; much more coloured
compounds are found among the sulphides of metals than among
their oxides; in the group NiCl2 — NiBr 2 — N il 2 the colour dee
pens; if the bromide of an element is not coloured, it is very doubt
ful that its chloride will be; a number of compounds can be named
that take on a colour when heated.
15.4. The Polar Bond and Electronegativity

The bonds in the molecules A2 and B 2 consisting of like atoms
are covalent. If the bond in the molecule AB is also covalent, Pa
uling suggests that the following equation should hold true:
Ea- b = V E x- aE b- b (15.12)
in whichÊ is the energy of the respective bonds, or to put it diffe
rently, in the expression
A2? = £a-b —V Ea-aEb-b (15.13)
AE should be equal to 0 (replacing the arithmetical mean by the
geometrical mean makes AE positive in all cases). The energy of
the unsymmetric bond calculated from equation (15.12) is always
less than experiment. This is explained by the fact that a covalent

bond between different atoms is always more or less polar. The degree
of polarity of a covalent bond can be judged by the deviation of
AE from zero, and this also indicates the ability of the atoms to
attract electrons to themselves (the character of the change in A ^
and in the dipole moment is the same, and the latter increases as
the degree of polarity of the bond rises).
%ionic bond
Fig. 15.4. Relationship between

degree of ionic character of the
bond and difference in electro
negativity (according to Pauling)
The value of AE depends on the difference in the electronegativity

x of the elements. Pauling found that this connection can be expres
sed by the approximate relationship
AE = const (x b —*xa)2 (15.14)
In the Pauling system such electronegativities are chosen that their
difference gives AE in electron-volts (const = l); the electronega
tivity of fluorine is assumed to be 4.0. If AE is to be obtained in
kilocalories, the value of the constant is put at 23.03. The value
of £ A-b can be calculated by combining equations (15.13) and
(15.14).
When xa = %b the bond is covalent and AE = 0; when xA xB
the bond is polar. The greater the difference between xa and xb*
the more polar the bond, if xa <C xb the bond can be considered
ionic. Thus as the difference in electronegativity increases, there
is a corresponding increase in the ionic character of the bond. The
curve in Fig. 15.4 shows the approximate relationship between the
difference in electronegativity of the elements and the ionic chara
cter of the bond. From this curve it follows that it is only in the
fluorides of the alkali and alkaline earth metals [( ^ b — %a ) i s 2 . 8
to 3.3] that the bond can be considered close to ionic.
We must again emphasize the approximate character of the con
ception of electronegativity (see p. 150) and particularly the funda
mental inaccuracy of equation (15.14) (see also p. 226). Actually,
*Ch. 16. C H E M I C A L B O N D I N COMPLEX COMPOUNDS 233
when the AB molecule is formed there may be a change in the num

ber of binding and antibinding electrons. Consequently, to connect
the interaction energy only with the partially ionic character of the
bond will not always be justified. Calculation of electronegativity
by the Pauling system may lead to substantial discrepancies between
the computed bond energy and experiment, especially when the
difference between the values of xa and xb is great. Nevertheless it
is convenient for approximate estimations and for systematization.
CHAPTER SIXTEEN
THE CHEMICAL BOND IN COMPLEX COMPOUNDS
16.1. Complex Compounds

Structurally, complex compounds are made up of an inner, coordi
nation sphere consisting of a central particle—a complexing ion or
atom—surrounded by ligands (addends) which are molecules or ions
of the opposite sign. In the formulae of complex compounds, the
coordination sphere is denoted by brackets. Examples of such com
pounds are K 4[Fe(CN)6], K JH g lJ , [Ag(NH3)2]Cl, K 2[Zn(OH)4] and
[Cr(H20) 6]Cl3. The number of ligands around the complexing particle
is called the coordination number. Ions situated beyond the coordina
tion sphere form the outer sphere of the complex.
The coordination sphere remains intact in solutions of complex
compounds. For example, the compounds cited above dissociate in
solution with the formation of the complex ions [Fe(CN)6]4~, [Hgl4]2“,
[Ag(NH3)2]+, [Zn(OH)4]2“ and [Cr(H20) 6P +.
The ideas about *the structure of complex compounds set forth
here form the basis of the coordination theory originated and deve
loped by Werner (Switzerland) in the latter half of last century.
The coordination theory, like Butlerov’s theory of the structure
of organic compounds, made it possible to establish the structure
of complexes long before the elaboration of physical methods of
determining structure, methods which fully confirmed Werner’s
conclusions.
Many complexes can be formed through the combination of ordi
nary substances, for example:
CuCl2+ 4NH3 = [Cu(NH3)4]C12
AgI + KI = K[AgI2]
There are also many other methods of synthesizing complex com
pounds.
Considerably more complex compounds are known than all other
inprganic substances, and they find practical application in the
most diverse fields. Complexes play a very important part in orga-
nic processes. For instance, haemoglobin and chlorophyll, substan

ces essential for the life of higher animals and plants, are both com
plexes. The complexing ion in haemoglobin is Fe2+; in chlorophyll,
it is Mg2+.
Among complex compounds there are acids, bases and salts, and
also substances which do not dissociate into ions, that is, non-ele
ctrolytes. Here are some examples:
Acids Bases Salts Non-electrolytes
H[AuC14] [Ag(NH3)2]OH [Ni(NH3)6l(N 03)2 [Pt(NH3)2Cl2]
H2[SiF6] [Cu(NH3)4](OH)2 Na3[AlF6] [IVi(CO)4]
The formation of complexes is particularly characteristic of ions
of the transition metals, very many complexes being known for
the ions Cu2+, Ag+, Au3+ and Cr3+ and likewise for ions of elements
of Group VIII of the periodic table.
The ligands in complex compounds are often ions of the halogens
and the ions C N ”, N C S", NOjj, O H ”, SOI", C 2 0 4 ~, CO 3 ", as well as
the neutral molecules H 2 0 , N H 3, N 2 H 4 (hydrazine), CrjH 5‘N (pyri
dine) and N H 2 — G H 2 — C H 2 — N H 2 (ethylenediamine).
The coordination number of some complexing ions is constant;
for example, in all Cr3+ and Pt4+ complexes it is six, but for most
complexing ions it can vary, depending on the nature of the ligands
and the conditions in which complexes are formed, for example, in
Ni2+ complexes coordination numbers 4 and 6 are found. The most
frequently encountered coordination numbers are six and four, and
there are very few complexes with a coordination number greater
than eight.
Some ligands have several groups in their molecules, which can
unite with the complexing ion. For example, the NH2 — CH2 —
CH 2 — NH 2 molecule contains two NH 2 groups which readily
unite with the ions Cu2+, Cr3+, Co3+, etc. Each of these ligands can
occupy in complexes the place of two ordinary ligands, such as
NH 3 or H 20. Ligands occupying several coordination places are
•called multidentate; depending on the number of places they occu
py, there can be bidentate, terdentate ligands and so on.
Complex compounds are classed as aqua-compounds, in which the
ligands are H 20 molecules, e.g., [Cr(H20)JC l 3 and [Ca(H20) 6]Cl2;
examines (ligands: NH 3 molecules), such as [Cu(NH3)4]S0 4 and
{Ag(NH3) 2]Cl; hydroxo-compounds (ligands: OH” ions), such as
K 2[Zn(OH)4] and Na2[Sn(OH)6]; and acido-complexes (ligands: acid
residues), such as K 4[Fe(CN)6] and K 2[HgI4]. There are also many
compounds of a mixed type, such as [Co(NH3) 4Cl2]Cl and
[Pt(NH 3)4(H 20 ) 2]Cl4.
The names of complex compounds are formed similarly to the
names of ordinary salts (NaCl — sodium chloride, K 2S 0 4—potas
sium sulphate, etc.), with the only difference that the ligands and
Ch. 16. CH E MI C AL BOND I N COMPLEX COMPOUNDS 235
degree of oxidation of the central ion are indicated. H20 and NH 3

.molecules are denoted ‘aquo’ and ‘ammine’, respectively. The folio-
-wing examples illustrate the nomenclature of complexes:
K2[PtCl6]
Potassium hexachloroplatinate (IV)
[Gr(H20 )6]Cl3
Hexaquochromium (III) chloride
[Pt(NH3)2Cl2]
Dichlorodiammineplatinum (II)
[Co(NH3)4(N 02)G1]C104
Chloronitrotetramminecobalt (III) perchlorate
K2[Zn(OH)4]
Potassium tetrahydroxozincate (II)
16.2* Isomerism of Complex Compounds

Isomerism is a wide-spread phenomenon among complexes, just
as it is among organic compounds. It was the study of isomerism
in complexes that first made it possible to establish their spatial
structure.
We shall now briefly consider the main types of isomerism in
complex compounds.
Ionization isomerism results from different distribution of ions
between the inner and outer spheres of the complex, for example
[Co(NH3)5Br]S04 and [Co(NH3)5S04]Br
[Co(en)2(N 02)Cl]Cl and [Co(en)2Cl2]N02
In the second example the letters 4en’ are an abbreviation for
ethylenediamine, NI12 — CH2 — CH2 — NH2.
Coordination isomerism is found in compounds in which both the
cation and the anion are complexing particles, and results from
different distribution of the ligands between them. Examples of
such isomers are the following:
[Cu(NH3)4][PtCl4] and [Pt(NH3)4][CuCl4]
[Co(NH3)6][Cr(CN)6] and [Cr(NH3)6][Co(CN)6]
Ligand isomerism is possible in cases where the ligand molecules
or ions can exist in isomeric forms. This type of isomerism is obser
ved, for example, when the ligands are the compounds
H3C ch3 ch3
NH2—C—CH2—NH2 and NH2—(1h —CH2—CH2—NH2

or
NH2—CH2—CH2—CH2—CH2—NH2 and CH3- N H - C H 2-C H 2- N H - C H ,
As in the case of organic compounds, there are also types of iso

merism involving different spatial arrangement of the particles—
cis—trans isomerism and stereoisomerism.
For complexes with a, coordination number of four, cis—trans
isomerism is possible when the four ligands are in the same plane.
[Pt(NH 3)2Cl2] is an example:
Cl. #. n h 3 nh 3 . .Cl
Pt Pt
C l-’ •nh 3 Cl •’ *'NH;
Cis isomer Trans isomer

A tetrahedral arrangement of the ligands makes cis—trans isome
rism impossible.
Stereoisomerism is illustrated by complexes of trivalent chro
mium containing two molecules of ethylenediamine. The coordina
tion number of Cr+3 is six. The ligands are situated at the vertices
Fig. 16.1. Diagrammatic representation of the structure of mirror isomers of

chromium-ethylenediamine complexes. Hydrogen atoms in ethylenediamine
molecule are not shown
of an octahedron at the centre of which is the Cr3+ ion. The ethyle

nediamine molecule, curved in shape, is attached to the chromium
ion by two NH 2 groups (as already noted, ethylenediamine occu
pies two coordination places). When there are two ethylenediamine
molecules in the octahedral complex two different configurations are
possible, as shown in Fig. 16.1. These forms are mirror images of one
another.
As in the case of organic compounds, stereoisomers of complexes
have identical chemical and physical properties. They differ only
in the asymmetry of their crystals and in the direction in which they
rotate the plane of polarized light.
Werner foretold the occurrence of stereoisomerism in complexes
on the basis of the coordination theory, and he himself synthesized
such compounds.
€ h . 16. C H E M I C A L B O N D LN C O M P L E X C O M P O U N D S 237
16.3. Explanation of the Chemical Bond in Complexes

on the Basis of Electrostatic Conceptions
Formulation of a theory explaining the formation of complex
compounds began in 1916-1922 in the investigations of Kossel and
Magnus (Germany), and was based on electrostatic conceptions. It is
evident that a complexing ion will attract ions of the opposite sign
and also polar molecules. On the other hand, the particles surroun
ding the complexing ion will be repulsed from one another, and the
repulsive force will be greater, as the number of particles grouped
round the central ion increases.
If one considers, as did Kossel and Magnus, that the interaction
of the particles in a complex is purely electrostatic and obeys Cou
lomb’s law, the bond energy between ligands and the complexing
ion can be calculated. In these calculations the complexing ion and
the ligands are regarded as undeformable, charged spheres. The
reader can easily carry out such calculations for such ions as [Hgl4]2“
^tetrahedral configuration), [AlFfi]3“ (octahedral arrangement of
the ligands), [Agl2]“ (linear structure) if he uses the values of the
ionic radii.
These calculations give values of the bond energy of the correct
order for complexes of the specified type, but for complexes in
which the ligands are polar molecules the results are worse. Howe
ver, they can be improved somewhat if the polarization effect is
taken into account (see p. 225).
Kossel and Magnus demonstrated by their calculations that when
the number of ligands was substantially increased, the repulsive
force between them became so great that the complexes were un
stable. It was found that for complexes to be sufficiently stable, the
coordination number must be two or three in the case of monovalent
complexing ions; four for bivalent ions, and four, five or six for
trivalent ions. These results are fairly close to what is observed in
reality.
Thus electrostatic conceptions indicated in principle the cause
of the formation of complex compounds, made it possible to
estimate theoretically their stability and approximately explain
the coordination numbers observed.
Nevertheless, the conception of complexes as aggregates consisting
of undeformable, charged spheres is a very crude model and cannot
explain many of their peculiarities. For one thing, electrostatic
conceptions cannot explain why a number of complexes with a coor
dination number of four have a plane configuration (complexes
of Cu2+, P t2+, etc.). Since, if a model with charged spheres is
used, a tetrahedral disposition of the four ligands about the comp
lexing ion \yould energetically be the most advantageous arran
gement.
Moreover, the charged-sphere model is completely useless for

explaining the magnetic properties of complexes. As we know, study
of the magnetic properties of a substance makes it possible to deter
mine the number of unpaired electrons (Appendix VIII). Since
according to Kossel and Magnus the interaction of complexing ions
and ligands does not change the electronic configuration, the ions in
the complex should have the same number of unpaired electrons as
the free ion. It has been found, however, that in complexes this
number may be different, the actual number depending on the nature
of the ligands. Thus the ion Fe2+ has four unpaired electrons, and'*
the same number is found in the complex [FeF6]4-, but in the ion
[Fe(CN)6]4“ all the electrons are paired.
Therefore, the present-day theory takes account of the quantum-
mechanical interaction of particles, along with electrostatic factors.
16.4. Quantum-Mechanical Interpretation

of the Chemical Bond in Complex Compounds
Several quantum-mechanical methods are now used for explaining
and calculating the chemical bond in complexes.
In the valence bond method it is assumed that there is a donor-accep
tor bond between the ligands and the complexing ion, a pair of
electrons being supplied by the ligands (see p. 196). Using this me
thod it has been possible to explain the structure and many proper
ties of a large number of complexes, including magnetic properties.
However, in a number of cases difficulties are encountered when
employing this method. It is likewise ill-suited for quantitative
estimations.
The crystal field theory, on the contrary, is based on a purely ele
ctrostatic model, but unlike the original theory of Kossel and Mag
nus it takes account of changes in the electron subshells of the com
plexing ion which take place under the action of the ligands. This
theory has proved extremely effective and it now finds considerably
more application than the valence bond method.
Naturally, the crystal field theory which proceeds from an ionic
model requires changes in cases where there is a noticeable propor
tion of covalent bond in the complex. When the proportion is com
paratively small, use is made of the ligand field theory in which the
presence of a covalent bond is taken into consideration by the intro
duction of certain corrections in calculations performed by the
methods of the crystal field theory. If there is a substantial propor
tion of covalent bond, the molecular orbital method is employed,
while taking account of the crystal field theory conceptions
(sometimes this approach is also called the ligand field theory).
The valence bond method, the crystal field theory and the mole
cular orbital method will now be discussed in greater detail.
Ch. 16. C H E M I C A L B O N D I N COMPLEX COMPOUNDS 239
16.5. Valence Bond Method

We know that formation of the ammonium ion, NHJ, involves
the unshared electron pair in the ammonia molecule (see p. 198),
and can be regarded as addition of the NH 3 molecule to a hydrogen
ion. Similarly, we can interpret the formation of ammines as the
addition of ammonia molecules to metal ions, for example:
Cu*- + 2NH3 = [Cu( :NH3)2]+
Zn2+ + 4NH3 = [Zn(: N H3)4]2+
According to the valence bond theory complexes result from for
mation of donor-acceptor bonds by unshared electron pairs in the
ligands. These electron pairs are held jointly by the ligand and the
central ion, occupying the free hybrid orbitals of the latter.
In the examples cited, the Cu+ and Zn2+ ions have a filled third
shell, but there are free s- and p-orbitals in the next shell. In
accordance with the principles set forth, two electron pairs from
NH 3 molecules fill sp hybrid orbitals in the [Cu(NH3)2]+ complex;
as we know, this form of hybridization causes a linear configura
tion of the particle. In the [Zn(NH3)4]2+ complex, four electron
pairs occupy sp3 hybrid orbitals, and the complex has a tetrahedral
configuration.
Not only s- and p-orbitals can be used in the formation of donor-
acceptor bonds in complexes, but d-orbitals as well. In such cases,
according to the valence bond theory, there is hybridization invol-
Table 16.1
The Type of Hybridization and the Structure of the Complex
Hybridization Configuration Ions
sp Linear Ag+, Hg2+

sp3 Tetrahedral Al3+, Zn2+, Co2+, Ti3+, Fe2\
Ni2+ (rarely)
s p 2d Square, planar Pt2+, Pd2+, Cu2+, Ni2+, Au3+
sp^d2 Octahedral Cr3+. Co3+, Ni2+, Pd4+, Pt4+
ving d-orbitals. Table 16.1 lists some forms of hybridization and

associated complex configurations. The last column in the Table
shows the ions in whose complexes the valence bond method assumes
the presence of the given type of hybridization.
The type of hybridization and the structure of the complex are
chiefly determined by the configuration of the central ion, but they
also depend on the nature of the ligands. We shall now consider
this, taking the Ni2+ ion as an example. This ion has a d8 configura-
240 PART III. STRUCTURE OF MOLECULES A N D CHEMI CAL BOND
tion. In accordance with Hund’s rule, the eight d-electrons are

distributed as follows in the quantum cells:
N i s * It i 11 l i t | ] t | T
When complexes are formed with Cl" ions which interact compara
tively weakly with Ni2+ ions, since the Cl" ion has a large radius,
the electron pairs occupy orbitals of the next electron shell. This
can be represented by the diagram
4s 4p
3d
'TrTTTTTT IUI U U TT
sp3-hybridization, tetrahedral configuration
The Ni2+ ion reacts more strongly with NH 3 molecules than with
Cl" ions. Accordingly, it adds six NH 3 molecules, forming the
[Ni(NH3)6]2+ ion, which has the configuration
___ 3d 4p 4d
[ N i ( M H 3 ) 6i [ T J
TTTTT TTF UD u u TTTTT
sp^-hybridization, octahedral configuration
The Ni2+ ion reacts still more strongly with CN" ions. In this
case two d-electrons of the Ni2+ ion are paired and the empty cell
thus formed is filled with an electron pair from the CN" ion:
4s 4p
3d ,------
[ N« ( C N) J ' | t l i t | | t | | t | | t | o m j
sp2d-hybridization, square planar configuration
From the examples cited we see that the number of unpaired

electrons in complexes formed by the ion of a certain metal is not
always the same. In [NiCl4]2" and [Ni(NH3)6]2+ the number is two,
but in [Ni(CN)4]2" there are no unpaired electrons. The first two
complexes are paramagnetic, while the third is diamagnetic.
From the above examples it is likewise evident that hybridiza
tion involving d-orbitals can be of two types—that in which the
outer d-orbitals are occupied (4d in the case of [Ni(NH3)6]2+) and
that in which the inner d-orbitals are occupied (3d in [Ni(CN)J2~).

Here is another example of ‘outer’ and ‘inner hybridization:
3d 4s 4d
[F c F j
4-
r 11m f m fmTTTTTI l l l l l l l I
‘Outer1 sp^-hybridization; complex is paramagnetic
[Fe(CN)6)4~fTTTTTTT Q i] ftllUIUI I T I I I I
‘Inner’ s/^dMiybridization; complex is diamagnetic
A great advantage of the valence bond method is that it can be

used to foretell the reactivity of complex compounds. To a great
degree this depends on the rate at which the ligands in a complex
are replaced in solution by other ions or molecules. Conditions favo
urable for the replacement of ligands are: (1) ‘outer’ hybridization
and (2 ) the presence of unoccupied ‘inner’ d-orbitals in the comple-
xing ion.
If there is ‘outer’ hybridization, the bond between ligands and
the complexing ion is weaker than in the case of ‘inner’ hybridiza
tion. Therefore, one of the ligands can separate from the complex
comparatively easily, being replaced by another particle in the
solution. Thus the complex [Fe(NH3)6]2+, in which there is ‘outer’
hybridization, is much more reactive than [Fe(CN)e]4”, in which
hybridization is ‘inner’.
If there is an unoccupied ‘inner’ d-orbital, the replacement mecha
nism is different. In that case it is possible for an additional particle
in solution to combine with the complex. After this combination,
one of the ligands may be split off. In this way one of the ligands in
the complex may be replaced.
The complexes [Cr(NH3)6]3+ and [V(NH3)6]3+ illustrate the above
rule. According to the valence bond method these complexes can
be' assumed to have the following configuration:
3d 4s 4p
[CrfNH3)6l m 11mini hd
[v(NH3)fi]3*n 111 ifirm rrn

16 3aK. 15648
242 P A R T III. S TRUCTURE OF MOLECULES A N D CHEMI CAL BOND
The second complex is much more reactive than the first, which is
explained by the presence of an unoccupied ‘inner’ d-orbital.
16.6. Crystal Field Theory

This theory is concerned with the action of the ligands on the
d-orbitals of the complexing ion. The shape and spatial arrangement
of the d-orbitals are shown in Fig. 4.7. In the free atom or ion the
energy of all the d-electrons belonging to the same electron subshell
is equal — these electrons occupy a single energy level. The ligands
that combine with a positive ion are either negative ions or polar
molecules with their negative end turned toward the complexing
ion. A repulsive force acts between the electron clouds of the d-ele-
ctrons and the negative ligands which increases the energy of the
d-electrons. However, the action of the ligands on the different
d-orbitals is not the same. The energy of the electrons in d-orbitals
situated close to the ligands increases more than that of electrons
in d-orbitals far removed from the ligands because under the action
of the ligands the energy levels of the d-orbitals are split.
We shall now consider the situation which arises when the ligands
are arranged in the form of an octahedron and in the form of a tetra
hedron round the complexing ion.
It is evident from Fig. 16.2 that in an octahedral environment, the
dz2- and d^-.^-orbitals, denoted dv, are arranged in such a way that
they are subjected to strong action by the ligand field. Electrons
occupying these orbitals have great energy. Electrons in the dxy-,
dxz- and dyz-orbitals (denoted de), on the contrary, have less ener
gy. The diagram in Fig. 16.3 shows the splitting of the energy levels
in the octahedral field.
When the complexing ion is in a tetrahedral environment of li
gands, it is evident from Fig. 16.4 that the dz2- and dx2- y2-orbitals
Ch. 16. C H E M I C A L BOND I N COMPLEX COMPOUNDS 243
have lower energy, while the dxy-, dxz- and dyz~orbitals have higher
energy. The resulting splitting of the energy levels is shown diagram-
inatically in Fig. 16.5.
The value of energy split A depends on the nature of the ligands
and the configuration of the complex. If the ligands are the same,
and they are at the same
. dz2,dX2.y2 (dr) distance from the centre of
the complexing ion, the
value of A for a tetrahed
ral configuration is 4/9 of
the value for an octahedral
configuration.
dxy>dx z , d y Z( d €)
The value of energy split
A can be calculated the
I oretically by the methods
i / I o n i n foi cetladh e d r a l
i /
&
<t> //
*5 I
/11
/
//
//
//
Free Ion
Fig. 16.3. Splitting of d-electron energy Fig. 16.4. Arrangement of coor
levels in an octahedral field dinate axes in a tetrahedral
complex
of quantum mechanics, and can be determined experimentally

by the absorption spectra of complex compounds.
As we know, the absorption spectrum in the visible and ultraviolet
regions is associated with the transition of electrons from certain
energy levels to others (see p. 146). A substance absorbs light quanta
whose energy is equal to the energy of the corresponding electron
transitions. The absorption spectrum, and consequently the colour,
of most complexes of d-elements is due to electron transitions from
a lower d-orbital to a d-orbital having higher energy. For example,
the complex [Ti(H20 )6]3+ has maximum absorption at wave number
v = 20,300 cm”1, which causes its violet colour. The Ti3+ ion has
only one d-electron, which in the octahedral complex can be trans
ferred from the de-orbital to the dv-orbital. The energy of the
16*
244 P A R T I I I . STRUCTURE OF MOLECULES A N D CHEMI CAL BOND
quanta corresponding to v = 20,300 cm -1 (which comes to 57 kcal

per g-ion) is equal, as stated above, to the value of A.
Thus the crystal field theory explains the fact, well known to
chemists, that the ions of elements of the groups of transition ele
ments are coloured, while ions having the configuration of the noble
gases are colourless. It likewise becomes understandable why Cu+
ions are colourless, while Cu2+ ions are coloured. The Cu+ ion has
a d10 configuration in which all the d-orbitals are filled, making
dxy> dxv dyi
& ie tr
/ JzL dxi-yi
/ / Ion in t e t r a h e d r a l
/ / fie ld
/ /
//
/■
Free ion
Fig. 16.5. Splitting of rf-electron energy levels in a tetrahedral field
the transition of electrons from one d-orbital to another impossible.

