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Useful relations Series expansions
x2 x3
At 298.15 K ex = 1+ x + + +
2! 3!
RT 2.4790 kJ mol−1 RT/F 25.693 mV x2 x3
(RT/F) ln 10 59.160 mV kT/hc ln(1 + x ) = x − + −
207.225 cm−1 2 3
kT 25.693 meV Vm 2.4790 × 10−2 1 1
m3 mol−1 = 1− x + x 2 − = 1+ x + x 2 +
1+ x 1− x
24.790 dm3 mol−1
x3 x5 x2 x4
sin x = x − + − cos x = 1− + −
Selected units* 3! 5! 2! 4!
1N 1 kg m s−2 1J 1 kg m2 s−2
1 Pa 1 kg m−1 s−2 1W 1 J s−1 Derivatives; for Integrals, see the Resource
section
1V 1 J C−1 1A 1 C s−1
1T 1 kg s−2 A−1 1P 10−1 kg m−1 s−1 d(f + g) = df + dg d(fg) = f dg + g df
1S 1 Ω−1 = 1 A V−1 f 1 f df df dg
d = df − 2 dg = for f = f ( g (t ))
g g g dt dg dt
* For multiples (milli, mega, etc), see the Resource section
 ∂y   ∂x   ∂ y   ∂x   ∂z 
Conversion factors  ∂ x  = 1/  ∂ y   ∂ x   ∂ z   ∂ y  = −1
z z z y x
θ/°C = T/K − 273.15*
dx n deax d ln(ax ) 1
= nx n−1 = aeax =
1 eV 1.602 177 × 10−19 J 1 cal 4.184* J dx dx dx x
96.485 kJ mol−1
 ∂ g   ∂h 
8065.5 cm−1 df = g ( x , y )dx + h( x , y )dy is exact if   =  
 ∂ y  x  ∂x  y
1 atm 101.325* kPa 1 cm−1 1.9864 × 10−23 J
760* Torr Greek alphabet*
1D 3.335 64 × 10−30 C m 1Å 10−10 m*
Α, α alpha Ι, ι iota Ρ, ρ rho
* Exact value
Β, β beta Κ, κ kappa Σ, σ sigma
Mathematical relations Γ, γ gamma Λ, λ lambda Τ, τ tau
π = 3.141 592 653 59 … e = 2.718 281 828 46 … Δ, δ delta Μ, μ mu ϒ, υ upsilon
Ε, ε epsilon Ν, ν nu Φ, ϕ phi
Logarithms and exponentials
Ζ, ζ zeta Ξ, ξ xi Χ, χ chi
ln x + ln y + … = ln xy… ln x − ln y = ln(x/y) Η, η eta Ο, ο omicron Ψ, ψ psi
a ln x = ln xa ln x = (ln 10) log x Θ, θ theta Π, π pi Ω, ω omega
= (2.302 585 …) log x * Oblique versions (α, β, …) are used to denote physical
exeyez…. = ex+y+z+… ex/ey = ex−y observables.
(ex)a = eax e±ix = cos x ± i sin x
18
PERIODIC TABLE OF THE ELEMENTS VIII
VIIA
Group 1 2 1 H 13 14 15 16 17 2 He
helium
I II
Period 1 hydrogen
III IV V VI VII 4.00
1.0079
1s1 1s2
IA IIA IIIA IVA VA VIA VIIA
3 Li 4 Be 5 B 6 C 7 N 8 O 9 F 10 Ne
lithium beryllium boron carbon nitrogen oxygen fluorine neon
2 6.94 9.01 10.81 12.01 14.01 16.00 19.00 20.18
2s1 2s2 2s22p1 2s22p2 2s22p3 2s22p4 2s22p5 2s22p6
11 Na 12 Mg 13 Al 14 Si 15 P 16 S 17 Cl 18 Ar
sodium magnesium aluminium silicon phosphorus sulf ur chlorine argon
3 22.99 24.31 3 4 5 6 7 8 9 10 11 12 26.98 28.09 30.97 32.06 35.45 39.95
3s1 3s2 3s23p1 3s23p2 3s23p3 3s23p4 3s23p5 3s23p6
IIIB IVB VB VIB VIIB VIIIB IB IIB
19 K 20 Ca 21 Sc 22 Ti 23 V 24 Cr 25 Mn 26 Fe 27 Co 28 Ni 29 Cu 30 Zn 31 Ga 32 Ge 33 As 34 Se 35 Br 36 Kr
potassium calcium scandium titanium vanadium chromium manganese iron cobalt nickel copper zinc gallium germa nium arsenic selenium bromine krypton
4 39.10 40.08 44.96 47.87 50.94 52.00 54.94 55.84 58.93 58.69 63.55 65.41 69.72 72.64 74.92 78.96 79.90 83.80
Period
4s1 4s2 3d14s2 3d24s2 3d34s2 3d54s1 3d54s2 3d64s2 3d74s2 3d84s2 3d104s1 3d104s2 4s24p1 4s24p2 4s24p3 4s24p4 4s24p5 4s24p6
37 Rb 38 Sr 39 Y 40 Zr 41 Nb 42 Mo 43 Tc 44 Ru 45 Rh 46 Pd 47 Ag 48 Cd 49 In 50 Sn 51 Sb 52 Te 53 I 54 Xe
5 rubidium st rontium yttrium zirconium niobium molybdenum technetium ruthenium rhodium palladium silver cadmium indium tin antimony tellurium iodine xenon
85.47 87.62 88.91 91.22 92.91 95.94 (98) 101.07 102.90 106.42 107.87 112.41 114.82 118.71 121.76 127.60 126.90 131.29
5s1 5s2 4d15s2 4d25s2 4d45s1 4d55s1 4d55s2 4d75s1 4d85s1 4d10 4d105s1 4d105s2 5s25p1 5s25p2 5s25p3 5s25p4 5s25p5 5s25p6
55 Cs 56 Ba 57 La 72 Hf 73 Ta 74 W 75 Re 76 Os 77 Ir 78 Pt 79 Au 80 Hg 81 Tl 82 Pb 83 Bi 84 Po 85 At 86 Rn
caesium barium lanthanum hafnium tantalum tungsten rhenium osmium iridium platinum gold mercury thallium lead bismuth polonium astatine radon
6 138.91
132.91 137.33 178.49 180.95 183.84