In the Cua+ ion (d9), one d-orbital is unoccupied. For the same reason
the ions Ag+, Zn2+, Cd2+ and Hg2+ which have a d10 electronic
configuration are colourless.
As already pointed out, the value of A, characterizing the strength
of the field produced by the ligands, depends on the nature of the
latter. Study of the spectra of complex compounds, making it pos
sible to determine the values of A, has demonstrated that ligands can
be arranged in the order of the descending strength of the crystal
field as follows:
C N ~> N 0 7 » th y le n e d ia m in e > NH3 > NCS~>
> H20 > F“ > COO-> O H -> Cl“ > B r -> I"
This series is for octahedral complexes but approximately the

same order is maintained for other configurations.
If the number of d-electrons in the completing ion does not exceed
the number of low-energy d-orbitals, they naturally occupy these
Ch. 16. C H E M I C A L B O N D I N - C O M P L E X C O M P O U N D S 245
orbitals. For example, the three d-electrons of the Cr3+ ion in the
octahedral field occupy the three low-energy d-orbitals
Jl iL
C r 3+ mTT
Because of this electronic configuration, trivalent chromium comple
xes are very stable since the electron clouds of the dE-orbitals are
situated between the ligands and weakly screen the charge of the
chromium nucleus. This explains why so many Cr3+ complexes are
known.
As we know, electrons are distributed among the quantum cells
(orbitals) in accordance with Hund’s rule: if there is a sufficient
number of cells, each will be occupied by a single electron (see
p. 57). This is explained by the fact that electrons repel from one
another and thus tend to occupy different orbitals. To transfer an
electron from an orbital where it is alone, to an orbital already
occupied by another electron requires the expenditure of a certain
amount of energy, P . The value of P can be determined by quantum-
mechanical calculation. Accordingly, if there are more electrons in
a complexing ion than low-energy orbitals, two situations may ari
se. If A < .P , the electrons of the central ion in the complex will
occupy the same orbitals as in the free ion; and the metal ion will
be in a state with a high spin; if A > P , the crystal field will cause
transition of electrons to occupied cells having a lower energy. As a
result of electron pairing the spin will diminish and the metal ion
will be in a state with a low spin.
In Table 16.2 are shown values of A for various complexes as
determined from their absorption spectra, calculated values of P
Table 16.2
Characteristics of Ions in an Octahedral Field
Configu p, Li A, Spin

ration Ion kcal/g-ion gands kcal/g-ion estate
d* Cr2+ 67.2 h 2o 39.7 High

Mn3+ 80.0 h 2o 60.0 Ditto
Mn2+ 72.8 h 2o 22.3 Ditto
Fe3+ 85.7 h 2o 39.1 Ditto
d® Fe2+ 50.3 h 2o 29.7 Ditto
CN- 94.3 Low
Co3+ 60.0 F- 37.1 High
- nh3 65.8 Low
d* Co2+ 64.3 h 2o 26.6 High
246 P A R T II I. STRUCTURE OF MOLECULES A N D C HE MI CAL B O N D
and information on the spin state of the ion obtained from magnetic
measurements. It is evident that the data given in the Table accord
with the principles set forth in the foregoing discussion. Thus the
crystal field theory establishes the quantitative relationship between
the magnetic and spectral characteristics of complexes—something
that cannot be done by the valence-bond method.
When the de- or d7-orbitals are not fully occupied, the symmetry
of the complex is disturbed, and this disturbance can be very great.
For example in the Cu2+ ion which has a d9 configuration, there is
only one electron in the d^-^-orbital in an octahedral environ
ment. This orbital screens the nuclear charge less than the others.
Fig. 16.6. Changes in ionic radii in Ca2+ —Zn2+ series

Therefore the four ligands located about it (see Fig. 16.2) will be
bound considerably more strongly than the other two, and the lat
ter can easily split off from the complex. Consequently the Cu2+
ion is characterized by a coordination number of four, and comple
xes have a flat structure.
The ligand field affects the most diverse properties of compounds
of the d-elements. It must be taken into account that ions in aqueous
solutions, as we shall see below, form hydrated complexes. In crystals,
ions are surrounded by their neighbours, i.e., they are in a state
similar to that found in complexes (it was from this that the term
‘crystal field theory’ originated). We shall confine ourselves to an
examination of regularities in changes in ionic radii.
As we know, the radii of ions of elements of the groups of transi
tion elements tend to diminish as the atomic number increases
(see p. 94). Nevertheless, although there is a general tendency to
diminution, the relationship between the ionic radius and the nuclear
charge is quite complicate. This relationship for the radii of doubly
charged ions of the Ca2+—Zn2+ group is shown in Fig. 16.6. This re
gularity is explained well by the crystal field theory. Actually, during
the transition from Ca2+ to V2+ the d-electrons occupy weakly scree-
Ch. 16. CHEMICAL BOND Iflt C O M P L E X C O M P O U N D S 247
ning d8-orbitals, which causes a marked decrease in the ionic radii

as the atomic number increases. In the Cr2+ and Mn2+ ions, the
strongly screening dv-orbitals are filled and the ionic radii do not
diminish as the atomic number grows, but increase. A similar pat
tern is seen in the following elements—electrons first fill unoccupied
places in the d8-cells and then in the dv-cells.
Since the size of the ionic radius affects the strength of the bond
in compounds, as well as acid-base properties and other characte
ristics, similar regularities are found in changes in these properties
in the series of d-elements.
16.7. Molecular Orbitals in Complex Compounds

Since the central particle in a complex is regarded as an ion in
the crystal field theory, results obtained in accordance With this
theory cannot be considered satisfactory when the bond between the
central atom and the ligands is far from being ionic. The inaccuracy
of the ionic model is evidenced by the sequence of the diminishing
strength of the crystal field (p. 244). In this series the CN” ion pre
cedes the F “ ion, although the F" ion is the lesser of the two, and on
the basis of electrostatics should be expected to have a greater
effect on the central ion.
The strictest approach to the problem of the nature of the bond in
complex compounds is to use the method of molecular orbitals (MO),
although this involves much greater difficulties than the crystal
field theory. Calculating the energy of the bond in complex com
pounds by the MO method requires high-capacity electronic com
puters. Calculations in accordance with the crystal field theory are
incomparably easier and are often resorted to in cases where they
are not strictly applicable, in order to obtain an approximate esti
mation.
In the MO LCAO variant, the wave function of the molecular
orbital is developed from the wave functions of the atomic orbitals
(AO) of the atoms forming the chemical bond. Every atom has many
AO—strictly speaking, an infinite number—and the more AO are
introduced into the MO, the more exact the calculation of the pro
perties of the bond by the MO LCAO method will be. However, it is
only AO whose energy does not differ too greatly that make an
appreciable contribution to the MO. The AO contribution to the MO
is characterized by the value of the coefficient ct with which the
AO wave function (p* enters the MO expression. For orbitals diffe
ring greatly in energy, the values of ct determined by the variational
method are very small, and roughly they can be considered equal
to zero-. Therefore, when developing the wave function by the MO
LCAO method, AO. which differ greatly in energy are usually not
included in the MO.
248 P ART I I I . STRUCTURE OF MOLECULES A N D CHEMI CAL BOND
This is easily explained. Assume that in atoms 1 and 2 there are

two AO which differ greatly in energy. Assume that the energy of
AO (1) is substantially lower than that of AO (2). This means that
an electron in AO (1) is attracted considerably more strongly by
the nucleus of atom 1 than is an electron in AO (2) by the nucleus of
atom 2. If electrons are added to the system of closely located nuclei
1 and 2, an electron that gets into AO (1) will be attracted by nucleus
1 and will not be shared, that is, it will not form a chemical bond.
When AO (1) has been filled with two electrons, the next electron
Fig. 16.7. Overlapping of s*- and

py-orbitals; ‘+* and *— signs
► of wave function
z
will go into AO (2), rather than a molecular orbital made up of

AO (1) and AO (2). Thus AO which differ greatly in energy do not
form a chemical bond.
This circumstance limits the set of AO introduced into the MO
wave function. Nevertheless, when examining complex compounds
the set is still very large. Take the case of an octahedral complex of
a d-element containing such simple ligands as F" ions: if AO are
limited to the orbitals of valence electrons, the MO expression must
include the orbitals of the central atom, namely, one 5-orbital, three
p-orbitals and five d-orbitals, i.e., nine orbitals in all, and four
orbitals from each ligand—one 5-orbital and three p-orbitals. Thus
the MO wave function for the given complex composed by the LCAO
method will consist of 33 terms. This means that the values of 33 c*
coefficients must be found, that is, that a system of 33 equations
with 33 unknowns must be solved. This is very difficult to do, even
with an electronic computer.
To simplify calculations when applying the MO LCAO method to
complex compounds, the symmetry of the atomic orbitals should be
taken into consideration. The fact is that all atomic orbitals, even
those close in energy, do not necessarily form a molecular orbital,
or in other words, form a chemical bond. Besides having close energy
levels, AO must have the same symmetry. In Fig. 16.7 is shown the
superimposition of s and p y atomic orbitals, the p y-orbital being
Ch. 16. CHEMICAL BOND I N COMPLEX COMPOUNDS 249
positioned perpendicular to the z axis, the line connecting the cen

tres of the atoms. Although there is mutual superimposition of the
orbitals, the overlap and exchange integrals are in this case equal to
zero since the contribution from the superimposition of the positive
part of the p y-orbital is cancelled by the contribution of the negative
part of the 5-orbital. The 5- and p y-orbitals have a different symmetry
with respect to rotation round the z axis; when rotated through 180°
the sign of the /^-orbital is reversed but that of the 5-orbital is not.
Moreover, the 5-orbital is superposed on itself when rotated through
any angle round the z axis, whereas the p y-orbital is only superpo
sed on itself when rotated through 180°. It follows that AO which
are unlike in symmetry need not be included in the MO, and this
greatly simplifies the expression for the MO wave function in com
plex compounds.
There is a rigorous mathematical theory treating the properties of
symmetry, which makes use of the conceptions and methods of the
section of higher algebra called the theory of groups. By means of the
theory of groups expressions are found for the wave functions of
molecular orbitals in complex compounds. In this book it is impos
sible to carry out this mathematical treatment, and we shall only
set forth its results for the case of the octahedral arrangement of the
ligands round the central atom which is characteristic of many com
plex compounds.
The molecular-orbital wave function, i|?mo» in a complex is a
linear combination consisting of the orbital of the central atom,
cpM, and a linear combination of certain orbitals of the ligands,
2 c(pL, which is called the ligand group orbital, the word ‘group’
indicating that the aggiegate of these linear combinations meets
the requirements of the theory of groups. Thus
li)MO= aq>M+ P 2 <;<PL (16.1)
in which a and are coefficients selected, like coefficients c, by the
variational method. Here we shall examine the results of the maxi
mum simplification of the MO LCAO variant in which the overlap
integrals are considered equal to zero, as in the Hiickel method (see
p. 206 and 211). It then follows from the conditions of normaliza
tion that
a 2 + P2 = 1
2 c2= i
and if the bond between the central atom and the ligands is fully
covalent
The plus sign in equation (16.1) corresponds to a binding orbital;

the minus sign, to an antibinding orbital.
250 P A R T I I I . STRUCTURE OF MOLECULES A N D C H E MI C A L BOND
Table 16.8
Molecular cr-orbitals in Octahedral Complexes
MO desig
nation AO of metal Group orbital of ligands
1
a lg ns -j/g- (a l + a 2 + a 3 + a 4 + a 5 + a e)
nPx
V2
tlu
npy
V 2 ' (a* - ° 4)
1
npz
1 /2
( n — \ ) d z2
l
2 *1/ 3- [2(a5 + afl) — (al + a2+ a3+ a4)]
eg
(n — l ) d x 2 _ y2
1
2 (a l + a 3 — a 2 — a 4)
In Table 16.3 are shown the orbitals of the central atom and the
group orbitals of the ligands which form molecular a-orbitals in
octahedral complexes. The letter a denotes orbitals of ligands, which
overlap with orbitals of the central atom according to the a type,
Fig. 16.8. Arrangement of coor

dinates and numeration of lig
ands in an octahedral complex
i.e., which are located along the lines connecting the ligands with
the central atom. For example, if the ligands are ammonia molecu
les, a denotes one of the sp3-hybrid orbitals of the nitrogen atom
(see p. 186) occupied by the unshared pair of electrons and directed
toward the metal atom. The indices correspond to the numeration
of the ligands given in Fig. 16.8.
Ch. 16. C H E M I C A L B O N D J N COMPLEX COMPOUNDS 251
The designations of the orbitals (aig, etc.) are taken from the
theory of groups, in which the types of symmetry (the so-called
irreducible representations) to which the aggregates of the group orbi
tals of the ligands belong are similarly designated.
When Table 16.3 and Figs 4.7 and 16.8 are compared it is easy to
see that the only orbitals of the ligands which are included in the
group orbital are those which overlap with the corresponding orbi
tal of the metal according to the a type. Thus the 5-orbital of the
metal overlaps equally with the orbitals of all six ligands, while
the /?x-orbital overlaps only with the orbitals of ligands 1 and 3,
and so on.
It is also evident from a comparison of Fig. 4.7 and 16.8 that metal
orbitals dxy, dxz and dyz cannot overlap with the ligand orbitals
according to the a type. These orbitals are denoted t2g. They can
overlap according to the jt type with those ligand orbitals which
have a suitable symmetry.
Many ligands, including NH3, H20 and halide ions, have no
orbitals with an energy close to that of the ^-o rb itals of the cen
tral ion. No jx bonds are formed between these ligands and the metal
atom.
Electrons occupying ^-orbitals in such complexes have an energy
differing but little from their energy in the unbound metal atom.
Approximately, it can be considered that they remain in their ato
mic orbitals. In a more rigorous treatment, it is considered that
electrons occupying dxy-, dyz- and d*z-orbitals in the free metal atom
are transferred to t2g nonbinding molecular orbitals when the com
plex is formed, the ^-orbitals differing little from the atomic orbi
tals in energy and the form of the electron cloud.
In Fig. 16.9 is given a diagram of the MO energy levels in the
octahedral complex. The transfer of an electron from a nonbinding
£2g-orbital to an eg antibinding orbital corresponds to energy change
A. Thus, whereas it is assumed in the crystal field theory that the
electron transfer, the energy A of which is determined from spectral
data, takes place between an AO of low energy (dxy, dxz, dyz) and an
AO of higher energy (dz2, dx2-^ 2), it is considered in the MO theory
that the transfer is from a nonbinding MO similar to the AO dxy,
dxzy dyzi to an antibinding MO formed from a dz2 or dx2_y2 AO.
A number of ligands, such as CO, CN", etc., have orbitals which
can overlap with dxyi dxz and dyz metal orbitals according to the n
type. These are the antibinding MO in such particles which are not
occupied by electrons. Orbitals of the CO molecule are presented
diagrammatically in Fig. 16.10. Orbitals occupied by electrons are
hatched. Similar MO are found in the CN" ion. The CO and CN"
particles are isoelectronic, they contain the same number of ele
ctrons and differ only in the charge on one of the atoms; for oxygen
Z = 8, while for nitrogen Z = 7. The MO in these particles which
252 P A R T I I I . STRUCTURE OF MOLECULES A N D CHEMICAL BOND
are occupied by unshared pairs of electrons are close to sp hybrid

AO. They form a bonds with metal atoms. On the other hand, the
unoccupied antibinding MO in CN“ or CO form jt bonds with the
dxy-, dxz- and d^-orbitals of metals. A diagram representing the
formation of these bonds is shown in Fig. 16.11. This results in a
flO o f metal MO o f complex orbi ia ls of

ligands
Fig. 16.9. Diagram of orbital energy levels in an octahedral complex. The cells
show the number of orbitals each level contains
very strong linking of the ligand with the central atom. Metal-li
gand bonds in complex cyanides and carbonyls (compounds of
metals with CO) are very stable. Since the /2g-orbital in such com
pounds becomes binding, its energy level falls and energy differen
ce A increases. This explains the position of CN“ in the series shown
on p. 244.
The transfer of the electrons of the metal to an antibinding orbital
of the ligand makes the bond between the atoms composing the li
gand less stable. This can be detected experimentally. Study of mole-
cular spectra and determination of the structure make it possible to

find the interatomic distance r0 and force constant k of the bond
(p. 147). The less the r0 and the greater k , the stronger the bond. It
has been found that the C — 0 bond in carbonyls is longer and k
Fig. 16.11. Diagram showing

formation of jt bond between
metal atom and CO molecule
Fig. 16.10. Diagrammatic rep

resentation of molecular orbitals
in CO molecule
1 —binding Jt-orbital; 2 —antibinding
Jt-orbital; 3— a-orbital occupied by
unshared electron pair
is less than in free CO. Thus for the CO molecule the values of r0
and k are 1.13 A and 18.6 dyne/cm, while for the G = O bond in
Ni(CO)4 they are equal to 1.15 A and 16.2 dyne/cm, respectively.
Fig. 16.12. Structure of

[Pt (C2H4)C13]- ion
A large number of complex compounds are known in which the

ligands are uncharged and in many cases non-polar molecules con
taining Jt bonds. Such compounds are called n-complexes. Examples
are K[Pt(C2H4)Cl3] (Zeise salt), Fe(C5H 5)2 (ferrocene) and Cr(C6H6)2
(dibenzenechromium). Since the ligands in these complexes are
completely non-polar, the formation of the latter cannot be explained
either by simple electrostatic conceptions or on the basis of the
254 P A R T III. S T R U C T U R E OF M O L E C U L E S A N D C H E M I C A L BOND
crystal field theory. Nevertheless, the bond in these compounds is

easily explained by the MO theory.
The configuration of the [Pt(C2H4)Cl3]~ ion as established by X-ray
analysis is represented in Fig. 16.12. The C2H 4 molecule is arranged
perpendicular to the plane in which the chlorine atoms lie. The
configuration of the molecular orbitals binding Pt and C2H 4 is
diagrammatically presented in Fig. 16.13. The AO are shown with
a solid line, and the MO formed from them, with a dotted line.
The hatching indicates that ele
ctrons occupy the AO. It is evi
dent that in this case both the
binding and antibinding jt-orbi-
tals of C2H 4 are involved in for
ming the bond. Development of
the wave functions of these orbi
tals was described on p. 216.
Comparison of Fig. 16.13 with
equations (14.46) and (14.47)
is advisable.
Fig. 16.13. Diagram showing forma

tion of bond in n complexes
1 —formation of bond through overlapping
of binding n MO of ligand and p-orbital
of metal; 2 —formation of bond through
overlapping of antibinding n MO of ligand
, and d-orbital of metal
In Fe(C5H5)2 and Cr(C6H0)2 the metal atom is located between

two flat, cyclic molecules, and these complexes are therefore said to
be sandwich compounds. In this case the bond is likewise folmed by
overlapping of the d-orbitals of the metal and the. lobes of the p-
orbitals of the carbon, which have different signs and from which the
MO in C5H5 and C6H6 are formed.
CHAPTER SEVENTEEN
THE HYDROGEN BOND
When hydrogen is joined to a strongly electronegative element it
can form an additional bond, though this is considerably less stable
than the usual valence bond. The ability of the hydrogen atom to
bind two other atoms in a number of cases was demonstrated by
M. Ilyinsky and N. Beketov in the ‘eighties of last century. Several
decades passed, however, before the conception of this bond, which
is intermediate between chemical and molecular, was finally esta
blished, receiving the name hydrogen bond.
Ch. 17. HYDROGEN BOND 255
The hydrogen bond is caused by the displacement of the electron

from the hydrogen atom, transforming it into a particle with unique
properties. If it is considered a cation, it (a) has no electron and the
refore, unlike other cations, is not repelled by the electron clouds of
other particles, and is only subject to attraction, and (b) is infini
tesimal in size, other ions being thousands of times larger than the
proton.
The more electronegative the atom to which hydrogen is joined, and
the smaller it is, the more strongly the hydrogen bond is manifested.
Consequently, it is characteristic, above all, of fluorine compounds as
well as compounds of oxygen, and to a lesser degree, nitrogen,
followed by chlorine and sulphur. The energy of the hydrogen bond
varies accordingly, depending in general on the species and state
of the atom to which hydrogen is joined, and also on the former’s
neighbours. Thus the energy of the hydrogen bond H...F (here and
below this bond is denoted with dots) is ~10 kcal, while the energies
of the H ...0 and H...N bonds are, respectively, ~ 5 and ~ 2 kcal.
Neighbouring electronegative atoms can likewise activate atoms of
CH groups to form a hydrogen bond, although the electronegativi
ties of carbon and hydrogen are almost the same. This explains the
appearance of hydrogen bonds in such compounds as HCN, CHF3
and the like.
As the hydrogen bond becomes stronger, relevant distances become
smaller. Thus, whereas in H20 the distance 0 — H is less than H...O,
the distance H — F and F...H are the same in hydrogen fluoride,
that is, the proton is situated midway between fluorine ions.
Through hydrogen bonds, molecules combine to form dimers and
polymers. The latter may have a linear, branched or ring structure.
Formic acid, for example, exists mainly in the form of a dimer, both
in the liquid and gaseous phase. The dimer has the following stru
cture, which has been established by electrono'graphy:
Association distinguishes water, ammonia, alcohols and many other

liquids from unassociated liquids, such as hydrocarbons. Associati
on causes a rise in melting points, boiling points and heats of vapo
rization, as well as changes in solvent properties, etc.
As noted above, the energy of the hydrogen bond is low, it is an
order less than the energy of the chemical bond. Therefore, raising
the temperature ruptures hydrogen bonds, although the process
extends, as a rule, over a comparatively long temperature range.
256 P A R T I I I . STRUCTURE OF MOLECULES A N D CHEMI CAL BOND
In carboxylic acids, for instance, association persists even during

vaporization.
In a series of similar substances it would be natural to expect a
rise in the melting and boiling points anA an increase in heats of
vaporization as the molecular weight increases, but as can be seen
Table 17.1
Melting Points, Boiling Points and Heats,
of Vaporization (at boiling point)
of Some Substances
Substance m. p., °C b. p., °C AHeap' kcal/mole
h 2o 0.0 100.0 9.75

h 2s - 8 5 .5 - 6 0 .7 4.50
H2Se - 6 4 .8 - 4 1 .5 5.1
H2Te -4 9 .0 - 2 .0 5.8
HF -8 3 .1 -1 9 .5 7.20
HG1 -1 1 2 - 8 4 .9 3.6
HBr -8 7 -6 6 .8 3.9
HI - 5 0 .9 - 3 9 .4 4.2
from Table 17.1, these values are higher for H 20 than for H2S,
and higher for HF than for HC1. This is because there are strong
hydrogen bonds between the H20 molecules, and between the HF
molecules. The scale of this effect is shown by the curve in Fig. 17.1.
The most convenient indicator for the hydrogen band is the boi
ling point since it is easily measured. On determining the boiling
point of an alcohol ROH and the corresponding mercaptan RSH we
would see that it is higher for ROH than for RSH. Ethers, even those
with a higher molecular weight than alcohols, are more volatile.
If water were not associated it would have a melting point of about
—100°C and a boiling point of about —80°C, which is evident from
a curve similar to that in Fig. 17.11 which can be drawn using the
figures given in Table 17.1
If one employs the method of comparative calculation and compa
res the boiling point in the groups HR (R = F, Cl, Br and I) and
H2R' (R' = 0, S, Se and Te) (Fig. 17.2), one can conclude from the
character of the deviation of the point for HF — H 20 and from the
fact that water vapour molecules are practically not associated, that
unlike water the association of hydrogen fluoride persists in the
1 These are approximate values, since such extrapolation does not take account
of certain peculiarities in the properties of compounds of elements of the second
period.
Ch. 17. HYD ROGEN BOND 257
vapour phase, otherwise one would expect all four points to lie on
a straight line. This shows the greater stability of the H...F bond as
compared to the H ...0 bond. This conclusion is confirmed by the
perceptibly lesser difference in the heats of vaporization of HF and
Fig. 17.1. Relationship between Fig. 17.2. Comparison of boiling points

heat of vaporization and molecular in HR and H2R' series
weight for hydrogen compounds of
Group VI elements
HC1 as compared to H20 and H2S (Table 17.1). Actually, (HF)n

molecules exist in hydrogen fluoride vapour which have the stru
cture
Although in most of the particles n = 4, there are some in which

n = 5 or 6.
All the examples considered so far are cases of an intermolecular
hydrogen bond. Often the hydrogen bond joints parts of a single
molecule, that is, the bond is intramolecular. This is characteristic
of many organic compounds. In most cases the hydrogen atom forms
part of a flat, six-membered ring. If the formation of such a ring is
hindered, an intramolecular hydrogen bond is not formed. Here are
17 3aK . 15648
258 P A R T III. STRUCTURE OF MOLECULES A N D CHEMI CAL B OND
some examples of an intramolecular hydrogen bond
Salicyl aldehyde Ortho-nitro- 2,6-Dihydroxy- Ortho-ethy-

phenol benzoic acid nyl phenol
Whereas in o-nitrophenol the hydrogen bond is intramolecular,

it is intermolecular in /z-nitrophenol since the hydrogen is remote
from the oxygen of the nitro group. The dissociation constant of
2,6-dihydroxybenzoic acid at 25°C is 5 X 10“2, which is 550 times
greater than the dissociation constant of 3,5-dihydroxybenzoic
acid. This is explained by the fact that in the latter compound the
Fig. 17.3. Environment of wa

ter molecule in ice
hydrogen bond is practically not manifested, which strengthens

the 0 — H bond in the carboxyl group. The ortho-ethynylphenol
molecule is noteworthy in that formation of the hydrogen bond
involves the Jt-electrons of the triple bond.
Taking into account the effect of the hydrogen bond has provided
a key to many facts which are otherwise difficult to understand.
Thus formation of salts of the type KHF2 and NaHF2 is explained
by the existence of the stable ion HF~, formed through the process
H2F2 H + + HF". Actually, the equilibrium HF + F ” ^ HF"
is displaced to the right (K298 = 5.1). The energy of the hydrogen
bond in F — H ...F" comes to 27 kcal/mole. The influence of the
hydrogen bond also makes it understandable why hydrofluoric acid,
unlike its analogues, HC1, HBr and HI, is not a strong atcid, its
dissociation constant being 7.2 X 10”4.
Hydrogen bonds are of great importance in determining the stru
cture of water and ice. A fragment of ice structure is shown in
Ch. 17. H Y D R O G E N B O N D 259
Fig. 17.3. Each oxygen atom in this structure is tetrahedrally bon

ded with four other atoms, and between them are arranged the
hydrogen atoms. Two of the latter are joinedo to the given oxygen
atom with a polar, covalent bond (d = 0.99 A); two others, with a
hydrogen bond (d = 1.76 A; Eo . . . h ~ 5 kcal/mole), that is, they
form part of two other molecules of water. This makes the structure
loose, not tightly packed. This accounts for the small density and
considerable porosity of ice. When ice is melted the hydrogen bonds
are broken to the extent of about 10%; and the molecules are drawn
somewhat together. Water is therefore slightly denser than ice.
Heating water causes it to expand, that is, it increases the volume,
hut on the other hand, it causes further disruption of the hydrogen
bonds, which should reduce the volume. As a result the density of
water exhibits a maximum at 4°C.
The hydrogen bond likewise plays an important part in processes
of solution, since solubility depends, for one thing, on the ability
of a substance to form hydrogen bonds with the solvent. Interaction
products called solvates are often formed, the solution of alcohols
in water being an example. This process takes place with the libe
ration of heat and a reduction in volume, which are indications of
the formation of compounds. In such cases it is impossible to speak
of the formation of solvates through electrostatic attraction by ions,
of the polar molecules of the solvent, since we have to do with the
solution of non-ionizing compounds. The absence of the influence of
a hydrogen bond can likewise explain cases where polar compounds
are insoluble in water. Thus polar ethyl iodide is a good solvent for
non-polar naphthalene, but itself is insoluble in such a polar sol
vent as water.
The problem of the nature of the hydrogen bond has not been
definitely solved. It is clear that this involves interdipole intera
ction, and the effect of polarization, as well as a donor-acceptor
mechanism. The difficulty of the quantum-machanical calculation
of the hydrogen bond is due to the fact that the error in calculation
is substantially greater than the value of the energy of the hydrogen
bond. Evidently the most promising results can be expected from
the molecular orbital method.
The hydrogen bond is encountered almost universally—in organic
crystals, since they contain C, H and O; in proteins, since they con
tain C, H and N; in polymers and in living organisms. It is sugge
sted that memory involves the storage of information in configura
tions containing hydrogen bonds. The ‘universality’ of the hydrogen
bond springs from the fact that water molecules are found everywhe
re, and each of them, possessing, as it does, two hydrogen atoms
and two unshared electron pairs, can form four hydrogen bonds.
17*
PART IV
THE STRUCTURE OF MATTER

IN THE CONDENSED STATE
CHAPTER EIGHTEEN
INTRODUCTION
18.1. Aggregate States

Matter can be in the solid, liquid or gaseous state, depending on
the distance between particles and on the forces acting between
them.
At a sufficiently low temperature a substance is in the solid sta
te1. The distances between the particles of a crystalline substance
are of the order of the size of the particles. The mean potential ener
gy of the particles is greater than their mean kinetic energy. The
movement of the particles making up crystals is extremely limited.
The forces acting between particles keep them close to equilibrium
positions, and consequently the probability of particles being
found in these spots is maximum. It is to this that crystalline bo
dies owe their shape and volume and great resistance to deforma
tion.
When crystals melt, a liquid is formed. A liquid differs from a
crystalline substance in that not all the particles are situated at
distances of the same order as in crystals; some of the molecules
are at substantially greater distances. In this state the mean kinetic
energy of the particles is roughly equal to their mean potential
energy.
The solid and liquid states are often grouped together under the
general term condensed state.
As a result of evaporation or boiling a liquid is transformed into
the gaseous state. In this state particles are at distances greatly
exceeding their size, and for that reason forces acting between them
are very small. Particles can move about freely. Whereas in a solid
all the particles form a single aggregate, and in a liquid, a great
number of large, stable aggregates, only particles consisting of two
to five molecules can be found in gases, and their number is usually
comparatively small. The mean kinetic energy of the particles of
a gas is substantially greater than their mean potential energy. Con
sequently forces of attraction between them are insufficient to hold
them near one another.
1 Helium is an exception. For a discussion of the amorphous state see p. 303.
The state and properties of an individual substance are determined

by the temperature and pressure. If the pressure is low and the tem
perature sufficiently high, the substance will be in the gaseous state;
at low temperatures it will be a solid, and at moderate temperatu
res, a liquid. Accordingly the phase diagram of a substance consists of
three fields (Fig. 18.1), corresponding to the crystalline (C), liquid
(L) and gaseous (G) states. These regions are separated from one
another by the melting (crystallization) curve Ob, boiling (condensa
tion) curve OK and sublimation
(desublimation) curve Oa. Point
O at which they meet is called P b
the triple point: at P = P 0 and
T = T q the substance exists in
three aggregate states simulta
neously. Point K at the end of
the boiling curve is called the
critical point: at P = PK and
T —T k the boiling liquid and
dry saturated vapour are abso
lutely indistinguishable.
Fig. 18.1. Phase diagram of a single

component system
From Fig. 18.1 it can be seen that at a pressure greater than P 0

but less than PK, for example, at P i (point c) isobaric'heating of a
solid will cause it to melt (point d). After all the substance has mel
ted, further heating will again cause the temperature to rise (process
de). At point e the liquid will boil and the temperature will again
cease to rise. When all the liquid has been vaporized, further heating
will overheat the vapour (process ef). The length of time at which the
temperature remainstconstant (points d and e) is determined by the
amount of the substance and its nature, assuming that other condi
tions remain unchanged. The greater the amount of substance and
the greater the heat of fusion and heat of vaporization, i.e., the
greater the bond energy in the solid and liquid phases, the more
protracted the isothermic ‘crossings’ of the Ob and OK curves
will be.
When P <1 P 0» for example, at P 2 (point g), a solid will be trans
formed directly into the gaseous state (at point h), i.e., sublimation
will take place. For most substances P 0 < 1 atm. In the case of the
few substances for which P 0 > 1 atm, such as carbon dioxide,
heating the crystals at P = 1 atm will cause sublimatioh.
Finally, at P > PK» for example, at P 3 (point &), further heating
after crystals have melted (point I) will put the substance into the
262 P A R T I V. S T R U C T U R E OF M A T T E R IN CONDENSED STA TE
supercritical state. This term simply emphasizes the fact that in

this region, for example, at point m, liquid and gas are indistin
guishable.
The higher the pressure, the higher the temperature at which mel
ting, vaporization and sublimation take place. A few substances
present exceptions, among them H20 and Bi; in their case increa
sing the pressure lowers the melting point.
We have considered the simplest phase diagram. If a substance
has several modifications the phase diagram is complicated, since
for each additional phase there is an additional region.
A direct or reverse sequence of the transformations ‘solid-liquid-
gaseous’ is not universal. It was pointed out that there are cases
where heating a solid converts it directly into the gaseous state.
There are likewise cases where the sequence is broken as the result
of a chemical change. Thus when crystalline MgC03 is heated it
will, at a certain temperature, decompose into MgO and C02, not
melt, the decomposition point being the higher, the greater the
C02 pressure. It is also impossible to condense N 02 from the gaseous
state into a liquid or solid since cooling or compression causes the
formation of a different substance—N20 4.
We return to the gaseous state. The higher the temperature and
the lower the pressure, the greater the independent motion of the
gas particles will be. At the limit where P ->0, the interaction bet
ween the particles, as well as their volume compared to the space
occupied by the gas become negligible. Such a gas is said to be per
fect.
Naturally the relationship between pressure P, temperature T
and volume V of a perfect gas is described by the simplest equation,
that of Mendeleev—Clapeyron
PV = nRT (18.1)
in which n is the number of moles of gas in volume V; T , the absolu
te temperature; and /?, the universal gas constant. For one mole
Pv = R T (18.2)
here v is the molar volume.
In conditions of moderate and high pressures and low temperatu
res, i.e., when compression involves a volume not occupied by the
particles themselves, equation (18.2) should be replaced by the
equation
P ( v - b ) = RT (18.3)
in which the correction term b takes into account the volume of the
molecules and their mutual repulsion at short distances. The equ
ation P (V — b) = const might be called Lomonosov’s equation
since it was he who first pointed out that “...the density of air1 under
great compression is not proportional to its pressure”, and explained
this fact as being due to the ultimate size of the particles. Actually,
it is only at the assumption that 6 = 0, that at a given temperature
PV = const or d = const' X P, where d is the density of the gas.
Moreover, in the nonperfect gas the forces of mutual attraction
of the molecules must be taken into account, that is, we must employ
an equation of the form
(P + Pint) ( v - b ) = R T (18.4)
in which the correction term Pint is introduced to take into account
the mutual attraction of the molecules, which is called internal
pressure. Equation (18.4) in which this term is directly proportional
to the square of the density is called Van der Waals’ equation. It is
one of the earliest and most thoroughly studied equations of the state
of real gases. In all, about 200 of these equations have been propo
sed, which shows the difficulty of giving a precise description of
the properties of real gases.
If such great energy is imparted to a gas that electrons begin to break away
from its molecules, there will be positively and negatively charged particles in
the space occupied by the gas. Thermal ionization takes place and as a result the
gas will become a conductor of electricity and will go over to the plasma state.
There is no sharp boundary between gas and plasma but plasma appears as
soon as the substance finds itself in a magnetic field because in this case the
motion of the charged particles becomes orderly.
18.2. Molecular Interaction

When studying the properties of substances much attention is
given to intramolecular interactions due to the action of valence or
chemical forces and characterized by saturation, great energy chan
ges and specificality, but along with this the interaction between
the molecules of a substance must be taken into account. In pro
cesses of gas expansion, condensation, adsorption, solution, etc.,
the action of these forces makes itself felt. They are often called
Van der Waals’ forces, which emphasizes that their existence expla
ins the difference between real gases and a perfect gas, the diffe
rence of the value of P int in equation (18.4) from zero.
Molecular interaction is of an electric nature; it differs from che
mical interaction in that it is manifested at comparatively great
distances and is characterized by the absence of saturability and
specificality, and by small energies. To give an idea of the magnitude
of these forces it can be noted that the heat of condensation of va
pour into liquid, which characterizes the energy of interaction bet
ween the vapour molecules, is relatively small, for example, for
1 Air was the only gas known in the first half of the 18th century.
264 P A R T I V. S T R U C T U R E OF M A T T E R I N C O N D E N S E D S T A T E
HI it is about 5 kcal/mole. The energy of intermolecular forces in

the liquid is of the same order. The energy of chemical interaction
is much greater; thus, the energy of the H — I bond exceeds
70 kcal/mole.
At comparatively large distances r between molecules where
their electron subshells do not overlap, only the action of the attra
ctive forces is manifested.
If the molecules are polar, electrostatic interaction is exhibited.
This is called the orientation effect, and the greater the dipole moment
(lx of the molecules, the greater the effect. Raising the temperature
should weaken this interaction since thermal motion tends to disturb
the relative orientation of the molecules. The attraction of polar
molecules decreases rapidly as the distance between them grows.
In the simplest case, theory (Keesom W., 1912) gives the following
relationship for the energy of the orientation effect:
2\i*N0
3RTr* (18.5)
in which N 0 is Avogadro’s number; R> the universal gas constant;

and T, the absolute temperature. This equation is quite exact at
high temperatures and low pressures, where the distance between
the dipoles is substantially greater than their length.
If the molecules of a substance are non-polar, there is no orienta
tion effect. But if molecules come into the field of neighbouring
particles (molecules, atoms or ions) they are polarized, and an indu
ced dipole moment appears. The easier the molecules are deformed,
the greater the induced effect (see p. 154). The interaction energy of
such molecules increases with the increase in \i and quickly dimini
shes with the increase in r, but it does not depend on temperature,
since the induction of dipoles takes place whatever the spatial
arrangement of the molecules. Theory (Debye, 1920) gives the
following relationship for the energy of the induction (deformation)
effect of two like polar molecules:
2a|i2
Uind — r« (18.6)
where a is the polarizability.

Intermolecular attraction is not limited to these two components.
The orientation and induction interactions are only part of the Van
der Waals attraction, and for many compounds they are the smaller
part. For such substances as Ne and A both components are equal
to zero—the particles of these elements are non-polar and their
electron shells are very rigid; nevertheless the noble gases can be
liquefied. This shows that there must be another component of
intermolecular forces. What is its nature?
Let there be two atoms of a noble gas. If the static distribution of

their charges is considered, it would seem that these atoms should
have no effect on one another. But experience and the quantum theo
ry have it that under all conditions, even at absolute zero tempera
ture, the particles in an atom are in constant motion. During the
movement of the electrons, distribution of the charges within the
atoms becomes unsymmetrical, resulting in the appearance of ^momen
tary dipoles. As molecules approach one another, the motion of
these dipoles ceases to be independent, thus causing attraction.
The interaction of momentary dipoles is the third source of inter-
molecular attraction. This effect, which is quantum-mechanical in
character, is called the dispersion effect, since the oscillations of
electrical charges cause the dispersion of light—different refraction
of light rays of different wavelength. The theory of the dispersion
effect was elaborated by London in 1930. From the above discussion
it follows that dispersion forces act between the particles of all
substances. Their energy is approximately expressed by the equation
3hv0a2
U di sp —
4r6
(18.7)
where h is Planck’s constant; v0, the frequency of oscillations corres
ponding to zero energy E 0, i.e., the energy at T 0 (the zero energy
of a vibrating particle is expressed by the relationship E 0 =
and a, the polarizability. The value of hv0 can be considered appro
ximately equal to the ionization potential.
Table 18.1
Values of C7>8-1060 (erg-cm6) Characterizing
the Orientation, Induction and Dispersion
Interaction Between Like Molecules
Interaction
Molecules
orientation induction dispersion
CO 0.0034 0.057 67.5

HC1 18.6 5.4 105
HBr 6.2 4.05 176
HI 0.35 1.68 - 382
nh3 84 10 93
h 2o 190 10 47
In Table 18.1 are given the components of Van der Waals’ forces
for certain substances. These figures show: (a) that the dispersion
effect is high in value and is foremost for non-polar and slightly
266 . P A R T I V. S T R U C T U R E OF M A T T E R I N C O N D E N S E D S T A T E
polar molecules; (b) that for strongly polar molecules the contri
bution of the orientation effect is large; and (c) that the induction
effect is usually of minor significance. It only becomes important
when polar molecules occur together with strongly polarizable mole
cules. Thus, as a result of polarization interaction, nitrobenzene
forms with naphthalene the molecular compound C6H 5N 02 ‘C^Hs.
A great many compounds of this type are known.
Adding together the orientation, induction and dispersion ener
gies and combining all the constants in conformity with equations
(18.5), (18.6) and (18.7) we obtain
the energy of intermolecular
attraction
U a, t r = - $ (18.8)
where
2\i*N0 2 , 3a2hv0
11 = -TTn Tr.- + 2a\JL‘
SRT 4~“
Thus the attractive forces are
inversely proportional to the
intermolecular distance to the
seventh power.
When distances between mole
cules are small and their electron
Fig. 18.2. Potential energy curves of

intermolecular interaction
subshells overlap intensely, i. e., when the electrostatic repulsion

of the nuclei and electrons becomes greater than their mutual attra
ction, the action of repulsive forces is manifested. Many facts point
to the existence of these forces, one of them being the small compres
sibility of liquids and solids.
The repulsive energy can be expressed to a first approximation
by the equation
UreP= -— (18.9)
which is similar to (18.8); here m is a positive constant—the repul
sion constant. It is evident from equation (18.9) that the repulsive
forces begin to be manifested at very small distances and increase
rapidly as r diminishes.
The total interaction energy between molecules is equal to
U = U a ttr + U r<p (18.10)
Ch. 19. C R Y S T A L L I N E S T A T E 267
or in conformity with (18.8) and (18.9)
= —£ + ■£ (18-ID
This equation is called the Lennard-Jones formula (1924)1. It cor
responds to the curves in Fig. 18.2. The minimum on the summary
curve represents energy of molecular interaction C/0 and equilibri
um distance between the molecules, r0. These curves resemble the
curves characterizing the relationship between the energy of ionic
interaction and the interionic distance (see Fig. 15.1). Nevertheless,
the curves in Figs 15.1 and 18.2 greatly differ in the quantitative
aspect since both the scale of the values of U and r, and the rela
tionships between energies Uattr and Urep and distance r are dif
ferent.
CHAPTER NINETEEN
THE CRYSTALLINE STATE
19.1. Characteristics of the Crystalline State

The word ‘crystal’ is always associated with the image of a poly
hedron of some kind. Crystalline substances, however, are characte
rized not only by their occurrence as bodies of definite form but by
their anisotropy or the vectorial character of their properties—the fact
that such properties as tensile strength, thermal conductivity,
compressibility, etc. depend on the direction through the crystal.
This is best explained by examples. A figure of exactly the same
form as a crystal of calcite (CaC03) can be made from glass, but the
difference between the calcite crystal and its glass model is easily
detected; calcite, unlike glass, exhibits double refraction (see Appen
dix IV). Here is another example. A ball can be fashioned from rock
salt which has all the appearance of a glass ball, but a simple test
will show that the ball has been made from a crystalline substance.
If the ball is heated and placed on a slab of paraffin wax, the para
ffin round the ball will melt unevenly because the thermal conducti
vity of an NaGl crystal is different in different directions.
We must now take up briefly some matters relating to the form
of crystals.
The forms of crystals are the subject of geometric crystallography.
This science began to develop in the 18th century. It is based on two
laws: the law of the constancy of interfacial angles and the law of ratio
nal indices.
1 Two limitations should be noted: (1) equation (18.11) can be considered
only as a first approximation since when r is of the order of 1 A, equations
(18.5) —(18.7) are not exact; (2) everything set forth above is only partially
applicable to molten salts and metals.
Fig. 19.1. Forms of quartz crystals
Fig. 19.2. Formation of face of an octahedral crystal consisting of unit cells of

cubic form
Fig. 19.3. Illustration of the law of rational indices. Sections OC and OC cut
off on z axis by faces CB and C'B' are in the ratio of 2 : 1
The first law, which was formulated by Rome de L’Isle (France)

in 1783, states that in all crystals of the same substance, the angles
between corresponding faces are equal. For example, in sodium chlori
de crystals all the angles between faces are equal to 90°. This does
not mean that crystals of the same substance always have the same
form. This can be illustrated by quartz crystals (Fig. 19.1)1. In spite
of the fact that the crystals presented in the Figure are different in
form, the angles between corresponding faces, e.g., a and b or b and
c are equal.
According to the law of rational indices, discovered in 1784 by
R. Haiiy (France), the faces of a crystal are always oriented in space
in such a way that the ratios of sections cut off on the three coordinate
axes of a crystal by one of the faces to sections cut off on the same axes
by another face are integers. Haiiy explained this as being due to the
fact that crystals are constructed of particles having the form of poly
hedrons. Fig. 19.2 shows the formation of the face of a crystal con
sisting of cubes, and the diagrams in Fig. 19.3 show how two faces
of a crystal constructed of cubes cut off sections OC and OC' on the
z axis, which are related to one another in the ratio of 2:1.
Haiiy’s law of rational indices which showed the discrete structure
of matter preceded Dalton’s investigations and influenced the for
mation of his views. The formulation of this law can be considered
an important landmark in the development of atomic-molecular
theory.
It goes without saying that the particles of which crystals con-,
sist—atoms, ions or molecules—are not cubes or parallelepipeds, but,
as we shall see below, are arranged in the crystals in an orderly
way, forming crystal lattices, which can be considered as consisting
of unit cells having the form of parallelepipeds.
Despite the great diversity of crystal forms they can be rigidly
and unambiguously classified. This systematization was introduced
by A. Gadolin, Russian Academician, in 1867; it is based on the
symmetry characteristics of crystals.
Symmetrical geometrical figures have one or several elements of
symmetry, they have centres, axes or planes of symmetry. Centre
of symmetry C is a point which bisects every straight line passing
through it and drawn to its intersection with the boundaries of the
figure (Fig. 19.4). The plane of symmetry divides a figure into two
parts, which are mirror images of one another (Fig. 19.5). The axis
of symmetry is a line such that the figure when rotated on it through
an angle of 360° coincides with itself n times (Fig. 19.6). The num
ber n is called the order of the axis. Axes are said to be of the second
order, third order, and so on (axes of the first order are not conside-
1 Figure 19.1 was taken from G. Boky's 4Kristallokhimiya\ Moscow, 1960;
the author stated that the illustration was taken from Rome de LTsle’s mono
graph (1783).
red because a figure will coincide with itself when rotated through
360° on any line). Besides the usual axes of symmetry, there are
inversion axes and axes of mirror rotation symmetry. For a figure to
coincide with itself when such axes are present, revolution round
the axis must be accompanied by rotation through 180° round another
axis which is perpendicular to the first (inversion), or by a mirror
reflection from the plane. Examples of figures having such axes of
symmetry are shown in Fig. 19.7.
In 1867 A. Gadolin mathematically demonstrated that 32 types
of crystal symmetry were possible, each of them characterized by a
B
Fig. 19.4. Figure with centre of Fig. 19.5. Figure with plane
symmetry C of symmetry AB
certain combination of symmetry elements. In Gadolin’s time some

20 of these types were known but today crystals of all 32 symmetry
types have been described.
According to symmetry types, crystals are divided into three
categories: lower, intermediate and higher. Crystals of the lower cate
gory have no axes higher than the second order; crystals of the inter
mediate category have one axis of a higher order, while those of the
higher category have several axes of a higher order. Categories are
divided into crystal systems.
The lower category includes three systems, namely, triclinicr
monoclinic and orthorhombic. Crystals of the triclinic system have
neither axes nor planes of symmetry; there may or may not be a
centre of symmetry. Examples of substances crystallizing in the
triclinic system are K 2Cr20 7 and CuS0 4 -5H20. In monoclinic crystals
(CaS0 4 -2H20, C4H 60 6—tartaric acid, etc.) there may be both an
axis and a plane of symmetry but there cannot be several axes or

planes of symmetry. Crystals of the orthorhombic system (BaS04,
MgS04 *7H20, etc.) are characterized by the presence of several
elements of symmetry, several axes or planes.
The intermediate category is divided into three systems which
are named according to the type of the principal axis (axis of higher
order): trigonal (has axis of the third order; examples: calcite, CaC03,
and dolomite, CaMg(C03)2); tetragonal (has axis of fourth order;
Fig. 19.7. Figures with mirror-ro

tation axis (a) and inversion axis
(b)
examples: Sn02, CaW0 4 and PbMoO^); and hexagonal {has axis of

sixth order; examples: quartz, S i02, as well as K N 0 3 and Agl).
The higher category has only one system—the cubic. The crystals
of this system, for example, CaF2, NaCl and NaC103, have several
axes of a higher order. Examples of crystals of the systems mentioned
are shown in Fig. 19.8.
A prerequisite for the formation of crystals of high symmetry is
that the particles of which they are composed should be symmetri
cal. Since most molecules, including a multitude of organic mole
cules are unsymmetrical, crystals of high symmetry constitute only
a small proportion of those known.
Many cases are known in which a substance exists in different
crystalline forms, i.e., differs in internal configuration and there
fore in physical and chemical properties. This is termed polymor
phism. Silicon dioxide, for example, is known in three modifications:
quartz, tridymite and cristobalite. At a certain temperature only
one of the polymorphic varieties of a substance is stable. At ordi
nary temperatures the stable form of S i0 2 is quartz; from 870 to
272 P A R T IV. S T R U C T U R E OF M A T T E R I N C O N D E N S E D S T A T E
1470°, tridymite; and above 1470°, cristobalite. Transition of an

unstable form to a stable form often takes place very slowly at a
low temperature. A substance can remain in an unstable, or as it
is called, a metastable state for a very long time1.
Fig. 19.8. Systems of crystals

L o w e r c a te g o r y
l — tric lin ic ( s tro n tiu m b ita rtra te S r [ C O O H ( C H O H ) s C O O ] a) ; 2— m o n o c lin ic (la c to s e
C i 2H * i O i t - H * 0 ) ; 3— o r t h o r h o m b i c ( s u l p h u r )
I n te r m e d ia te c a te g o r y
4 — t r i g o n a l ( s o d i u m p e r i o d a t e t r i h y d r a t e N a I 0 4 - 3 H 20 ) ; 5 — t e t r a g o n a l ( c a s s i t e r i t e S n O * ) ;
6 — hexagonal (n e p h e lin e N a A lS i0 4 )
H ig h e r c a te g o ry
7— c u b ic (ro c k s a l t N a C l)
Many gaseous substances which crystallize on cooling, do so with

the formation of several modifications. Extensive high pressure
investigations carried out during the last few decades have revealed
that the formation of crystalline modifications at high pressures is
likewise typical. This shows that polymorphism is a widespread
phenomenon. It has been found that there are seven modifications
1 The term ‘metastable state’ denotes the state of a system which is unstab
le but, nevertheless, can persist for a lengthy period. For example, a mixture
of hydrogen and oxygen does not react at room temperature but the slightest
spark causes it to explode. At ordinary temperatures such a mixture is in a me
tastable state.
of K N 03, eight modifications of Na 2S 0 4 and 16 modifications of

naphthalene.
Another phenomenon frequently observed among crystalline
bodies is isomorphism, which is the property of atoms, ions or mole
cules to replace one another in a crystal lattice, thus forming mixed
crystals. For example, the colourless crystals of potassium alumi
nium sulphate, KA1(S04)2 *12H20, and the violet crystals of chro
me alum, KCr(S04) 2 ‘12H20, have the same octahedral form. If a
solution containing both substances is evaporated, crystals are for
med which contain both aluminium and chromium. Mixed crystals
are formed in the same way when a solution containing KA1(S04)2«
• 12H20 and RbAl(S0 4) 2 -12H20 is evaporated. Mixed crystals
are completely homogeneous mixtures of solid substances, they are
solid solutions by replacement.
Therefore, it can be said that isomorphism is the ability to form
solid solutions by replacement.
We have cited examples of the most perfect isomorphism—that of
substances which are similar in chemical composition, type of
chemical bond, crystalline form, structure and valence of their ele
ments and the dimensions of the particles which replace one another.
If some of the indications of similarity are not present, and espe
cially if there is only one, say congruent structure, isomorphism is
imperfect, which may be manifested in limited miscibility, or there
may be no isomorphism of any degree. Thus the isomorphism of
SrS0 4 and KBF 4 is imperfect; crystals of NaCl and KC1 are not
isomorphous although they are similar in chemical composition,
type of bond, form and structure; here the difference in the radii of
Na+ and K + makes itself felt. The difference in atomic radii and
type of chemical bond explains why crystals of CuGl and CuZn are
not isomorphous despite their similar structure and form.Although
rNa+ « 7*cu+» NaCl and CuCl are likewise not isomorphous, their
similarity is cancelled by the substantial difference in the polarization
properties of the cations.
19.2. Study of Crystal Structure

The regular form of crystals is brought about by the orderly arran
gement of their components—atoms, ions or molecules. This arran
gement can be represented in the shape of a crystal lattice—a spatial
framework formed by intersecting straight lines. At the points of
intersection are the centres of the particles forming the crystal.
Such ideas about the structure of crystalline bodies were expressed
long ago by many investigators, including Lomonosov, who explai
ned the properties of saltpetre on this basis. But it was only in the
20 th century that it became possible to demonstrate this
experimentally and begin studying the internal structure of crystals.
18 3aK. 15648
274 P A R T IV. S T R U C T U R E OF M A T T E R IN CONDENSED STATE
This followed the discovery in 1912 by Laue, Friedrich and Kni-

pping (Germany) of the diffraction of X-rays by crystals. This is
the basis of X-ray diffraction analysis.
The wavelength of X-rays is of the same order as the dimensions
of atoms. For that reason a crystal, consisting as it does of orderly
arranged particles, is a natural diffraction grating for X-rays.
Let us consider the passage of a beam of monochromatic1 X-rays
of wavelength X through a crystal. Because of the great penetrating
power of X-rays, the greater part of the radiation passes through the
crystal. However, a certain proportion of the radiation is reflected
Fig. 19.9. Derivation of Bragg—Vulf equation
from the planes in which the atoms comprising the crystal lattice
are located (Fig. 19.9). The reflected rays interfere with one another
which brings about their mutual intensification or damping. It is
evident that the result of interference depends on the path diffe
rence 6 of rays reflected from parallel planes. There will be intensi
fication when 6 is a multiple of the wavelength; in that case the
reflected waves will be in phase. As can be seen from Fig. 19.9 ray
S t reflected from atomic plane P t traverses a shorter path than ray
S 2 reflected from neighbouring plane P 2. The difference 8 between
these paths is equal to the sum of the lengths of sections A B and
BC. Since AB = BC = d sin 9 , it follows that 6 = 2d sin 9 , where
d is the distance between the reflecting planes and 9 , the angle
formed by the incident ray and the plane. Intensification of the
reflected radiation takes place when
nX = 2d sin 9 (19.1)
where n is a whole number. Equation (19.1) was derived in 1913,
simultaneously and independently by W. Bragg (England) and
1 Monochromatic radiation consists of rays of one wavelength.
Yu. Vulf (Russia); it is the principal relationship employed for

determining crystal structure.
If the crystal is oriented with respect to the incident X-ray in
such a way that relationship (19.1) is fulfilled, the reflection of the
ray can be observed. If condition (19.1) is not met, there is no refle
ction. Thus if 6 is equal to half the wavelength, rays reflected from
neighbouring planes P 4 and P 2 will antiphase an(l will be
damped; if 8 = 3/2 X, a ray reflected from plane P{ will be damped
by a ray reflected from plane P 3y and so on.
Obviously very many planes can be drawn through the atoms in a
crystal lattice, but as can be seen from Fig. 19.10 the density with
which many planes are filled with atoms is small, and reflections
from them will therefore be weak. Moreover, when the distance d
between neighbouring planes is less than X/2, equation (19.1) is not
Tig. 19.10. Planes drawn thro

ugh atoms in crystal lattice
fulfilled whatever the value of (p. Consequently, only a few planes

give bright reflections. The intensity of the reflections is not the
same in all cases but depends on the number of atoms per unit area
of the reflecting plane, and on the scattering power of the given spe
cies of atom.
At present it is the rotation method that is mostly employed for
X-ray studies of crystals. In this method the crystal is secured to a
rod in the centre of a cylindrical chamber, on the inside wall of
which a photographic film is fastened (Fig. 19.11). The crystal is
slowly rotated by a clockwork mechanism. A beam of monochromatic
X-rays is passed into the chamber from the side, perpendicular to
the axis of rotation.
The X-rays are reflected when the crystal is in positions which
make possible fulfilment of relationship (19.1), and the reflections
are recorded on the film. The X-ray pattern obtained consists of
rows of dots which are traces of the reflected rays. To determine
crystal structure, several X-ray patterns are made with the crystal
in different positions relative to the axis of rotation.
As in the electron diffraction technique (see pp. 142-144), the
method of trial and error can be used for interpreting X-ray pat
terns — this was the method used in the first investigations in this
18*
field. A certain structure is assigned to the test substance, and the

X-ray pattern calculated by means of relationship (19.1), and com
pared with that obtained experimentally, taking into account the
Fig. 19.11. Diagrammatic representation of a chamber for producing X-ray dif

fraction patterns by the crystal rotation method
intensity of the different reflections. Naturally this method is only

applicable when the structure of the substance is fairly simple.
Other techniques, which cannot be gone into here, must be used for
Fig. 19.12. Diagram of arrangement for producing Debye powder patterns

l— X -ray tu b e ; 2— d ia p h ra g m ; 3— s u b s ta n c e under e x a m in a tio n ; 4 —p h o t o g r a p h i c f ilm
establishing complex structures. In many cases deciphering X-ray

patterns is very difficult, and the calculations involved are extremely
laborious; they are performed today with electronic computers.
It is often difficult to obtain the comparatively large crystals
required for the rotation method. In such cases the powder method
of Debye and Scherrer is employed. In this case (Fig. 19.12) the beam
Ch. 19 . C R Y S T A L L I N E S T A T E 277
of X-rays is passed through a cylinder moulded from fine crystals

of the test substance. Among the large number of tiny crystals in the
powder there are always some whose orientation satisfies equation
(19.1), and which, therefore, produce reflections. The X-ray pat
terns obtained are called Debye powder patterns. Experimentally
the powder method is simpler than the rotation method, but deciphe
ring the powder pattern is usually more difficult; for some types of
crystals complete establishment of the structure by this method is
impossible.
Reflection of X-rays from crystals results from the interaction
of the radiation and the electrons. Therefore, the centres of the atoms
determined roentgenographically are the ‘centres of gravity’ of the
electron subshells. In many-electron atoms these centres practically
coincide with the nuclei, but in light atoms the position of the nuclei
can differ perceptibly. Since hydrogen ions—protons—have no
electron subshells, their position cannot be established by X-ray
diffraction analysis, and the method of neutron diffraction is used
instead. Beams of neutrons are obtained from an atomic reactor.
Unlike X-rays, neutrons do not interact with paired electrons but
they are reflected by atomic nuclei.
Electron diffraction technique is also employed for studying
crystal structure. Since electrons are retarded by matter to a much
greater degree than are X-rays, in the electron diffraction study of
solid bodies, either a small layer of the test substance is used or
the diffraction of electrons is investigated when reflected from a sur
face. The latter method is particularly valuable in that it enables
study of thin surface layers such as films of oxides, nitrides and other
compounds on metals.
The structure of several tens of thousands of crystalline substances
has now been studied by methods of diffraction analysis and fresh
information is being published every month. By means of these
methods investigators have succeeded in establishing the general
foatures of a molecular structure which is apparently the most com
plex in nature, the structure of the substance governing the heredity
of living organisms.
i9.3. Types of Crystal Lattices

Crystal lattices are divided into several types according to the
kind of particles at the points of the lattice and the character of the
bond between them.
At the points of atomic crystal lattices are neutral atoms joined
by covalent bonds. There are comparatively few substances with
an atomic lattice, among them diamond, silicon and some carbides
and silicides, that is, compounds of other elements with carbon
and silicon. In these solid bodies all the atoms are bound, one to
278 P A R T IV. S T R U C T U R E OF M A T T E R IN CONDENSED STA TE
another, in the same way. It is impossible to distinguish individual

molecules in the structure of the atomic crystal and the entire cry
stal can be considered a giant molecule. Since covalent bonds are
very strong, substances having atomic lattices are always hard,
high-melting and non-volatile; they are practically insoluble.
At the points of molecular crystal lattices are molecules. Most
substances with a covalent bond form crystals of this type. Solid
hydrogen, chlorine and carbon dioxide and other substances which
are gases at ordinary temperature form molecular lattices. The
crystals of most organic compounds are also of this type. This means
that very many substances with molecular crystal lattices are known.
The molecules at the nodes of the lattice are bound together with
intermolecular forces, the nature of which was discussed in Chap
ter Eighteen (see pp. 263-267). Since intermolecular forces are
weaker than chemical bonds, molecular crystals are low-melting,
quite volatile and fairly soft. Substances whose molecules are non
polar have particularly low melting and boiling points. Thus crystals
of paraffin wax are very soft although the covalent C — C bonds in
the hydrocarbon molecules of which the crystals consist are as
strong as the bonds in diamond. Crystals formed by the noble gases
should also be classed as molecular, consisting of monoatomic
molecules, since valence forces play no part in their formation; the
bonds between the particles are of the same character as in other
molecular crystals, which results in the comparatively large intera
tomic distances in these crystals.
At the points of ionic crystal lattices are positive and negative
ions arranged alternately; such lattices are characteristic of com
pounds of elements which differ greatly in electronegativity, such
as the fluorides of the alkali metals. As in the case of atomic cry
stals, individual molecules cannot be distinguished in ionic crystals
because there is no preferred ion with which a given ion of opposite
sign interacts; the entire crystal can be considered a giant molecule.
The bonds between the ions are strong, and consequently ionic
compounds have high melting points and low volatility; they have
great hardness but in this respect they usually stand below substan
ces with an atomic lattice. Attention must be called to two points.
Firstly, hardness and a high-melting point are not necessarily asso
ciated with ionic forces. Hardness and melting points are frequently
lower than in substances with an atomic lattice. Secondly, many ionic
crystals are made up of polyatomic ions, such as SO2-, NO3, [Hgl4]2",
[Cu(NH3)4]2+, [A1F6]3“, etc. While the bonds between the particles
forming the lattice are ionic, the atoms within the complex ions
are usually joined with covalent bonds.. Since complex ions are of
large size, it follows that when the charges are equal the forces
acting between particles in the lattice will be considerably weaker
than in a lattice consisting of monoatomic ions. Consequently the
melting points and hardness of substances containing polyatomic

ions are lower. For example, NaCl has a melting point of 801°C,
while the melting point of NaN03 is only 311°C.
The crystal lattices formed by metals are said to be metallic.
At the points of these lattices are positive metal ions, and the valen
ce electrons can move between them in different directions. The
aggregate of electrons is sometimes called electron gas. Such a stru
cture of the lattice brings about the high electric and thermal con
ductivity, and high plasticity of metals—mechanical deformation
does not rupture the bonds and destroy the crystal, since the ions
of which it is composed float, as it were, in a cloud of electron gas.
19.4. Some Crystal Structures

As was already pointed out, the particles in crystals are arranged
in a definite order, thus forming a crystal lattice.
Like the outer form of crystals, crystal lattices can be classified
according to their symmetry. In 1890, long before the development
of experimental methods for studying structure, Y. Fedorov worked
out such a classification mathematically. He dermonstrated that
230 variants of the combination of symmetry elements were possible
for crystal lattices. These combinations are called Fedorov symmetry
groups. There are many more combinations of symmetry elements
for crystal lattices (230) than for the outer forms of crystals (32),
because of the presence of additional elements characterizing inner
symmetry.
Any crystal lattice can be considered as consisting of unit cells.
The unit cell is the smallest part of a crystal having all the structural
characteristics of the given lattice. The diagram in Fig. 19.13 repre
sents the crystal lattice of metallic sodium in which one of the unit
cells is hatched. The unit cell is a parallelepiped, by moving which
in the direction of each of the coordinate axes x , y and z, arranged
parallel to the edges of the figure, the crystal lattice can be constru
cted. This operation resembles bricklaying. The lengths of the
edges of the unit cell, denoted by the letters a, b and c (correspon
ding to the coordinates x , y and z) are called the parameters of the
cell. The unit cell can be fully characterized by stating the lengths
of the edges of the parallelepiped, the angles between them and the
coordinates of the atoms in the cell; the latter are often expressed
in fractions of the corresponding parameters of the cell. By way of
example, the unit cells of copper and sodium are shown in Fig. 19,14.
In the structures described below, some atoms are shown for greater
clarity in addition to those that form the unit cells.
When considering the structure of crystalline substances a distin
ction must be made between structure and structural type. Structural
type relates to the relative arrangement of the atoms in space with
280 P A R T I V . S T R U C T U R E OF M A T T E R I N CONDENSED S TA TE
out specifying the distances between them. When characterizing

the structure of a substance, the parameters of the unit cell must
Fig. 19.13. Crystal lattice of sodium (one unit cell hatched)
be stated in addition to the structural type. A structural type takes

its name from a substance having this type of structure. A great
Fig. 19.14. Unit cells of crystal lattices

(a) copper (fa c e -c e n tre d c u b ic ) ; (b ) s o d iu m ( c u b e - c e n tr e d c u b ic )
number of structures may belong to the same structural type. Thus

many metals form crystals of the magnesium structural type. Some
structural types characteristic of inorganic substances are described
below.
We begin with structures of metals. As was noted in Chapter
One, packing of the particles in most metals is as close as possible
(see p. 16). There are two variants of the closest packing of spherical
bodies—cubic and hexagonal.
Consider the case of a single layer of balls—the closest arrange
ment will be that shown in Fig. 19.15. Four of the fourteen balls are
in contact with six others. We now place balls above and below this
layer. It is evident that to obtain the packing, the balls of the upper
and lower layers must be arranged in such a way that they fill the
hollows between the balls of the middle layer. Two layers of balls
are shown in Fig. 19.16; the balls in the bottom layer are indicated
with dashed lines, the hollows they fill are black. It can be seen that
Fig. 19.15. Close arrangement Fig. 19.16. Fragment of close

of balls on a plane surface packing of two layers of balls
half the hollows in the top layer are left unfilled, these hollows are
hatched. When arranging the third layer, two patterns are possible:
the balls can be placed in the black hollows or in those that are
hatched. In the first case we have hexagonal packing (Fig. 19.17a),
and in the second, face-centred cubic (Fig. 19.176). In both cases the
extent to which the space is filled with the balls is the same, and
comes to 74.05%. The coordination number of the atoms in both
structures is twelve. Copper is an example of a metal having a face-
centred cubic structure, while magnesium is an example of a metal
having a hexagonal structure.
Some metals have a different type of crystal lattice—the cube-
centred cubic (the unit cell of such a lattice is shown in Fig. 19.146).
An example of a substance having this structure is the a-form of
iron which is stable at room temperature. It can be seen from
Fig. 19.146 that in this case the coordination number is eight. The
structural types of the lattices of various metals are given in Tab-
le 19.1.
(a ) (b)
Fig. 19.17. Closest packing of balls
(a) h e x a g o n a l; (b) c u b ic
Fig. 19.18. Crystal structure of caesium chloride

L ig h t b a lls a re C s+ io n s ; d a r k b a lls a re G 1 -. C sC l u n i t c e ll is s h o w n to th e le ft
Fig. 19.19. Crystal lattice of NaCl

L ig h t b a lls a re N a+ io n s ; d a rk b a lls a re C l-
Table 19.1
Crystal Structures of Metals
Li Be
3 2
Na Mg
3 2
K Ca Sc Ti V Cr Mu F<‘ Co Ni Cu Zn
3 1; 2 i;2 2; 3 3 3 1;3 1:2 1; 2 1 2
Rb Sr Y Zr Nb Mo Tc Ru Jib Pb Ag Cd
3 1 2 2; 3 3 2; 3 2 1; 2 1 1 1 2
Gs Ba La Hf Ta w Re Os Ir Pt Au Hg
3 3 1;2 2; 3 3 3 2 1; 2 1 1 1 —
L a ttic e ty p e s : 1 — fa c e -c e n tre d c u b ic ; ; “2 —■ h e x a g o n a l ; 3 — cube-
c e n tre d c u b ic .
Caesium chloride has a structure similar to the cube-centred cubic

lattice of metals, it is represented in Fig. 19.18. Unlike metal lat
tices, this structure consists of two kinds of particles, Cs+ and Cl"
ions. The coordination number for both ions in this structure is
eight. It can be seen from Fig. 19.18 that around the Cs+ ion are
eight Cl" ions. In the same way around each Cl" ion are arranged
eight Cs+ ions, though only one is shown in Fig. 19.18, that in the
centre of the unit cell. The others are outside this cell.
Sodium chloride has a different structure (Fig. 19.19). The arran
gement of one kind of ions, e.g. the Cl" ions, is the same as in the
face-centred cubic lattice, these ions are at each corner of the cube
and in the centre of each face. In the centre of the cube is a sodium
ion, while the other sodium ions are arranged in the centre of each
edge. Together they form the same kind of a lattice as the chloride
ions. The coordination number for both the Cl" and Na+ ions is
six; around each ion of one sign are arranged six ions of the opposite
sign, thus forming a regular octahedron.
The NaCl and CsCl structural types are quite common among
inorganic substances; all the alkali halides, for instance, have stru
ctures of these types. In ordinary conditions caesium bromide and
iodide have the same type of lattice as the chloride; other alkali
halides have structures of the NaCl type. At very high pressures it
is natural to expect a polymorphous transition to the more compact
CsCl structure. This was recently established experimentally for
many alkali halides.
A number of elementary substances (silicon, germanium, grey
tin) have crystal lattices of the diamond type. The unit cell of such
a lattice is represented in Fig. 19.20. In diamond each carbon atom
is joined with covalent bonds to four other atoms. The unit cell of
this lattice is constructed by adding four atoms to the fourteen atoms
284 P A R T I V. S T R U C T U R E OF M A T T E R IN CONDENSED ST A T E
of the face-centred cubic lattice. The additional atoms are inside

the cube at the centre of tetrahedrons formed by atoms at alternate
corners of the cube and their three nearest neighbours in the centres
of the faces. The coordination number of the atoms in the diamond
lattice is four.
The sphalerite structural type (sphalerite is a variety of zinc sul
phide, ZnS) is similar to the diamond lattice. The sphalerite lattice
is represented in Fig. 19.21. This structure can be obtained from the
Fig. 19.20. Structure of diamond Fig. 19.21. Structure of spha

lerite
S m a ll b a lls a re Z n * ^ io n s ; la rg e
b a l l s a r e S 2-
diamond structure by replacing half the carbon atoms by zinc ions

and the other half by sulphur ions (compare Fig. 19.20 and 19.21).
The sphalerite structural type is characteristic of binary compounds
in which the total of valence electrons of the two elements is the
same as in carbon. Examples of compounds having this structure
are SiC, BN (cubic form), A1P, InAs, InSb, GaAs and CuCl. The
total of outer electrons of the atoms forming these compounds is eight.
The structure of wurtzite, another variety of ZnS is shown in
Fig. 19. 22. In the lattice fragment represented in the Figure, the
atoms of one element are arranged at the corners of the hexagonal
prism, in the centres of its upper and lower faces and inside three
of the six triangular prisms composing the hexagonal prism. The
atoms of the other element are on the lateral edges of all six trian
gular prisms, and also inside three of the six triangular prisms where
there are atoms of the first element.
The particles in wurtzite are arranged in such a way that each
atom of one element is tetrahedrally surrounded by four atoms of
the other element. Thus, as regards the nearest neighbouring atoms
there is no distinction between the structures of wurtzite and spha
lerite. The difference between these lattices is that like atoms in
sphalerite are arranged in the same way as in the face-centred cubic
lattice, and like atoms in wurtzite, in the same way as in the hexa
gonal lattice.
The structure of ice is similar to that of wurtzite. If atoms of
zinc and sulphur in wurtzite are replaced by water molecules., we
obtain their position in ice. A fragment of this structure is shown
in Fig. 19.23. Each molecule is connected with four others by hydro
gen bonds. The hydrogen bonds in the H 20 molecule have a tetra
hedral orientation owing to the tetrahedral arrangement of the sp3-
Fig. 19.22. Structure of wurt- Fig. 19.23. Structure of ice

zite
L a rg e b a lls a re S ’- io n s; s m a ll b a lls
a re Zn*+
hybrid orbitals of the oxygen atom, two of which form a covalent

bond with the hydrogen atoms, while the other two are occupied
by unshared electron pairs which attract the hydrogen ions of neigh
bouring H 20 molecules. In Fig. 19.23 the black circles show the
position of the hydrogen, and the hatching, the region where the
negative charge is concentrated.
It can be seen from Fig. 19.23 that there are voids in the crystal
line of ice, which accounts for its low density which is less than
that of water. The voids in the crystal lattice of ice can be filled
with other molecules, for example, CH4, H 2S, the single-atom mole
cules of the noble gases, etc. This results in the formation of a sort
of chemical compound. Compounds formed by the filling of the
voids in a crystal lattice by molecules of other substances are called
clathrates.
As we shall see later, fragments of the crystalline structure of
ice persist in water, which explains many of the latter’s properties.
We have described some structural types of compounds of the
general formula AB—an atom of one element is combined with an
atom of another element. We shall now consider two examples of
the structure of compounds having the formula AB2.
286 P A R T I V . S T R U C T U R E OF M A T T E R IN CONDENSED S TA TE
In the structure of fluorite, CaF2 (Fig. 19.24), eight fluorine atoms

at the corners of a cube are in the environment of fourteen calcium
ions, eight of which occupy the corners, and six, the centres of the
faces of a larger cube. In the representation of the unit cell there
are more calcium ions than fluoride ions. However, all the fluoride
ions belong solely to the given cell because they are within it. All
the calcium ions, on the other hand, belong not only to the given cell
but also to neighbouring cells. The ions at the corners of the cube
simultaneously ‘serve’ eight cells; those situated at the centres of
Fig. 19.24. Structure of fluorite Fig. 19.25. Structure of rutile

L ig h t b a lls a re oxygen a to m s ; d a rk b a lls
L i g h t b a l l s a r e F ““ i o n s ; d a r k b a l l s
a r e C a 2+ ; e a c h f l u o r i d e io n is te tr a - a re T i
h e d r a lly s u rr o u n d e d b y to u r c a lc iu m
io n s . I n th e d ia g ra m o n ly one te t
rah e d ro n is in d ic a te d w ith d o tte d
lin e s
the faces, two neighbouring cells. It follows that for eight fluoride
ions, there are, on an average, 8 X 1/8 + 6 X V2 = 4 calcium
ions, which corresponds to the formula CaF2.
Another widely encountered structure of compounds of the gene
ral formula AB2 is that of rutile, T i0 2 (Fig. 19.25). The titanium
atoms in the rutile unit cell form a body-centred rectangular paralle
lepiped with a square base—a distorted cube. The oxygen atoms
are arranged on the diagonals. In this structure, each titanium atom
is surrounded by six oxygen atoms which form a regular octahedron
round it, while each oxygen atom is in the centre of an isosceles
triangle defined by three titanium atoms. Thus the coordination
numbers of titanium and oxygen in this structure are six and three,
respectively.
Such are some examples of crystal structure. Other types of crystal
structure are taken up in inorganic chemistry courses.
19.5. Energetics of Ionic Crystals

Since the chemical bond between the particles in many inorganic
compounds is close to ionic, this type of crystal lattice is of great
interest for inorganic chemistry. Particularly important is the value
of the lattice energy U0 which is measured by the work that must be
done to remove the ions composing the crystal to an infinitely great
distance from one another. This value is usually expressed on the ba
sis of a gram-molecule of the substance in question.
Th§ value of the lattice energy determines the strength and solu
bility of crystals and other properties. As we shall see later
(p. 298), if the energy of the crystal lattice is known, it is possible
to find the solvation energy, that is, the energy of interaction of the
ions of a dissolved substance and the molecules of the solvent. The
solvation energy determines, to a great extent, the behaviour of
substances in solutions.
The lattice energy can be found from experimental data or calcu
lated theoretically. We shall first consider the theoretical calcu
lation.
An equation for calculating the lattice energy was first derived
by M. Born. It has the form
cr„_a s s £ ! f ( l _ i ) (19.2)
It will be noticed that this equation differs from Born’s formula

for calculating the energy of ionic molecules [equation (15.8), p. 223]
in the presence of the factors A^0 and a. The first quantity is Avoga-
dro’s number, and is introduced into the equation to obtain the
energy per mole of substance1. Quantity a is known as the Madelung
constant, Madelung being the investigator who in 1918 first calcula
ted this quantity for NaCl. This constant is introduced into equation
(19.2) because in the crystal lattice, unlike the ionic molecule, each
ion instead of interacting with a single ion of the opposite sign,
interacts with a large number of positive and negative ions at diffe
rent , distances from the ion under consideration. The principle of
calculating the Madelung constant can be explained, using sodium
chloride as an example.
Let us consider one of the sodium ions in the NaCl lattice (see
Fig. 19.19). The nearest neighbours of each sodium ion are 6 chloride
ions at a distance r. The Coulomb energy of interaction with these
ions will be
1 The lattice energy is a positive quantity, since by definition it is the

energy of the disruption of the lattice. The energy of formation of the lattice from
free ions has the same value, but with the sign reversed.
Next, 12 sodium ions are arranged around the chosen sodium ion
at a distance of r Y 2. £>ince these ions have the same sign as the ion
under consideration, the interaction energy is written
e2
u2= 12——=■
2 r "l/2
The following neighbours of this particular ion are 8 chloride ions
at a distance of r ]73, which gives the contribution to their interac
tion energy
e2
u3= —8
7 V3
In the general form the expression for the Coulomb interaction energy
can be written in the form
Uc = ut + u2 + u3+ . . .
This is a convergent series1; by calculating a sufficient number of
members in the same way as was done for u2 and u3 it will be
found that
Ucu = —a —r
= —1.7475 — r
It follows that the Coulomb interaction energy of one ion with all
the other ions in the sodium chloride lattice is a times greater than
the interaction energy of two single-charged ions at a distance r.
Thus Madelung constant a for NaCl is 1.7475. This quantity can be
calculated in the same way for other crystal lattices. Values of the
Madelung constant for some crystal structural types are given in
Table 19.2.
Table 19.2
Madelung Constants a for Some Types of Crystal Lattices
C o o rd in a tio n
S tru c tu ra l ty p e F o rm u la num ber a
Sodium chloride NaCl Na C; Cl 6 1.7475

Caesium chloride CsCl Cs 8; Cl 8 1.763
Sphalerite ZnS Zn 4; S 4 1.638
Wurtzite ZnS Zn 4; S 4 1.641
Calcium fluoride CaF2 Ca 8; F 4 2.520
Rutile T i0 2 Ti 6; 0 3 2.408
A comparison of equation (15.8) for calculating the bond energy

in the ionic molecule and equation (19.2) for calculating the crystal
1 A series is said to be convergent if the algebraic sum of its members app
roaches a limit as the number of members is increased.
lattice energy shows if the comparatively small change in r 0 during

the transition of gaseous molecules into the crystalline state is negle
cted, it can be considered that the energy of forming a crystal from
ions is a times greater than the energy of forming a corresponding
number of ionic molecules. It can be seen from Table 19.2s that the
Madelung constant is greater than unity. Therefore, a substantial
amount of energy is liberated when a crystal is formed from ionic
molecules and, conversely, converting a crystal into gas consisting
of molecules requires a substantial expenditure of energy. Consequen
tly ionic crystals have high melting points and high heats of suhli-
mation.
Repulsive energy of the electron shells UB (Born repulsion)
rapidly diminishes as the distance between the particles increases,
and therefore, when calculating its value, one need only consider
the interaction of the ion with its nearest neighbours, at distance r.
Therefore, one can write
U = UC+ UB= - a (19.3)
Quantity B is determined from the conditions of the equilibrium
of forces at distance r 0 as was described on p. 222; as a result equa
tion (19.2) is obtained.
Born repulsion factor n in equation (19.2) is found from crystal
compressibility data. We shall now consider the principle of this
calculation.
Crystal compressibility x is understood to mean the relative redu
ction in volume per unit of applied pressure P, that is,
dV
dP (19.4)
When a crystal is compressed, the ions are brought closer together,
distance r in equation (19.3) decreases. It is evident that the molecu
lar volume of the crystal is proportional to the cube of the interionic
distance, that is,
V = pr3 (19.5)
Proportionality factor P can easily be found from the geometric
relationships if the structural type of the crystal is known; for cry
stals of the NaCl type P = 2N.0.
A decrease in r causes a change in the potential energy, dU.
Equating this change to the work performed by the pressure, PdV
we have
—dU = PdV
Therefore,
19 3aK . 15648
and
dP d2U
dV dV 2
(19.6)
It thus follows that crystal compressibility determined by relation
ship (19.4) can be expressed through the second derivative of the
potential energy of the ions with respect to volume.
Since we have relationships (19.4) and (19.6), and an expression
for the potential energy of the ions in the crystal is also known
(19.3), compressibility can be represented as a function of r0 and n.
Performing the appropriate operations, which are not difficult but
cumbersome and are therefore omitted here, gives for crystals of the
^4+B" type the expression
184
71 = 1 (19.7)
xae2
from which n can be calculated if x and r 0 are known. Crystal com

pressibility x can be determined experimentally. This determination
made it possible to find the values of n given on p. 222. We shall now
consider calculation of lattice energies from experimental data.
This requires knowledge of energy changes in a number of processes,
which in the aggregate can be represented by a diagram called the
Born—Haber cycle. Let us examine this cycle, taking sodium chlori
de as an example:
AH subl Na
Na(cryst) -> Na(gas) ■> Na+(gas)
26.0 118.0
-1 8 0 .4 —U0
\ NaCl(cryst)-:
+
1/zEbnd - E , Cl
V2Cl2(gas) ->■ Cl(gas) _ 90 j -» Cl-(gas) ............f ...............:
28.9
^ form
- 9 8 .4
In this cycle we pass from solid metallic sodium and gaseous chlo
rine (lefthand part of diagram) to crystalline sodium chloride (right-
hand part of diagram) in two ways. The first consists in conver
ting sodium and chlorine into the ions Na+ and Cl" and forming
crystalline sodium chloride from them. In accordance with the
definition of the energy of the crystal lattice, energy is liberated
when NaCl is formed from gaseous ions, which is equal in absolute
value to Uq. T o obtain sodium ions metallic sodium must be vapo
rized, which involves expenditure of the heat of sublimation, AH SUbi-
This’c'an*be determined by thermochemical methods. The atoms must
then be ionized, necessitating the expenditure of ionization energy
/ Na which can also be measured (see p. 60). To obtain chlorine ions
the bond in the Cl2 molecule must first be broken. To obtain one
chlorine atom requires the expenditure of 1l2Ebnd (for the determi

nation of this quantity see p. 148). The electron detached from
the sodium atom must then be joined to the chlorine atom, which
takes place with the liberation of the energy of electron affini
ty, E cl.
The second way proceeds directly from sodium and chlorine to
crystalline sodium chloride. The heat effect in this process is com
paratively easy to measure, it is the heat of formation of sodium
chloride from the elements, AH fOTm-
According to Hess’s law the heat effect of a given process is inde
pendent of the way in which the process takes place but is deter
mined solely by the initial and final conditions of the system. Since
in both variants of the process considered, the initial and final con
ditions are the same, the total heat effect in the first variant is.
equal to the heat effect in the second. Therefore, we can write
form = Ai/subl “1“ I Na V2^6nd -^Cl ^0
from which
Uq = —Ai7form + h H subl + ^Na + 1/2-£'bnd —^CI (19.8)
All the quantities in the righthand part of this relationship can be
measured, and therefore the value of U0 can be found. In the Born—
Haber cycle for NaCl given above the energy effect is given for all
the relevant processes (in kcal).
The quantity most difficult to measure, among those included in
the Born—Haber cycle, is electron affinity E. For that reason the
cycle was first used for determining electron affinity rather than
finding crystal lattice energies. In that case the lattice energy was
calculated theoretically by the Born method. Later, when experi
mental methods were developed for determining electron affinity,
it was found that values of E calculated from theoretical values of
lattice energies were quite close to experiment. Thus the theoreti
cal calculation of U0 for ionic crystal lattices gives correct values.
The difference between theoretical values of U0 and values calculated
from the Born—Haber cycle comes to several per cent for alkali ha
lides, but it is greater for salts of multivalent metals. This can be
explained by the presence of a certain proportion of covalent bond
in these compounds.
Values of the crystal lattice energy for some compounds determi
ned on the basis of experimental data are given in Table 19.3. It can
be seen that for salts consisting of singly charged ions these values
are of the order of 200 kcal/mole; for compounds containing multiply
charged ions, they are considerably greater. Parallelism in the vari
ation of U0 in series of similar compounds is illustrated by the
comparison in Fig. 19.26.
19*
292 P A R T IV. S T R U C T U R E OF M A T T E R I N CONDENSED S T A T E
Table 19.3
Crystal Lattice Energies of Some Substances,
kcal/mole
Anion
Cation
F- c i- Br- I- 02- S2-
Li+ 247 202 191 177 703

Na+ 219 180 177 165 623 —
K+ 194 169 162 153 563 ___
Rb+ 186 164 158 149 544 —
Cs+ 179 156 151 144 527 —
Be2+ 826 713 692 670 1080 ___
Mg2+ 689 595 577 553 940 778

Ca2+ 617 525 508 487 842 722
Sr2+ 580 504 489 467 791 687
Ba2+ 547 468 463 440 747 656
Z n2+ 718 642 633 620 970 852
Cd2+ 662 598 593 563 911 802
H g 2+ — 624 624 630 940 842
Pb2+ 590 534 528 497 850 732
M n2+ — 589 555 542 920 841
Cu2+ — 660 652 — 990 890
Both the experimental and theoretical methods of finding lattice

energy require information to obtain which involves certain diffi
culties. Thus in order to calculate the Madelung constant it is neces
sary to know the crystal structure of the substance, which must be
determined by the difficult interpretation of crystal X-ray patterns;
it is likewise necessary to know the value of compressibility x, whose
measurement is a problem for high-pressure technology and can be
performed by only a few laboratories. For that reason the equation
suggested by A. Kapustinsky for calculating lattice energies is exten
sively employed; the only information required for calculating U0
by this method is the ionic radii.
A. Kapustinsky noticed that the Madelung constant for diffe
rent salts was aprroximately proportional to the number of atoms
in their molecules. He also suggested that the Born repulsion factor
n should be considered the same for all compounds and that inte
rionic distance r0 should be replaced by the sum of the radii of
cation and anion. Equation (19.2) then had the form
ZcZfl ^ n
U0 = A (19.9)
rc~\~ra
Ch. 20. L I Q U I D A N D A M O R P H O U S S T A T E S 293
where A is a constant; 2rc, the number of ions in the formula of the

salt (e.g., for CaCl2, 2,n = 3); zc and za, the charges of cation and
anion, and r c and ra, their radii. From crystal-lattice energy data,
A was found to equal 256.1 (assuming that U0 was expressed in
kcal/mole).
In spite of the fact that the Kapustinsky equation contains a num
ber of simplifications not found in Born equation (19.2) the results
tlx, kcal/mole
Fig. 19.26. Relationship of lattice energy of calcium compounds (Ui) and stron
tium compounds (Un)
are as exact. This is evidently because inaccuracies introduced by

the simplifications are largely compensated by the deviation of the
theoretical values of U0 obtained with the Born equation from the
true values. It was noted above that this deviation is due to the
presence of a certain proportion of covalent bond in all crystalline
compounds. Calculations with the Kapustinsky equation are extre
mely simple, and it is therefore widely employed in many fields
of science.
CHAPTER TWENTY
LIQUID AND AMORPHOUS STATES
20.1. Structure of Liquids

The liquid state of aggregation is intermediate between crystalli
ne and gaseous (see Fig. 18.1). Consequently at high temperatures
the properties of a liquid approach those of a non-perfect gas in
294 P A R T I V. S T R U C T U R E OF M A T T E R IN CONDENSED S TA TE
which collisions of the molecules are very frequent, while at low

temperatures they approach the properties of a crystalline substance.
If a liquid is heated under increasing pressure (otherwise it would be
vaporized) a state will be reached in which vaporization of the liquid
does not involve the expenditure of energy. This is called the criti
cal state and it corresponds to the critical temperature and pressure,
which are different for different substances (see Fig. 18.1). At the
critical point all the properties of a liquid and the corresponding
vapour—energy, density, etc.—become identical. Consequently, if
a liquid is heated while under its critical pressure, it will, on
reaching the critical temperature, be in no way distinguishable from
its vapour—neither in the character of the movement of its particles,
nor in its structure, nor in any other property1.
On the other hand, as the temperature is lowered, the similarity
of a liquid to a gas (the terms gas and vapour are to be considered
synonymous) gradually gives way to its similarity to the correspon
ding crystals. This is maximal near the crystallization point, alt
hough even there the two states remain distinguishable12. But the
change in properties of a substance as it solidifies or melts is not, as
a rule, very great. This is seen from the data for several metals in
Table 20.1, giving changes in volume v, heat capacity c and compres
sibility faotor x on melting, as well as heats of fusion AH fUS.
Table 20.1
Characteristics of Some Metals at Their Melting Points
m .p ., Vl « - Vcr .1 0 0 ci q---------
~ ccr • 1ftft x/rr
IQ
—x__
cr
Metal —— Iw ------------- -- 1UU AW/ u S,
°C v cr kcal/g-at
ccr *cr
Cd 321 4 .7 3 .4 3 0.0 1 .4 7
Hg -3 9 3 .6 -2 .4 6 .4 0.55
Pb 327 4 .8 7 .7 8 .3 1.20
Sn 232 2 .8 -6 .3 21 1 .6 6
It should be emphasized that the close similarity in the proper

ties of liquids and crystals near the melting point is not limited to
metals but is found in widely diverse substances, and this applies
to many other properties besides density, heat capacity and compres
sibility factor. Thus for most substances the change in volume du-
1 The similarity of gas and liquid is likewise seen in the fact that under high
pressure some mixtures of gases separate into layers, as do some liquids which
are only partially soluble in one another.
2 Experiments have shown that even under very high pressures the distin
ction in properties of a substance in the liquid and crystalline states does not
disappear.
ring crystallization comes to approximately 10%. This means that

the inter-particle distance changes by only ~ 3 % , that is, that the
arrangement of particles in a liquid closely resembles their arrange
ment in the corresponding crystals. The closeness of the heat capa
city of the melted and solidified substance indicates the similarity
of the thermal motion of particles in liquid and solid bodies. Their
energetic similarity at the melting point is borne out by the fact
that unlike heats of vaporization AH vap, heats of fusion AH fUS are
small. Thus for HI, AH vap = 5 kcal/mole, but AH fUS is only 0.7
(cf also Table 20.1). In other words, the ordered arrangement of the
particles which is characteristic of crystals is only partially lost in
liquids, at least near the crystallization point. Conceptions based
on the close resemblance of liquid to crystal were first propounded
by Ya. Frenkel (1934).
The presence of a spatial ordering of the molecules in a liquid is
confirmed by many other facts, including results of experiments in
the scattering of light, X-rays, neutrons and electrons.
Debye liquid X-ray patterns at temperatures close to crystalliza
tion points are similar to the X-ray patterns of the corresponding
crystals, differing only in the blurring of the rings, which increases
as the temperature rises.
The results of the X-ray analysis of liquids can be explained by
assuming that their structure is either an accumulation of ultramic-
roscopic, considerably deformed aggregates, or a continuous network
in which the elements of the structural order are restricted by their,
nearest neighbours.
The first assumption signifies that an enormous number of ‘crystal
islets’, termed cybotaxes, are divided by areas of randomly arranged
particles. These groups have no distinct boundaries but merge with
the areas of randomly arranged particles; they shift about, losing
some particles and adding others, and can disintegrate and form
again. As in a gas, they move about spatially; as in crystals, they
oscillate about a position of equilibrium. Heating reduces the time
the cybotaxes spend in a ‘settled way of life’, while cooling has the
reverse effect. It follows that this conception of a liquid is a synthe
sis of the conceptions of gas and crystal—a combination of an or
derly arrangement of molecules in a small volume and an unordered
arrangement in the entire volume.
The second assumption amounts to a conception of a quasicrystal
line structure in which each molecule in a liquid is surrounded by
neighbours arranged about it in almost the same way as in a crystal
of the same substance. But in the second layer there is a departure
from orderlines, and this increases with the distance from the mole
cule first considered. In other words the deviation from an orderly
arrangement increases regularly in proportion to the distance from
the given molecule; the order in liquids is short-range, whereas in
296 P A R T I V . S T R U C T U R E OF M A T T E R I N C O N D E N S E D S T A T E
crystals it is long-range: a strict repetition of the structural element—

ion, atom, groups of atoms or molecules—in all directions. Thus at
temperatures close to the crystallization point a liquid is a distorted
crystal in which long-range order has been lost. Figuratively spea
king, a crystal can be likened to a roadway of paving blocks, and a
liquid, a roadway of cobblestones.
It is hard to say which viewpoint is preferable. What is important
is that there is a certain order in liquids, which is manifested the
more distinctly, the closer the temperature to the melting point;
it is likewise important that both hypotheses explain the properties
of liquids.
Study of the scattering of X-rays in liquids consisting of polyato
mic molecules has demonstrated not only the orderly arrangement of
the molecules but also a certain tendency in the mutual orientation
of the particles, which is pronounced in the case of polar molecules,
and enhanced by a hydrogen bond.
Although in general the arrangement of particles in a liquid chan
ges insignificantly during crystallization, there are some substances
in which the arrangement of the particles remains practically unchan
ged, while in others it becomes substantially different. In the latter
case, where crystallization requires considerable ‘repacking’ of the
particles, the process is more difficult. Delay in crystallization
makes possible supercooling of the liquid, i.e., cooling of the liquid
to below the melting point. Since the possible degree of supercoo
ling, other conditions remaining the same, will be greater, the less
‘isostructural’ matter there is in the liquid and solid states, the ten
dency to supercooling provides indirect evidence concerning the
structure of a liquid.
The unordered motion of the molecules in a liquid continually
changes the distances between them. It can be said that the structure
of a liquid is statistical in character. This is the essential distinction
between a liquid and crystals. The statistical character of the ordered
arrangement of the molecules in a liquid leads to continual fluctu
ations—deviations not only from the mean density but also from the
mean orientation, since the molecules can form groups in which a
certain orientation predominates. The less the value of these devi
ations, the more frequently they occur.
Water and aqueous solutions of electrolytes are the liquids of
greatest importance for inorganic chemistry, and we shall therefore-
take up their structure in greater detail.
20.2. The Structui€ of Water

It was pointed out previously (see p. 258) that in the crystal lattice
of ice, H 20 molecules are joined together with hydrogen bonds.
The crystal structure of ice is far from being the closest packing. If a
calculation inverse to that on p. 16 is performed, and the density of

ice determined for the closest packing, proceeding from the radius-
of the H 20 molecule in the structure of ice as determined by X-ray
analysis (1.38 A), we obtain the value 2.0, more than twice the
actual density (0.9).
Bernal and Fowler (Great Britain) who were the first to conduct a
thorough X-ray diffraction study of water, demonstrated in 1933
that there remain in water fragments of ice structure—‘crystalline
islets’ (see p. 295). This phenomenon is more pronounced in water
than in most other liquids. The greater part of the molecules in water
still have the tetrahedral environment found in the ice structure.
The mean coordination number of water molecules is close to four,,
at temperatures of 2, 30 and 83°C it is equal, respectively, to 4.4,
4.6 and 4.9. The greater part of the hydrogen bonds joining the H20
molecules in the ice lattice persist in water—the proportion of dis
rupted hydrogen bonds at temperatures of 0, 25, 60 and 100°C is
respectively 9, 11, 16 and 20%.
As a result of the presence of elements of a crystalline structure in
water, along with the high value of the dipole moment of the H20*
molecule, water has a very large dielectric constant e = 79.5 at
25°G. This means that the interaction between charged particles in
water is approximately 1/so as strong as in a vacuum. Consequently,
all ionic compounds dissociate in aqueous solution. Unlike solvents
with a sinaller value of e, water causes practically complete disso
ciation. In aqueous solution many compounds with a polar bond in
the molecules, such as hydrogen halides, H2S, cadmium salts, etc-.,
are also ionized; in such compounds the degree of dissociation may
be less than 100%.
20.3. Solutions of Electrolytes

Most inorganic compounds are ionized to a considerable extent in-
solution. The presence of ions in solutions of electrolytes makes them
capable of conducting an electric current, accelerates exchange re
actions and explains many other properties.
What causes the dissociation of a dissolved substance into ions is
solvation—the vigorous interaction of the ions with the molecules
of the solvent. Hydration, that is, the interaction of the ions with
water, is a special case of solvation.
Mendeleev was the first to call attention to the importance of the
interaction between the dissolved substance and the solvent. In the
‘sixties of the last century he advanced a chemical theory of solutions,
according to which aqueous solutions contain a number of hydra
tes—unstable compounds of the dissolved substance with water which
change into one another. Employing physicochemical methods of
investigation, Mendeleev demonstrated the existence of such com-
298 P A R T I V . S T R U C T U R E OF M A T T E R , I N C O N D E N S E D S T A T E
pounds in the liquid phase in the systems H 2S 0 4 — H 20, C2H5OH —

H 20 and CH3OH — H 20. Actually, we now know that in solutions
there are comparatively stable molecular groups formed by molecu
les of the solvent and solute. As the polarity of the solvent dimini
shes, the tendency to form molecular compounds declines. Frequently
the bond between solvent and solute is maintained in the solid
state; crystalline solvates are often formed during crystallization
from solution, examples being CuS04*5H20, CaCl2 •6H20 ,‘LiC104
•4CH3OH and AlBr3*C6H6. Whereas water of crystallization is
found in many salts, ‘alcohol of crystallization’ is not frequently
encountered and ‘octane of crystallization’, for example, very rare
ly indeed.
The conception of the hydration of, ions introduced in 1890 by
I. Kablukov and V. Kistyakovsky, which combined Arrhenius’s
theory of electrolytic dissociation with Mendeleev’s chemical theory
of solutions was very effective in explaining the properties of solu
tions of electrolytes.
The intensity of the interaction of ions with water molecules can be
characterized by the heat of hydration AH h —the amount offbeat
liberated during the transition of a gram-ion (6.023 X 1023 ions) from
a vacuum into aqueous solution. The value of AH h can be found from
experimental data, and there are also methods of theoretical calcu
lation.
In order to calculate AH h from experimental data the energies of
the crystal lattices U0 and the heats of solution of the salts A H S must
be known. The process of the solution of a salt can mentally be divi
ded into two stages: disintegration of the crystal lattice into ions
which is accompanied by the absorption of heat equal to U0, and
hydration, resulting in the liberation of heat A//^. It is evident that
the heat effect when a salt is dissolved is equal to the algebraic sum
of these quantities:
AH s = AH h + U0
from which
- A H h = U0 - AH 8 (20.1)
As we know the energy of the crystal lattice in many salts has
been found (see p. 292), and heats of solution are comparatively easy
to determine experimentally. Having this information, the heat of
hydration can be determined by means of equation (20.1). The
change in entropy during hydration A*Sh can also be determined,
and from the equation
A A H h - TA Sh
it is possible to compute the change in the isobar-isothermal poten
tial during the transition of ions from a vacuum into solution. The1
1 It is now accepted that heat liberated is indicated with the minus sign.
quantity AGh is frequently called the energy of hydration. The energy

and heat of hydration differ by several per cent and AGh is replaced
in rough calculations and comparisons by the value of AH h, which
is known for a large number of ions.
When calculating with equation (20.1) we obtain the sum of the
heats of hydration of both the cations and anions forming the salt.
To find the heats of hydration of the individual ions, this value
must be divided into the cation and anion components. Finding the
correct method of dividing is quite difficult. In 1953 A. Kapustins-
ky, S. Drakin and B. Yakushevsky demonstrated that isoelectronic
ions having charges of different signs, e.g. Na+ and F“, K + and Cl^,
etc., differ little in properties in aqueous solutions, and that the
difference diminishes as the size of the ions increases. Therefore the
characteristics of the ions Cs+ and 1“ can be determined with suffi
cient precision by dividing summary values for Csl in half. Having
Table 20.2
Characteristics of the Hydration of Ions with
Noble-Gas Configuration (AG^, kcal/g-ion)
dM—H20 , AGh
Ion n A
A
Li+ 4 2.02 -9 1 -4 8 -121

Na+ 6 2.42 -8 1 -4 3 -9 7
K+ 6 2.79 -6 7 -3 9 -7 9
Rb+ 8 2.92 -7 2 -3 8 -7 4
Cs+ 8 3.10 -6 6 -3 6 -6 6
Be2+ 4 1.62 -3 7 9 -2 1 6 -577
Mg2+ 6 2.04 -2 8 7 -1 9 0 -4 5 0
Ca2+ 6; 8 2.42 -2 3 4 -171 -3 7 3
Sr2+ 8 2.60 -2 1 4 -1 6 3 -341
Ba2+ 8 2.74 -1 8 8 -1 5 8 -3 1 0
AF+ 6 1.88 -6 7 0 -4 4 8 -1091
Y3+ 9 2.44 -5 0 7 -3 8 4 -8 5 0
La3+ 9 2.74 -4 5 7 -3 5 5 -771
these values, it is possible to obtain values for other ions. Energies

of hydration of some ions, calculated in this way, are given in the
last column of Table 20.2.
It can be seen from Table 20.2 that energies of hydration of singly-
charged ions are of the order of 100 kcal; those of doubly-charged
ions, ca. 300-600 kcal; triply-charged ions, ca. 800-1100 kcal. Thus
the energetic effect of hydration is no less significant than the effect
of the ordinary chemical reaction. It is likewise evident from the
300* P A R T IV. S T R U C T U R E OF M A T T E R I N C O N D E N S E D S T A T E
Table that the energy of hydration of ions in a group of the periodic

system diminishes, which is due to the increase in the ionic radii.
In Fig. 20.1 is shown the parallelism in changes in the heats of
hydration of singly- and doubly-charged ions with noble-gas confi
guration.
Most of the ions in a solution are firmly bound with the water
molecules surrounding . them, forming hydrated complexes. These
not infrequently persist in the crystalline hydrates formed when the
Fig. 20.1. Relationship of heats of hydration of M+ and M2+ cations
salt is crystallized from the solution. For example, X-ray structural

analysis of such compounds as MgCl2-6H20 , Na2S 0 4 *10H2Or
KA1(S04)2-12H20, N iS04-7H20 and Nd(Br03)3-9H20 shows that
they contain the complex ions [Mg(H20 )6]2+, [Na(H20 )6]+r
[Al(H20)el3+, [Ni(H20 )6]2+ and [Nd(H20)9]3+.
In Table 20.2 are given the coordination numbers of the ions n r
according to the data of X-ray structural analysis of the crystalline
hydrates, as well as the distances between the ions and the surroun
ding water molecules, dM-H2o Precise X-ray studies of the stru
cture of solutions recently conducted by A. Skryshevsky and
A. Dorosh have shown that the same distances between ions and the
surrounding H 2(^ molecules are maintained in the liquid phase.
Thus hydration can be divided into primary, the interaction of
the ion with the nearest water molecules, and secondary, interaction
with more distant water molecules.
The first equation for the theoretical calculation of the energy of
hydration was proposed in 1920 by M. Born. This equation proceeds
from the assumption that the energy of the transition of an ion from
a vacuum into aqueous solution can be represented as the difference
in the work of charging particles in these media. For simplicity’s
sake the ion is considered a conducting sphere of radius r.
The work of charging the sphere can be calculated if one pictures
the charge being transferred from infinity to the surface of the sphe
re in small portions dq{, dq2, . . . . The work dA performed in crea
ting an additional charge dq on the surface of a sphere whose charge
is already equal to q, is expressed by the relationship
The total work performed in increasing the charge from 0 to q

will be equal to
<7
4= = £ (20.2)
0
For a charge placed in a medium with dielectric constant e, it
follows that
jf2_
A 2er (20.3)
Subtracting equation (20.2) from equation (20.3) gives the equ
ation for the energy of hydration:
<20-41
Since for ions, q = ez, equation (20.4) can be written in the form
(20.5)
The Born equation (20.5) gives values for the energy of hydration
which, as a rule, differ from experiment by several tens of per cent.
Such a discrepancy is natural as a consequence of the grossly appro
ximate character of the model employed in its derivation—the ion
is considered to be a charged conducting sphere, and the solvent, a
continuous medium with dielectric constant e. Nevertheless, the
equation is very simple and makes it possible to judge correctly of
the order of the value AGh, and is therefore extensively employed.
During the last few years an increasing number of investigations have been
published which show that the interaction of ions with molecules of a solvent
is to a great extent due to quantum-mechanical factors, and is similar in nature
to the formation of a coordination bond in complex compounds.
This treatment of solvation was initiated by the work of Bernal and Fow
ler already mentioned (see p. 297), in which attention was called to the fact that
the hydration energy of cations is close in value to the sum of ionization ener
gies 2 ^ corresponding to the conversion of a neutral atom into an ion. Bernal
and Fowler therefore assumed that the interaction of ions with a solvent consi
sted, for the most part, in the return of the missing electrons to the ion so that
a neutral particle was formed.
This approach was developed by V. Mikhailov and S. Drakin, who worked out
a method of calculating the energy and entropy of hydration, and the values
obtained agree well with experiment.
Since an ion in solution mainly forms a solvated complex by interacting with
the unshared electron pairs of the solvent’s donor atoms (usually oxygen or
nitrogen), the rest of the solvent molecule has comparatively little effect on the
solvation energy. For that reason the solvation energies of ions in different sol
vents having the same electron-donor atom are very close. For example, the sol
vation energies of the Li+ ion in H20 , GH3OH, G2H5OH and HCOOH are
—121.0, —120.0, —119.0 and —120.0 kcal/g-ion, respectively.
The donor-acceptor interaction of the hydrogen ion with a solvent is parti
cularly vigorous. It firmly combines with one molecule of the solvent, forming,
in aqueous solutions, the hydroxonium ion, H30 +.
When considering the solvation of ions, one must take into account the effect
of the penetration of the ion into the structure of the solvent, resulting in the
formation of a void in the solvent and the rupture of intermolecular bonds. For
aqueous solutions the energy of formation of such a void, A vd, can be calculated
approximately from the energy of the hydrogen bond, Z?o...H> which is equal to
5 kcal/g-ion, according to the equation
A vd = 0 . 9/i£ o ...h ( 20. 6)
where n io the coordination number of the ion, and the factor 0.9 takes account
of the fact that about 10% of the hydrogen bonds in water are already ruptured
at 25°C (see p. 297). As we know, each H20 molecule can form four hydrogen
bonds with its neighbours. Since one side of the molecule is turned toward the
ion, only two hydrogen bonds can be formed with the molecules of another
hydrated shell. Inasmuch as a bond is formed between two molecules, 1/2 the
energy of the bond is assigned to each, whence the product nEo...n appears in
the expression.
From what has been said it follows that the expression for the hydration
energy of an ion AG/i can be represented by the sum
+ (2°-7)
where Ag \ and AG),1 are the energies of primary and secondary hydration, respec-
tively.
The Born equation (20.5), which does not take account of the donor-acceptor
interaction of the ion with the solvent, does not give a precise result when cal
culating the complete hydration energy, but it is quite suitable for calculating
the secondary hydration energy. To calculate AG^1, the radius of the hydrated
complex (radius of the ion plus diameter of the water molecule) must be substi
tuted in equation (20.5). Having found A vd and AG^1, and knowing the experi
mental values of AGh, AGj can be calculated by means of equation (20.7).
Results of such calculations are given in Table 20.2, from which it is evident
that AG£ is usually substantially greater than AG),1. The curve in Fig. 20.2
shows thalÂG\ for ions of the noble-gas type is proportional to
Everything said above about the solvation of ions relates to very dilute solu
tions. When we go over to solutions of medium and high concentration the pic
ture is considerably more complicated. Here the interaction of the ions with one
another is superimposed on the interaction of the ions with the solvent. In
the case of low concentrations of the electrolyte, this is manifested in the for
mation around ions, of ionic atmospheres of ions of the opposite sign. In more con-
Ch, 20. L I Q U I D A N D A M O R P H O U S S T A T E S 303
centrated solutions associations of solvated ions are formed — ion-pairs, trip

lets, etc. Finally, in very concentrated solutions, there is not enough solvent
for the formation of solvated shells and the ions are dcsolvated. In this con
nection K. Mishchenko and A. Sukhotin introduced the concept of the boundary
of complete solvation, the concentration of a solution in which there is still
sufficient solvent for the formation of the first solvated spheres. If the boundary
of complete solvation is crossed there is a sharp change in many of the pro
perties of solutions.
Many investigators are working today to elaborate a quantitative
theory of the liquid state, but great difficulties are involved—theore
tical analysis is complicated by the circumstance that in a liquid
Fig. 20.2. Relationship between primary hydration energy and total ionization
energy
vigorous interaction of the particles is combined with great disor-

derliness. This prevents, for one thing, the use of simple models
like those which rendered such good service in the formulation of
the theory of gaseous and solid states — the concept of a perfect
gas as a substance in a state of utmost rarefaction with complete
disorderliness of its particles, and the conception of the perfect cry
stal in which great density is combined with the complete orderli
ness of the particles.
20.4. The Amorphous State

Amorphous substances are distinguished from crystals by their
isotropism, that is, like liquids, the value of a given property is the
same no matter in what direction it is measured within the substan
ce. An amorphous structure, like the structure of a liquid, is chara
cterized by short-range order. For that reason the transition of an
amorphous substance from the solid to the liquid state is not marked by
a sudden change in properties, which is a second important distin

ction between an amorphous solid and a crystalline solid. Thus
unlike a crystalline substance which has a melting point (m.p.) at
which there is a sudden change in its properties (Fig. 20.3a), an
amorphous substance is characterized by a softening range, Ta — Tb,
and a continuous change in properties (Fig. 20.3&). This range maybe
of the order of tens or even hundreds of degrees, depending on the na
ture of the substance. The presence of a softening range over which
an amorphous substance is in a plastic state, is direct evidence of the
Fig. 20.3. Change in volume of crystalline substances (a) and amorphous sub
stances (b) when heated
structural nonequivalence of its particles. As a consequence there

is only gradual rupture of the bonds when heated. This nonequiva
lence is not very great, however, this being indicated indirectly by
the fact that the heat of converting an amorphous body into a crys
talline body is inconsiderable.
There was good reason for mentioning this particular transforma
tion, since an amorphous body is less stable than a crystalline one.
Therefore, in principle, any amorphous body should crystallize
and this process should be exothermic. For that reason the heat of
formation of an amorphous body is always less than the heat of
formation of a crystalline body from the same initial substances.
Thus the heats of formation of the amorphous and crystalline modifi
cations of B20 3 from the elements are —301 and —306 kcal/mole,
respectively. This example confirms the insignificant difference in
the structure of amorphous and crystalline substances, and the com-
rrl£nsurability of the heat of transformation (in the given case it is
equal to 5 kcal/mole) with the heats of fusion confirms the simila
rity of the amorphous to the liquid state.
There are often amorphous and crystalline forms of the same sub
stance. Thus there are amorphous forms of a number of elements,
such as sulphur, selenium, etc., and oxides—B20 3, S i0 2, Ge02,
etc. Nevertheless, it has been impossible to crystallize many amor

phous substances, including most organic polymers.
In practice the crystallization of amorphous substances is very
rarely encountered since structural changes are hampered by the
high viscosity. Therefore, unless recourse is had to special treat
ment, for example, to the lengthy action of high temperature, the
transition to the crystalline state proceeds at an infinitely slow rate.
In such cases it can be considered that the amorphous state is, practi
cally speaking, quite stable.
When likening an amorphous body to a liquid and regarding it as
a supercooled liquid which has solidified because of its great visco
sity, it should be remembered that unlike liquids, neighbouring
particles in an amorphous substance practically do not change pla
ces with one another. The high viscosity of melts impedes the move
ment and reorientation of molecules, which blocks the formation of
crystal nuclei. For that reason when liquids (melts) are quickly
cooled they solidify in the amorphous rather than the crystalline
state.
Since silicate glasses are typical amorphous bodies, the amorphous
state is often termed vitreous, a glass being understood to be an amor
phous (that is, uncrystallized), solidified melt. Because of their
extremely high viscosity glasses can remain without visible signs of
crystallization for thousands of years.
Polymers are also amorphous substances, but they differ from ordi
nary amorphous bodies in that they are formed from monomers by
the chemical combination of molecules, not by lowering of the tem
perature. Another distinction is that when they undergo transition
from the amorphous to the crystalline state, crystallization involves
only certain areas—the large size of the molecules impedes a high
degree of orderliness. It is difficult for the large and interlaced mole
cules to arrange themselves symmetrically in space.
The more symmetrical the particles themselves, the more sym
metrically they are arranged and the smaller the bond between them
in the liquid state, the more reason there is to expect that cooling
the liquid will cause its crystallization. Actually, molten metals,
the arrangement of whose atoms approximates close packing, readily
crystallize, while molten silicates often pass into the vitreous state.
Organic compounds containing hydroxyl groups, e.g., glycerol, un
like hydrocarbons do not, as a rule, crystallize on solidifying—the
affect of the hydrogen bonds is too great.
APPENDICES
I. Determination of the ratio e/m for an electron. Let us consider

a beam of electrons that passes between the plates of a plane capa
citor (Fig. 1.1). The force that acts on the electron in the electric
field is equal to
f el = eE = e ~ (1.1)
where e = charge of electron
E — field strength in the capacitor
V = voltage across the plates
d = distance between the plates
This force imparts acceleration to the electron in the direction

perpendicular to the original direction of the electron beam
fel = ™ea (1.2)
where a = acceleration of electron
me = mass of electron
APPENDICES 307
It follows from equation (LI) and (1.2) that

e V
rrip d
For a period of time t during which the electron is between the
plates, the beam is displaced by a distance y; the value of y is deter
mined by the relation
y = Y at* (1.3)
Obviously, t = Uv
where I = length of plate
v = velocity of the electron
Hence,
1 eVl2
y 2 medv2 (1.4)
The distance y can be determined from the distance A B on the
screen.
However, equation (1.4) contains the velocity of the electron v
which is as yet unknown. It can be found from the deviation of the
electron in a magnetic field. It is convenient to use such a field
that would compensate for the deviation of the electron in the ele
ctric field so that the direction of the electron beam remains unchan
ged; then
f el = fmagn (L5)
where f magn — force of the magnetic field acting on the electron.
Current i = ev corresponds to the moving electron. Hence,- accor
ding to electrodynamics, the magnetic field will act on the electron
moving perpendicular to the field with a force
fmagn = iH = evH (1.6)
where H = intensity of the magnetic field.
From equations (1.6), (1.5) and (1.1) it follows that
evH = e âr
or
In this way in equation (1.4) the only value remaining unknown is

e!me which can now be determined.
II. Characteristics of wave motion. Interference and diffraction
of waves. The wave process is characterized by the following para
meters: the wavelength X, its amplitude a (Fig. II .1), and the rate
of propagation u. Since in a unit of time the wave covers a distance
equal to w, the number of waves that fit into the distance u is equal
20*
308 APPENDICES
to u/X; this value, equal to the number of vibrations per unit of

time, is called frequency and is denoted as v; hence,
Since the frequency of electromagnetic vibrations for visible and

ultraviolet radiation is very great, it is usually more convenient to
use the value v, the wave number
As can be seen, v is equal to v multiplied by the constant u (i.e.,

v is directly proportional to v and wave number is thus a measure
of frequency). The wave number is the number of waves per centi
metre.
Figure II. 2 shows the wavelengths of different kinds of electro

magnetic radiation and the energies of the quanta of radiation (per
mole) corresponding to different wavelengths.
The phenomenon known as interference is the effect produced when
a train of waves is reinforced or weakened by another one (i.e., when
intensification or extinction takes place). Reinforcement is observed
when the waves are in phase, i.e., when the rises and falls of both
waves coincide. On the contrary, when the rise of one wave coincides
with the fall of the other one (the waves are out-of-phase), the waves
cancel each other out, i.e., extinction takes place.
Diffraction is the phenomenon that occurs when waves meet obsta
cles; it is due to the separation of the wave into several groups of
waves that interfere with each other. Such an obstacle may be a
APPENDICES 309
diffraction grating (Fig. II.3) with a great number of clear slits equi
distant from each other and of a size that is of the same order as that
of the wavelength.
106 103 1 10'3 10'6 10'9 10'tz

slj cm i i i i i i i j____ i____ i____i_____i____i____ i— i— i— i— i— L— i—
, 10'9 10~s W'3 1 fO3 10s to9
E} k n a L j m n l e ■ ■ ■ ■ ■ ■ ■ ■____i___ i____i____i___ i— i— i— i— i— i— i— l_
R otation o f Vibrations\ Transitions N uclear

molecules of a to m s / of electrons in processes
in molecules) inner she U s
' Transitions o f electrons
in o u te r shells of atom s
and molecules
Fig. 11.2. Wavelengths and energies of different kinds of electromagnetic radia
tion
Fig. II.3. Schematic drawing of formation of diffraction pattern when radiation

passes through a diffraction grating
Since the wave can pass the grating only through the clear slits,
each slit becomes an independent source of waves which propagate
in all directions and interfere with each other. As seen in Fig. II.4,
from the point of view of simple geometry, it follows that the waves
310 APPENDICES
are in phase in the directions defined by the relation
where n = 1, 2, 3, . . .
0 = angle between the given direction and the line perpendi
cular to the grating
K = wavelength
d = distance between the clear slits of the grating
Intensification of the waves occurs in the directions that satisfy
this relation. On the contrary, in directions defined by similar rela-
Fig. 11.4. Interference of waves on passing through a diffraction grating
tions containing half-integer values of n (n = V2, 3/2, 5/2, • • •)>

the waves are out-of-phase as a result of which extinction of the
waves occurs.
The diffraction pattern is observed on a screen placed in front of
the grating. A diffraction pattern (the periodic change in illumina
tion along the screen) is shown schematically in Fig. II.3. It is evi
dent that the wavelength k can readily be determined by measuring
the angle 0 if the distance between the clear slits in the grating is
known.
The diffraction pattern can be observed not only when the wave
passes through a grating, but also when it is reflected from the latter.
Diffraction gratings for visible light (k = 4000-7600 A) are produ
ced by incising on a glass plate (with the aid of a diamond point)
very many parallel lines close to each other; there are 1000 and more
lines per 1 mm of length. Naturally occurring crystals are used as
diffraction gratings for X-rays. The particles in crystals are arran
ged in a strictly definite order and the distances between the layers
of the atoms are of the same order as the wavelengths of X-rays.
APPENDICES 311
III. Construction of the Schrodinger equation. The equation for

a monochromatic electromagnetic wave is
d2a d2a d 2a / 2 ji \ 2 A /jit /i\
w + w + -m + \ — ) a = 0 (IIL1)
where a = amplitude
X = wavelength
To obtain the Schrodinger equation, in this relation a is substi
tuted by the ty-function and X by its value as determined from the
de Broglie relation X = hip, where p — m6mentum of the particle
(p = mu); thus we obtain
^ + $ + J3 -+SV’i>-° <ni-2>
The kinetic energy of the particle E h is related to the momentum
by the equation
F — p2
Eh- 2 ^
Hence,
d2yp d2^ 8j i 2mEk _ 0 ,ttt ^
The resulting equation does not take into account the potential
energy of the particle; it is the equation for the motion of a free
particle. When a particle moves in a potential field, its total energy
E is equal to the sum of the kinetic energy and the potential energy
U which depends on the coordinates x , y, and z
E = & + U (*>V>*) (HI.4)
Hence,
p2= 2m [E— U (x, y, z)] (III.5)IV
.
By substituting the value of p 2 from equation (III.5) into equation
(III.2) we obtain
d2ty ,52t i d2v | 8n2m p j, ( „\i , _ n
dx2 ^ dy2 1 dz2 '
or
/i2 / d ty , d2v
1 9 * \ \-TI ( t yt
ii
Z) *Y
ih —
— "Fih
8ji2m \ dx2 1 dy2 1 Y
IV. Polarization of light. Light is t ansverse electromagnetic

vibrations. This means that the electric and magnetic fields vibrate
perpendicularly to the direction of propagation of the waves. These
vibrations are in different planes passing through the axis of the
light beam (Fig. IV, 1).
The light emitted by different sources (sun, candle, electric lamp,
etc.) is not polarized, that is, the vibrations are in all possible planes.
312 APPENDICES
Light is said to be polarized if the vibrations of the field are in a sin

gle plane. The plane of the vibrations of the magnetic field is called the
plane of polarization.
Polarized light can be produced by different methods. One exten
sively employed method makes use of the optical properties of
crystals.
When light is passed through a crystal in a direction which does
not coincide with its optical axis, the light ray is split into two pola
rized rays issuing from the crystal at a certain angle to one another
Fig. IV. 1. Vibrations of electric and magnetic fields in electromagnetic wave
(Fig. IV.2)1. This is because the refraction indices in the crystal are
different for vibrations taking place in different planes. This is very
pronounced in crystals of the mineral calcite (CaC03), the angle
between the rays being 6.5°. Large transparent crystals of this mine
ral, known as Iceland spar, are used for obtaining polarized light.
Fig. IV.2. Passage of light ray

through crystal of Iceland spar
In order to produce polarized light one of the rays issuing from

the crystal must be separated from the other. The optical system
known as a Nicol prism is often used for this purpose.
The Nicol prism (Fig. IV.3) is a crystal of Iceland spar sawn in
half along the diagonal and cemented with a substance whose refra
ctive index is greater than that of Iceland spar. In this prism one of
the rays undergoes complete internal reflection and is diverted to
thes lateral face which is painted black to absorb the light. The
1 This phenomenon is not observed in crystals of the cubic system.
APPENDICES 313
second ray falls on the joint at a different angle and passes through
the prism. Thus by passing light through a Nicol prism we obtain
a beam of polarized light. If a second Nicol prism is placed in the
path of this ray and rotated 90° about its axis, the polarized light
will not pass through. Thus by means of a second Nicol prism—the
analyzer—it is possible to determine the direction of the plane of
polarization of the light. When examining optically active substan

ces, the test substance is placed between two Nicol prisms and the
angle through which the plane of polarization is rotated is deter
mined.
V. Derivation of relationship describing electron diffraction by
molecules. To obtain a relationship expressing the intensity of an
electron stream scattered through a certain angle 0 to the initial

direction of the electron beam, we shall consider the diffraction of
electrons by a diatomic molecule consisting of atoms A and B, whose
nuclei are at a distance r apart.
Since the wave properties of electrons are manifested in their dif
fraction, the electron stream in the given case can be considered a
beam of wavelength A,. Mark and Wierl (Germany) who were the
first to apply electron diffraction for the study of molecules emplo
yed unchanged the theory of X-ray scattering previously formulated
by Debye.
Let a plane wave fall on molecule AB (Fig. V.l). When the wave
impinges on atoms it is scattered in all possible directions, and the
314 APPENDICES
scattered waves interfere with one another. Let us consider the inter
ference of the scattered waves in a direction making a certain angle
0 with the initial direction of the electron beam OA.
As in the example with the diffraction grating (see Appendix II)
the intensification or damping depends on the phase difference p
which is connected with the path difference 6 of the rays according to
the relationship
2nd
X (V.l)
As can be seen from Fig. V.l
6 = AM — AN
where A M and A N are projections of the section AB, equal to 7%
on the directions of the incident and scattered rays.
If sections A E and A S, equal to r, are laid off on the above dire
ctions, and a straight line is drawn through points S and E, it will
form angle q) with the line A B . By means of this auxiliary constru
ction, 6 can be expressed through r and angles 0 and q). Actually,
the difference between the projections of AB on directions OA and
AP can be replaced by the difference between the projections on the
direction A B , of sections equal to AB which are laid off on directions
OA and AP. Then 6 is equal to the difference between the projections
of A E and A S on AB. But since the difference between the proje
ctions of two sides of a triangle on any direction is equal to the pro
jection of the third side on the same direction, 6 will be equal to
the projection of ES on AB, i.e., E S cos q). It is evident that
ES = 2r sin —
whence
q
6 = 2r sin y cos q>
It follows that the phase difference of waves moving in the dire
ction AP will be expressed by the relationship
4nr . 0
p = —j —sin y cos q)
or
p = sr cos q) (V.2)
where
(V.3)
The amplitudes o|)A and of waves scattered by atoms A and B

depend on the intensity of the interaction of the incident waves with
the atoms. In the given case, where we are examining the diffra-
APPENDICES 315
ction of fast electrons, the amplitudes can be considered proportio

nal to the nuclear charges Z. The intensity of the electron stream is
proportional to the square of the amplitude of the resulting wave
(we are considering de Broglie waves, and the electron density is
determined by the q u an tity ^ 2).
When waves interfere, vector addition of
their amplitudes takes place (Fig. V.2). Angle
p between the vector directions is equal to
the phase difference.
It follows from Fig. V.2 that the square of
the amplitude of the wave resulting from
interference is expressed by the relationship
^ S= Â + ^B + 2l|>A%lCOSp (V.4)
Fig. V.2. Vector addition of amplitudes
Since the intensity of the electron stream I is proportional to i|)2,

while \|)A ~ ZA and ~ Zb ( ~ is the proportionality sign), we
can write
I *** Z \ -|- Zb 2Za Zb cos p (V.5)
In order to make the following operations less cumbersome we shall
consider only that part of the total intensity of the electron stream
which is dependent on p. For it we can write
I ~ ZA Zb cos p
It is necessary, however, to take account of the fact that phase
difference p which we have calculated depends on the position of
the molecule relative to the electron stream. Any arrangement of
the molecules in a gas is possible, and therefore in.order to find the
actual intensity of the electron stream the mean value of cos p must
be calculated for all possible positions of the molecules.
The position of the molecule relative to the incident wave when
the value of angle 0 is fixed can be determined byvthe angle <p. The
refore, we must calculate the mean value of cos p for all possible
values of cp from 0 to jc. It is determined by the relationship
[ cos p dp
cos p = (V.6)
I dp
in which the integrals must be taken within such limits that angle (p
ranges from 0 to ji .
We integrate. According to (V.2)
dp = —sr sin <p d (p
316 APPENDICES
Therefore,
wt
j dp = — j sr sin (p d(p — sr cos y = —2sr
o
JT
j cos p dp = — j cos (sr cos <p) sr sin <pd(p
o
We calculate the last integral, putting cos = u; then cfo =
= —sin cp d cp, and consequently
JI JC Jl
— j cos (sr cos <p) sr sin (p dcp — J cos (sru) sr da — sr smJ sru)
= sin (sr cos q>) = [ —sin sr —sin sr]= —2 sin sr
It follows that the mean value of cos p will be

------- sin sr
cos •p = -------
cr
(V.7)
and the expression for the intensity of the scattered electrons will
have the form
sm sr
I ~ Za^b ■ (V.8)
For a polyatomic molecule containing n atoms of two species,
I and //I, we must carry out a vectorial summation of the amplitudes
of the de Broglie waves for the electrons scattered by each atom.
This is achieved by constructing the appropriate polygon of vectors
in exactly the same way as the triangle in Fig. V.2 was constructed.
The line closing the polygon gives the resulting amplitude. In this
case the total intensity of the scattered electrons is expressed by
the relationship
(v -9)
i i
Summation is carried out for all interatomic distances, including
r = 0, when the neighbour of the atom under consideration is taken
to be this atom itself; in that case (sin sr)/sr = 1. Thus, as the result
of such summation there appear the members Z\ and Z2m which were
omitted for the time being when expression (V.8) was derived.
The theory set forth here is concerned only with coherent (elastic)
electron scattering by molecules, in which the molecules do not pass
into an excited state. But during the bombardment of molecules
APPENDICES 317
by fast electrons, incoherent (inelastic) scattering of the electrons

also takes place, in which the electrons give up part of their energy
to the molecules, transferring them to an excited state. In this case
wavelength k of the de Broglie waves of the electrons incident on
the molecules changes. Incoherent scattering gives a solid background
without maxima, which quickly attenuates as parameter s increases.
The actual curve of the change in the intensity of scattered electrons
Fig. V.3. Distribution of intensity of

electron stream scattered by Br2 mo
lecules
1 —coherent scattering (curve displaced
upward); 2 —incoherent scattering; 3 —sum
mation curve
is determined by the sum of the coherent and incoherent scattering.

Such summation for electrons scattered by Br2 molecules is shown
in Fig. V.3.
When electron diffraction pictures are interpreted visually, inco
herent scattering need not be taken into account since the eye senses
disturbances in the regular decrease in the blackening of the photo
graphic plate and easily perceives maxima and minima correspon
ding to the curve of coherent scattering.
VI. Moment of inertia. Imagine that a material point rotates
about an axis (Fig. V I.1). The velocity of a moving body is the deri
vative of the distance traversed with respect to time
~~~di
__ ds
( VI A)
In the case of rotary motion the element of the distance ds can be

expressed as the product of the radius r of the circle along which
the material point is moving by the value of the angle d(p corres
ponding to the distance ds travelled
ds = r dcp (VI.2)
Then instead of (VI.l) we can write
dip
318 APPENDICES
The magnitude dyldt, denoted by the letter co, is the angular velo
city. Thus
v = rco (VI.4)
The kinetic energy of a moving body is defined by the equation
Taking account of (VI.4) we have

jp mr2a)2 /CD2 /\ tt c\
= = ~ (VI.5)
The magnitude I = mr2 is called the moment of inertia. Equation
(VI.5) relates to a material point. The moment of inertia of a body
Fig. V I.2. Rotation of dia

tomic molecule
is the sum of the moments of inertia of the material points compo

sing the body
/= 2 » irf (VI.6)
We next find the relationship between the energy of rotary motion
the angular momentum M -- mvr and the moment of inertia I =

= mr2. Since
M 2 = m2v2r2
it is evident that
Erot — M 2/2mrz
Thus
ETot = M 2l2I (VI.7)
Let us find an expression for the moment of inertia of a diatomic
molecule consisting of atoms A and B whose nuclei are at a distance
r apart (Fig. V I.2).
APPENDICES 319
Free rotation of the molecule will take place about an axis pas
sing through the centre of mass perpendicular to the line connecting
the atomic nuclei. Let the distances of the nuclei from the axis of
rotation be a and b (Fig. VI.2). During rotation the following rela
tionship should be met
mAa = mBb (VI. 8)
where mA and mB are the masses of atoms A and B. Furthermore,
a+ b= r (VI.9)
By solving the set of Equations (VI.8) and (VI.9), we find expres
sions for a and b
rm B ^ rm A
mA -\-m B * mA -p mB
(VI.10)
In accordance with (VI.6) the moment of inertia of molecule AB
will be
/ = mAa2 + mBb2
Substituting the expressions for a and b, we obtain
/ = [mA mB /(mA + raB)] r2 (V I.ll)
The magnitude mAB, defined by the relationship
wab = mAmB /(mA + mB) (VI. 12)
is called the reduced mass. Thus
/ - m*ABr2 (VI. 13)
VII. Expressions for wave functions of hybrid orbitals.
sp-hybridization
% -- 0l>. ^pv); $2 = — (t|>, - % x)

spMiybridization
=■ + V 2 % X)
= ITT ( ^ ~ ^ ^ )
sp^hybridization
1 1
320 APPENDICES
VIII. Electron spin and magnetic properties of matter. If a mag

net with ‘magnetic charges’ at the ends equal to q is placed in a mag
netic field, a pair of forces (a couple) will act on it, that strives to
position the magnet along the lines of force of the magnetic field
(Fig. V III.1). The moment of the forces M that turns the magnet is
equal to
M = qlH sin cp (VIII.1)
where I = length of magnet
H = magnetic field strength
(p = angle formed by the magnet and the lines of force of the
field
The product ql = iimagn is called the magnetic moment.
As is known from the course of general physics, on a current-
carrying circuit in a magnetic field, there also act a pair of forces
Fig. VIII. 1. Interaction of magnet Fig. VI11.2. Current-car

and field rying circuit
that strive to place it perpendicular to the field. Therefore, the

circuit has a magnetic moment; it can be shown that in the given
case (Fig. V III.2)
\lmagn = i S (VIII.2)
where i = current
S = area of the circuit
The magnetic moment is a vector directed perpendicular to the
plane of the circuit. The motion of a charged particle along a closed
path is similar to that of the current in the circuit; it also gives rise
to a magnetic moment. If, in accordance with the Bohr theory, it is
assumed that the electron moves in an orbit, the orbital magnetic
moment of the electron can be calculated with the aid of equation
APPENDICES 321
(V III.2). For the first Bohr orbit the magnetic moment is equal to
(VIII.3)
where e = charge of electron
h = Planck’s constant divided by 2jt
mc = mass of electron
c = velocity of light
l^magn characterizes the value of the projection of the orbital magne
tic moment of the electron; it is equal to 0.927 ‘lO-20 erg/gauss and is
called the Bohr magneton.
The projection of the orbital magnetic moment of the electron on
the direction of the magnetic field is equal to the product of the
Bohr magneton and the magnetic quantum number m. It is evident
that when m is equal to zero, the projection under consideration is
also equal to zero.
An electron has an inherent magnetic moment due to its spin. The
magnitude of the projection of this magnetic moment is equal to one
Bohr magneton; it can have a positive or negative sign depending on
the direction of the spin in space. This projection cannot be equal to
zero.
The composition of magnetic moments of electrons in a molecule
is carried out by the rule of addition of vectors. When all the ele
ctrons of a molecule are paired, the resultant magnetic moment is equal
to zero.
The presence or absence of a resultant magnetic moment in a mole
cule can easily be determined by the interaction of the given sub
stance with a non-uniform magnetic field. If the molecules of the
substance have a magnetic moment, the substance is said to be paramag
netic; it is attracted by the magnetic field. In the absence of a magnetic
moment in the molecules, the substance will be diamagnetic; it is repel
led by the magnetic field. The resultant magnetic moment of the mole
cules can be determined by the intensity of the interaction of the
substance with the magnetic field.
Oxygen is a paramagnetic substance. On approaching a pole of a
strong magnet to the surface of liquefied oxygen, it is readily noticed
that oxygen is attracted to the magnet. Determinations have shown
that the magnetic moment of the 0 2 molecule corresponds to the
presence of two unpaired electrons.
IX. Calculation of the absorption spectra of poly methylene dyes.
As was pointed out on p. .194, the jr-electrons in] a chain of
carbon atoms containing conjugated double bonds, which can be
denoted —(CH = CH)m—, are not localized, but can move freely
along the chain. The conditions of movement of an electron in such
a polymethylene chain (the —CH= radical is called methylene or
methene) correspond quite closely to the model of the unidimensional
21 3aK . 15648
322 APPENDICES
potential well (see p. 39-43). By means of this model it is possible to

calculate fairly accurately the absorption spectra of a number of
compounds represented by the general formulae
The path along which the electron can travel is indicated by a dotted
line.
The energy of an electron in a unidimensional potential well is
expressed by the relationship
8mea*
In the above compounds the number of delocalized jx-electrons is
equal to 2 m. In the unidimensional potential well these electrons
will occupy m first energy levels, for which n is equal to 1, 2, . . .
. . ., m; on each energy level there will be two electrons with oppo
site spins.
Since all the energy levels for which n ^ m will be filled, the
transfer of energy to a molecule occupying a level where n = m
will cause its transition to the next level for which n = m + 1.
The molecule will absorb quanta corresponding to the energy of the
given transition
E= — Em
Wavelength % of the radiation corresponding to this energy can be

calculated by means of the relationships
E = hv; v= y
Dimension a of the potential well is equal to the length of the chain

of atoms along which the electron can move. If the distance between
the carbon atoms in the polymethylene chain is taken to be 1.40 A
and the ends of the ‘well’ are considered to be at a distance of one
APPEND1CE8 323
bond length beyond the nitrogen atoms, the calculation gives the
results cited in Table IX .1. It can be seen that the calculated values
of A are quite close to the observed absorption maxima for compounds
of the specified types.
Table I X .1
Wavelengths of Radiation Absorbed by Compounds A and B
W a v e le n g th o f m a x im u m a b s o rp tio n ;
e x p e rim e n ta l v a lu e s , A W a v e le n g th c a lc u la
m te d fro m p o te n tia l
w e ll m o d e l, A
com pound A com pound B
2 4250 _ 3280
3 5600 — 4540
4 6500 5900 5800
5 7600 7100 7 0 .6 0
6 8700 8200 8330
7 9900 9300 9600
X. Solution of homogeneous sets of linear equations. Sets of

equations of the type
011*1 + fl1 2 * 2 + • • • + O'i n x n = 0
021*1 + 022*2 + • • • + 02 nx n = 0 .
0 J ii* i + 0 n 2 * 2 + • • • + & nnx n = 0 >
where xt, x 2, . . xn are unknowns and aii9 a12 . . . are coefficients

the first subscript corresponds to the number of the equation, and)
the second, to the number of the term) are said to be homogeneous.
For such systems two types of solutions are possible. In the first
,place it is clear that all the equations will be satisfied if all the
unknowns are equal to zero: z t = 0, x 2 = 0, . . xn = 0. This is
called a zero or trivial solution, and it is always possible. If a set of
equations has no other solution, the equations are said to be incom
patible.
There are also sets of equations for which a non-trivial solution
is possible. For such a solution to be possible, the determinant com
posed of the coefficients must be equal to zero. This is expressed as
follows:
011012 • • • 0 1 n
021fl22 • • • 0 2 n n
0n l0/i2 • • • 0nn
21*
324 APPENDICES
Next let us see what is meant by a determinant. Let there be a

table composed of magnitudes of some kind, for example
fa b
A table consisting of magnitudes arranged in the- form of rows and

columns is called a m a t r i x . The magnitudes forming a matrix are
called its e l e m e n t s . If the number of rows is equal to the number of
columns, the matrix is s q u a r e .
A determinant is a quantity calculated from the values of the
elements of a square matrix according to a certain rule. A determi
nant is denoted by drawing vertical lines on both sides of the corres
ponding matrix.
If a matrix consists of four elements, the determinant is the diffe
rence of the products of the elements forming the diagonals, for
example
a b
a d — be
c d
Before considering the calculation of more complex determinants,

we must introduce the concept of the m i n o r . The minor is a determi
nant obtained from a given matrix by crossing out a line and a co
lumn containing some chosen element.
Now we can formulate a rule for calculating any determinant.
To find the value of the determinant each element of the first line
must be multiplied by the corresponding minor and the products
added algebraically, writing the product with the plus sign if the
element is odd (1st, 3rd, etc.) and with the minus sign if it is even.
If necessary, values of the minors can be calculated in the same way.
Thus, in the long run, any determinant 's expressed by the algebraic
sum of the products of its elements. As an example we can take the
calculation of the determinant used in the treatment of the buta
diene molecule by the Hiickel method:
H 0 0
X 1 0 1 1 0
1 x 1 0
X 1 X 1 —1 0 X 1
0 1 x 1
0 1 *11 0 1 X
0 0 1 i
X 1 1 1 X 1 0 1
X2 — X - l •fl
0 x
O
1 X 1 X
= x 2 (x2 — 1 ) — x x - ^ x 2 + 1 = x k— x 2— x 2— x 2 + 1 = x * — 3x2+ 1
VALUES OF UNITS OF MEASURE
AND PHYSICAL CONSTANTS USED
IN THE BOOK IN THE SI SYSTEM OF UNITS
1 A = 10-io m
1 g = 10~3 kg
1 kcal = 4.184xl03 J
1 eV = 1.G02X 10“19 J
1 erg = 10“7 J
Electron charge e = 1.602xlO -19 G
Velocity of light c = 2.998xl08 m/s
Planck’s constant h=^ 6.625Xl0~34 J-s
Bohr magneton \^magn —9.273 XlO-24 a-rn2
NAME INDEX
Arens, J., 95 Dulong, P., 67

Arrhenius, S., 298 Dumas, J., 117, 118
Avogadro, A., 12, 13, 15, 16, 128,
154, 265, 287
Einstein, A., 29, 30
Balmer, J., 21
Bartlett, W., 62 Faraday, M., 14
Beketov, N., 254 Fedorov, Y., 279
Bergman, T., 116-117 Fittig, R., 116
Bermal, A., 297, 301 Fowler, C., 297, 301, 302
Berthollet, P., 116 Franck, J., 24
Berzelius, J., 117 Frankland, E., 118
Biron, E., 87 Frenkel, Y., 295
Bohr, N., 25, 26, 27, 35, 114, 321, 325 Friedrich, H., 273
Boky, G., 92, 269
Born, M., 222, 223, 287-292, 300, 301,
302 Gadolin, A., 269, 270
Bdrnstein, E., 95 Gapon, E., 18
Brackett, 21 Gerhardt, C., 117, 118
Bragg, W., 274 Germer, L., 34
Butlerov, A., 118-120, 122, 137, 186, Gibbs, J., 91
233 Goldsmidt, H., 95
Gomberg, M., 115, 116
Guggenheim, E., 15
Clapeyron, B., 262
Compton, A., 31, 33 Haber, F., 290, 291
Coolidge, A., 173 Hamilton, W., 161
Coulomb, C., 14, 101, 171, 214, 227, Hartree, D., 36
237, 288 Hauy, R., 269
Hegel, G., 117
Heisenberg, W., 18, 35
Dalton, J., 12, 269 Heitler, W., 159, 168, 169, 172, 174
Davisson, 34 Hertz, G., 24, 28
Davy, H., 117 Hess, G., 291
Debye, P., 156, 264, 276, 277, 295, Hiickel, W., 206, '211-221, 249, 324
313 Hund, F., 57, 177, 209, 240, 245
De Broglie, M., 35, 134, 316, 317
Dewar, J., 195
Dorosh, A., 300 Ilinsky, M., 254
Drakin, S., 299, 302 Ivanenko, D., 18
N A M E I NDEX 327
James, H., 173 Petit, A., 67

Pfund, 21
Planck, M., 22, 23, 25, 29, 30, 36, 58,
Kablukov, I., 298 265, 325
Kapustinsky, A., 292, 293, 299 Prue, G., 15
Karapetyants, M., 8, 75
Keesom, W., 264
Kistyakovsky, V., 298 Rome de L’isle, 269
Knipping, 273-274 Roothan, 173
Kolos, W., 173 Rutherford, E., 12, 18, 25
Kossel, W., 100, 151, 237, 238 Rydberg, J., 22
Lande, A., 222 Scherrer, P., 276

Landolt, H., 95 Schrodinger, E., 35, 39, 43, 159, 160,
Langmuir, I., 152 311
Laplace, P., 161 Shchukarev, S., 87
Laue, M., 273 Skryshevsky, A., 300
Le Bel, J., 122, 123 Slater, J., 178, 182, 214
Lennard-Jones, G., 267 Sommerfeld, A., 27, 35, 47, 320
Lewis, G., 152, 175 Stark, J., 20
Liebig, J., 120, 122 Stoletov, A., 28
Lomonosov, M., 12, 120, 262, 273 Sukhotin, A., 303
London, H., 159, 168, 169, 172, 174,
175, 265
Lyman, T., 21 Tartakovski, P., 34
Thiele, F., 193
Thomson, G., 34
Madelung, 287 Thomson, J., 17, 203
Magnus, H., 237, 238
Mark, H., 313
Mendeleev, D., 11, 64, 65, 69, 74, Van der Waals, 263, 264, 265
89, 113, 262, 297 Van’t Hoff, J., 122
Mikhailov, V., 302 Vulf, Y., 275
Millikan, R., 13, 29
Mishchenko, K., 303
Moseley, H., 67, 68 Werner, A., 233, 236
Mulliken, R., 149, 204 Wierl, 313
Wohler, F., 120
Wurtz, C., 116
Newton, I., 117
Yakushevsky, B., 299
Paschen, F., 21
Pasteur, L., 123, 126
Pauli, W., 55, 76, 83, 170, 174, 207 Zeeman, P., 20
Pauling, L., 150, 178, 182, 232, 233
SUBJECT INDEX
Acids, 99 Atoms, 11, 12, 17

heteropolyacids, 100 many-electron, 27, 54
isopolyacids, 100 mass and size of, 15
oxygen-containing, 99, 101, 231 mesatoms, 11
Acid and basic properties, 99, 101-112 planetary model of, 25
Actinide contraction, 106 positronium, 11
Actinides, 66, 71, 84 radioactive, 11
Addenda, 233 Avogadro number, 12-15, 128, 154
Aggregate states, 91, 260-263 265, 287
gaseous, 200
liquid, 293-303
plasma, 263 Bases, 99
solid, 260 Benzene, structure of, 192-195
transformations of, 262 Bohr magneton, 321, 325
see also Amorphous state; Cry Bond, see also Bonds
stalline state banana, 202
Amorphous state, substances in, 303- conjugated double, 194, 321
305 covalent, 152, 159-221, 229, 231-
crystallization of, 304, 305 232
softening range of, 304 donner-acceptor, 196-201, 302
stability of, 305 double, 126, 148, 187, 193, 194
see also Glass; Isotropism; Poly heteropolar, 151
mers homeopolar, 152
Amphoteric compounds, 99, 103 in electron deficient molecules,
Angstrom unit, 17 201
Angular momentum, 25, 26, 53, 147 ionic, 151, 221-233
Anisotropy, 267 polar, 231-233
Atomic nucleus, 10, 17, 18, 166 polar covalent, 152, 228
Atomic number, 11 single, 126, 129, 148, 176, 187
Atomic radii, 91-93 triple, 148, 187, 190, 197, 198
of metals, 92 see also Chemical bond; Theory
of nonmetals, 93 Bond-dissociation energy, 133, 151,
Atomic spectra, 19-27 167
of hydrogen, 20, 27 Bond order, 212
of lithium, 59 calculation of, 220
of other elements, 21 Bonds
originatioQ of, 57-60 ji bond, 188, 191, 193, 216, 217,
Atomic structure, 25-27, 28, 39, 116 253
Atomic units, 36 a bond, 187, 191, 217, 252
Atomic volume, 65, 66 O — H bond, 101, 151
Atomic weight, 12, 15 hydrogen bond, see below
SUBJECT I N D E X 329
R — H bond, 98, 101, 112, 186 stability of, 237

R — 0 — bond, 98, 101, 103, 112 see Isomerism: Cis-trans isome
Born-Haber cycle, 290, 291 rism; Stereoisomerism; Chemi
Born repulsion factor, 222, 289 cal bond
Complex ions, 227
Complexes
Carbon, 122, 123, 177, 182, 209, 217 jt-complexes, 253, 254
Cathode rays, 15, 17 hydrated, 300
Chemical affinity, 116 sandwich, 254
Chemical bond, 92, 210 Complexing atom, 233
basic characteristics of, 127-137 Complexing ion, 233
length, 127, 135, 210 Compton effect, 28, 31-33
strength (energy), 101, 125, Computer, electronic, 60, 174, 247,
133, 210, 223, 287 248, 276
valence angles, 127, 129 d-Contraction, 97
basic types of, 149-159 Convergent series, 288
conceptions of, 116-118 Coordinates
Chemical bond in complex compounds, cartesian, 46
245 polar system, 46
electrostatic explanation of, 237- Coordination number, 97-98, 233,
238 234, 297
quantum mechanical interpreta Coordination sphere, 233
tion of, 238 Copper atom, radius of, 16
Chemical elements, 11, fly-leaf Coulomb’s law, 26, 101, 222, 225, 227,
Chemical interaction, 151, 174-179, 237
263 Covalent bond, see Bond
attractive force, 63, 166 Crystal compressibility, 289, 290, 292
dispersion effect in, 166, Crystal lattices, 230, 269, 273
265 classification of, 279
induced effect in, 264 stability of, 231
orientation effect in, 264 types of, 277-279
repulsive forces, 63, 166 atomic, 277
Chemical structure ionic, 278
spatial isomerism, 122-127 metallic, 279
structural isomerism, 120-122 molecular, 278
theory of, 28, 116-120 unit cells of, 279
see also Atomic structure, Mole Crystal lattice energy, 287-294, 298
cular structure methods of determining
Chlorophyll, 234 experimental, by Born-Ha
Cis-trans isomerism ber cycle, 290-292
in complex compounds, 236 theoretical, by Born equa
in organic compounds, 126, 157, tion, 287-290
190 by Kapustinsky equa
Classical mechanics, 35, 39 tion, 292-293
Clathrates, 285 Crystal structure, study of, 273-27 7
Close packing, 16, 98, 281, 305 by electron diffraction, 277
cubic, 281, 282 by neutron diffraction, 277
hexagonal, 281, 282 by X-ray diffraction, 274
Colouring of compounds, 231 powder method, 276
Comparative calculation, methods of, rotation method, 275
75-76, 128, 135, 227, 256, 257, long-range order of, 296
293, 300 Crystal structures of:
Complex compounds, 233-254 caesium chloride, 283
classes of, 234 diamond, 283, 284
coordination bond in, 237, 301 fluorite, 286
nomenclature, 235 ice, 285
reactivity of, 241 metals, 280-283
330 SUBJECT I N D E X
cube-centered cubic, 280, 281 gratings for, 34, 274

face-centered cubic, 280, patterns, 139-144
281 study of molecules by, 313-316
hexagonal, 283, 284 diatomic, 313
rutile, 286 polyatomic, 316
sodium chloride, 283 Electron diffraction by molecules,
sphalerite, 284 313-317
wurtzite, 284 Electron gas, 279
Crystalline hydrates, 301 Electron mass, 18
Crystalline state, characteristics of, Electron pairs, 152, 175, 178, 196
267—303 shared, 152, 198
anisotropy, 267 unshared, 187, 200, 250, 302
isomorphism, 273 Electron shells and subshells, 55, 64
metastability, 272 designations of, 55
polymorphism, 271-272 energy of electrons in, 82, 83
Crystallography, 267 filling of, 76-85
classification in, 269-272 max. number of electrons in,
symmetry types, 271 55, 83
laws of, 267-269 Electron spin, 53, 57, 170, 174
and magnetic properties, 320-321
De Broglie waves, 33-35 Electron states, 39, 46, 48, 55, 56,
137, 162
Debye powder patterns, 277 degenerate, 46, 53
Degree of dissociation, 99 designations for, 49
Degrees of freedom, 45 see Ground state; Excited state
Delocalization of electrons, 192-194, Electron-volt, 24
252 Electronegativity, 149-151
Dewar structures, 195 Pauling’s system of, 150
Diamagnetism, 241, 321 polar bond and, 231-233
Dielectric constant, 101, 153 scales, 150
Diffraction, 28, 33, 34, 307-309 Electronic structure, 17
Diffraction gratings, 34, 274, 310, 313 and properties, 102-112
Dipole moment, 133, 153-157, 225, of atoms in the ground state, 76-
228 80
Donnor-acceptor bond, see Bond of hydrogen, 46
Dulong and Petit’s rule, 67 periodic law and, 64-114
Electronography, 139, 255
Effective charges, 158-159, 201 Electrons, 11, 17-18
Einstein’s law binding and antibinding, 208-211,
for photoelectric effect, 29 233
of interdependence of mass and energy of (potential and kinetic),
energy, 30 23, 26, 37
Electrolytes, solutions of, 297-303 isolation of, 17
ji-Electron, 39, 194, 212, 217 paired and unpaired, 175-179,
Electron affinity, 63, 112, 149, 224, 186, 197, 321
291 ratio of charge to mass, 17, 306-
Electron bombardment, 24-25, 316 307
Electron charge, 13, 17 Energy levels, 24, 41, 43, 54, 55, 59,
Electron cloud, 47, 48, 50, 162, 172, 145
183-185 in hydrogen atom, 24, 59
perturbation of, 193 in lithium atom, 58-59
Electron deficient molecules, 201-203 Equations (also Formulae)
Electron density, 47, 158, 162, 172, Bohr equation, 25
195 . Born equation (energy of hydra
Electron diffraction analysis, 34, 138- tion), 301, 302
145 Born equation (interaction of
camera for, 34, 138 ions), 223
Bom equation (lattice energy), examples of, 213-221

287, 292, 293 fields of, 211-212
Bragg-Vulf equation, 274 conceptions in, 212-213
Gompton-effect equation, 31-33 their calculation, 220-221
Hund’s rule, 57, 177, 209, 240
Debye equation, 264 Hybridization, 182-187, 190-192
Einstein’s equation (mass and and structure of complexes, 239-
energy), 30 . 241
Einstein’s equation (photoele see also Orbitals, hybrid
ctric effect), 29 Hydration, 101, 297
Kapustinsky’s equation, 292-293 energy of, 299
Keesom equation, 264 calculation of, 300-303
Lennard-Jones formula, 267 heat of, 298
Lomonosov’s equation, 262 of ions, 298-300
Men delee v-Clapeyron equation, primary, 300
262 secondary, 300
Moseley’s formula, 67 Hydrides, 99, 107, 228
Pauling equation, 231 Hydrogen atom
Planck’s formula, 22, 30 energy levels of electron, 24, 53,
Rydberg formula, 22 76
Schrodinger equation, 35, 39, mass, 16
43, 159-166 place in periodic table, 76, 109
Secular equations, 165, 214, 215 properties, 70, 103, 109
Van der Waals’ equation, 263 radius, 101
Excited state, 58, 137 spectrum, 20-21, 27
structure, 26, 46, 77
wave function, 180, 181
Fedorov symmetry groups, 279 Hydrogen bond, 254-259
Formulae association by, 255
empirical, 119, 121 energy of, 255
structural, 119-132, 156-157, 189- in ice, 258, 296-297
202 intermolecular, 257
see Equations intramolecular, 257
Free radicals, 115, 116, 193, 203, 213 occurrence of, 259
Free valence index, 212, 213, 220, 221 Hydrogen bridge, 202
Functional groups, 121, 193 Hydrogen molecule, Heitler-London
calculation of, 159, 168-174
Hydroxonium ion, 302
Gas
perfect, 262, 263, 303
real, 263 Ice, structure of, 258, 285, 296-297
see Noble gases Impulse, see Momentum
Gram-atoms, 12, 15 Induction effect, 119, 120
Gram-molecule, 12 Interference, 28, 33, 308
Ground (or normal) state, 57, 162 Ion-pairs, 303
Gibbs’ function, 91 Ionic bond, see Bond
energy of, 221-224
Ionic crystals, energetics of, 287-293
Haemoglobin, 234 Ionic molecules, 221
Halogens, 103, 108, 109, 224, 297 potential energy curve, 222
Hamiltonian operator, 161-165, 214 Ionic polarization, 225-229
Haiiy’s law of rational indices, 269 effect on properties, 229-231
Heisenberg uncertainty principle, 37, supplementary effect, 228
52 Ionic radii, 91-97
Hess’s law of heat effect, 291 of isoelectronic ions, 94, 96
Hiickel method, 211-221 relative sizes, 94
application of table of values, 95
Ionization energy, 60-62, 149 Madelung constant, 287, 288-289, 292

variation of, 85-87 Magnetic moment of electrons in mole
Ionization potential, 60 cules
Ions, 11, 115, 229 inherent, 321
anions, 95, 115, 227-228 orbital, 320-321
cations, 35, 115, 228 resultant, 321
isoelectronic, 94 Mass number, 16, 18
Isomerism of complex compounds, . Mendeleev’s periodic system, 11, 64t
235-236 65, fly-leaf
cis-trans, 236 representations of, 74
coordination, 235 long-form table, 1st vari
ionization, 235 ant, 72
ligand, 235 long-form table, 2nd va
stereoisomerism, 236 riant, 73
Isomerism of inorganic compounds, schematic, 70
122 short-form table, fly-leaf
Isomerism of organic compounds, 120- structure of, 69-74
127 groups and subgroups, 71-
spatial, 122-127 72, 102-112
geometric, 126-127 periods, 69-71
optical, 122-126 Metallic state, theory of, 44
structural, 120-122 Metastable state, 272
isotope, 122 Method
metamerism, 121 Born, 287-290, 291
position, 121 comparative calculation, 75-76,
skeletal, 120 96, 110, 128, 135, 136, 155,
tautomerism, 121 . 227, 256, 257, 293, 300
Isomorphism, 273 electron pair, 178, 196
Isotopes, 18 Hiickel, 206, 211-221, 249, 324
Isotropism, 303 Mendeleev, 74, 89
molecular orbital, 196, 203-211,
247-254
Kossel diagram, 100, 102, 112, 229 Mulliken, 149
Pauling, 150, 231-233
physical, for determining molecu
Lanthanide contraction, 92, 97, 105 lar structure, 137-149
Lanthanides, 66, 71, 84 rotation, 275
Laplacian operator, 161 valence bond, 178, 239-242
Law of probability, 169 valence diagram superposition,
Ligand, 233, 234, 237, 250 195-196
Light variational, 162, 169, 249
dual nature of, 28-30, 33-34 Molar volume, 13
polarization of, 125, 311-313 Molecular interaction, 263-267
see Diffraction, Interference attraction, energy of
Light quantum, 22, 58, 243 dispersion, 265
see Photon induction, 264, 265
Liquid state, 293-303 orientation, 264, 265
theory of, 295, 303 repulsion, energy of, 266
Liquids, structure of, 293-297 total energy, 266-267
at critical point, 294 see Van der Waals’ forces
at crystallization point, 294 Molecular orbital method, 203-211
cybotaxes in, 295 Molecular orbitals, 204
quasicrystalline, 295 in complex compounds, 247-254
short-range order of, 295 in diatomic molecules, 206-211
statistical character of, 296 Molecular spectra, absorption
X-ray analysis of, 295, 296 calculation for polymethylene
Logarithmic scale, 83 dyes, 321-323
types of Pauli exclusion principle, 55, 76, 83,

electron transition, 146 170, 174, 207
rotational, 146 Pauling system, 150, 232-233
vibrational, 146 Penetration effect, 85, 86, 228
Molecular structure, methods of de Periodic law, 64-114
termining formulation of, 64-69
chemical, 137, 235-236 predicting properties by, 74-76
physical, 137-149, 233 significance of, 113-114
electron diffraction, 138- Periodicity, 57, 64, 65-69
145 secondary, 87-89
spectral, 145-149 Phase diagram, regions of
X-ray diffraction, 149, condensation, 261
273-277 crystalline, 261
Molecules, 11, 115 sublimation, 261
polarity of, 153-157 supercritical, 262
potential energy curve, 166-168 Photoelectric effect, 28, 33
stability of, 229 Photon, 15, 28
structure of, 115-133 momentum of, 31, 33
Moment of inertia, 317-319 Planck’s constant, 22, 25, 29, 30, 265
expression for diatomic mole Plasma, 263
cule, 318-319 Polar bond, see Bond
Momentum, 31, 32, 37 and electronegativity, 231-233
see Angular momentum Polarizability
Moseley’s law, 69 coefficient, 154, 225, 264
of ions, 225-227
of molecules, 154, 264
Neutron diffraction, 34, 277 Polarization of light, 125, 311-312
Neutrons, 11, 18 Polymers, 305
Newton’s law, 35, 117 Polymorphism, 271
Noble gases 62, 70, 91, 95, 103, 111, Protons, 11, 18, 101, 143, 228, 255,
112, 115, 151, 226, 228 277
Nodes, 50-51
Normalization, 43, 162, 182, 219, 249
Normalization factor, 43, 170 Quantization rules, 27, 35
Nucleus, see Atomic nucleus Quantum cell, 57, 240, 295
Quantum-mechanical explanation of
atomic structure, 39-63
Operator, 161 footnote of hydrogen atom, 46-50
Hamiltonian, 161 chemical bond in complexes, 238
Laplacian, 161, 162 covalent bond, 159-221
Orbitals, 47, 57 Quantum mechanics, 35-39
binding (antibinding), 206-208, equation for, see Schrodinger
216-220 equation
designations for, 49 laws of, 39-45, 46
• hybrid, 182-186, 319 Quantum number, 45, 47, 49
ligand group, 249 atomic, 50-57, 207
see Molecular orbitals magnetic, 52
Orbits, stationary, 25 orbital, 52
Oxidation-reduction reactions, 90, 112 principal, 52
Oxidation state (or number), 89-91 spin, 53, 55
Oxidizing agents, 90-91, 102-112 molecular, 207
Oxygen, 77, 93, 98, 108, 177, 210,
321
Rare gases, see Noble gases
Reducing agents, 90-91, 102-112
Paramagnetism, 210, 321 Resonance
a-Particles, 12, 18 nuclear, 158
theory, 195-196 high-resolution, 19

Roentgenograms, 292 optical, 19
Rome de L’Isle, law of, 269 Spin, see Electron spin
Rydberg constant, 21, 27 high and low, 245
parallel and antiparallel, 57, 170,
174, 209
Schrodinger equation, 35-39, 59 Spontaneous fission, 81
construction of, 311 Stark effect, 20, 53
solutions of Stereochemistry, 122
approximate, 59, 159, 160 Stereoisomerism, 122-127
by perturbation technique, in complex compounds, 236
169 in organic compounds, 126
by using approximate fun Structural formulae of benzene, 192-
ctions, 160-166 195
by variational method, Structure determination, see Mole
162, 169, 204 cular structure
exact, for single-electron Superconductivity of metals, 193
system, 159-160
for square-well models, 39-46
one-dimensional, 39-43 Theory
three-dimensional, 43-46 Arrhenius, 298
Screening constant, 69, 85 Bergman-Bertholett, 116-117
Screening effect, 68, 83, 85, 86, 226 Berzelius, 117
Secular determinant, 165, 214, 219 Bohr, 25-27
Secular equations, 165, 214, 215 Bohr-Sommerfeld, 27, 35, 47, 320
Simple substances, 64, 91 Butlerov, 118-120, 123, 137, 186,
Solutions, see Electrolytes 233
Solvates, 259 crystal field, 242-247
crystalline, 298 Davy’s, 117
Solvation of ions Debye, 264, 313
energy of, 287 Dumas-Gerhardt, 117-118
in concentrated solutions, 302-303 Frankland, 118
in dilute solutions, 301-302 Heitler and London, 174-179
Spectra Keesom, 264
absorption, 145 Kossel and Magnus, 237-238
emission Ligand field, 238
band, 19-20 London, 265
continuous, 19 Mendeleev, 297, 298
line, 19-20 of groups, 249, 251
see Atomic spectra, Molecular spectra of liquid state, 303
Spectral lines, 19, 20 of metallic state, 44
multiplets, 20 Planck’s, 22-23
singlets, 20 Resonance, 195-196
Spectral series, designations of, 22,. 49 Rutherford, 25
diffuse, 22 Werner, 233
fundamental, 22 Thermodynamic potential, 91
principal, 22 Transition elements, 70, 71
sharp, 22
Spectral series of hydrogen
Balmer, 21 Uncertainty principle, Heisenberg, 52
Brackett, 21 Uranium atom, mass of, 16
Lyman, 21
Paschcn, 21
Pfund, 21 . Valence, 118, 174-179
Spectral terms, 22, 23, 68 explanation of orientation of,
ground, 60 179-187
Spectrographs, 19 Valence bond method, 178, 239-242
SUBJECT I N D E X .335
Valence electrons, 151, 152, 179, 191, Wave number, 20, 22. 308
193 Wave properties of particle, 28-39
Van der Waals’ forces, 263-265 Wavelength, 20, 23
components of, 265 Compton, 33
Waves, nature and characteristics of,
28-33, 307-310
Water, structure of, 258-259, 285, Work function, 29
296-297 Wurtz-Fittig synthesis, 116
Water of crystallization, 299
Wave functions, 36, 47-49, 169, 172,
179, 204, 217, 249 X-ray diffraction analysis, 16, 149,
angular component, 47 273-279
radial component, 47 X-rays, 16, 58, 67, 69
representation, graphical, 180,
182, 183-185
symmetrical and antisymmetri- Zeeman effect, 20, 53
cal, 173, 205 Zero-point energy, 41, 91, 167, 168
Wave mechanics, see Quantum mecha
nics
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MENDELEEV’S PERIODIC
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MENDELEEV’S PERIODIC

M3J|ATEJIbCTB0 BblCLUAR U1H0J1A*

TRANSLATED FROM THE RUSSIAN

MIR PUBLISHERS *MOSCOW

Revised from the 1970 second Russian edition

Treated in this book arc present-day ideas on the structure

© English translation, Mir Publishers, 1978

Part III. THE STRUCTURE OF MOLECULES AND THE CHEMICAL

Chapter Fourteen. Quantum-Mechanical Explanation of the Covalent Bond 159

Appendix IV. Polarization of L ig h t ........................................................... ?,H

In our days, often called a period of scientific and technological

1 Information about the energetics of chemical reactions, necessary for

problems cannot be treated quantitatively. Accordingly, the purpose

The ancient Greek philosophers already supposed that matter consists

1.2. The Avogadro Number

it is possible to calculate the number of atoms of helium obtained

where V = voltage applied to the plates

wherefrom ed can be found if the mass of the droplet is known (the

that the quotient obtained on dividing a faraday by the value of

Hence, the mass of the hydrogen atom, for instance, is

We see that the mass of an electron is extremely small as compared

The study of the spectra of chemical elements constitutes the expe­

2.1. Principle of Operation of Spectrographs;

Fig. 2.1. Schematic drawing of spectrograph

according to their wavelength. Visible and ultraviolet radiation

The use of high-resolution spectrographs has shown that these bands

2.2. The Atomic Spectrum of Hydrogen

Fig. 2.3. Atomic spectrumof hydrogen in §|

2.3. The Spectra of Other Elements

In 1889 Rydberg (Sweden) found that the wave numbers of a parti­

where n2 > n x. The numerical values of these functions are called

For the hydrogen atom, Z = 1; for the single-charged H e+ ion,

The energy levels of the electron in the hydrogen atom is schema­

Fig. 2.4. Energy levels of the electron in the hydrogen atom

(eV) is the energy gained by an electron which is accelerated by an

2.5. History of the Development of the Concepts

1 The angular momentum of a particle* rotating in a circle with a radius r

The value h!2jx is designated as h (crossed h), then

By substituting the known values into equation (2.12), we find that

eie2 dr— eie2

Substituting the values of v and r from equations (2.11) and (2.12),

The value of R calculated from this equation coincides with that

In 1905 Einstein showed that the photoelectric effect could be

Fig. 3.1. Dependence of the

have the maximum energy. Obviously, the energy of such electrons

3.2. The Law of Interdependence of Mass and Energy

be infinitely large). Therefore, all the mass of a photon is dynamic,

This equation expresses the interdependence of the momentum2*

3.3. Compton Effect

According to the law of conservation of energy, the kinetic energy

Then equation (3.11) can be written as

If we differentiate the ratio v = c/X, we obtain

Substituting in equation (3.15) the value V by c/X and the value of Av

3.4. De Broglie Waves

of their wave nature. Therefore, the conclusion can be drawn that

The study of the spectra of chemical elements constitutes the expe

In 1889 Rydberg (Sweden) found that the wave numbers of a parti

The energy levels of the electron in the hydrogen atom is schema

Fig. 4.4. Three-dimensional square poten

From equation (4.16) it can be seen that raU is approximately pro

1 By spin or self-rotation is only meant that the electron possesses an inhe

Fig. 5.4. Schematic representation of Men

In these methods, the same as in Mendeleev’s method, the physi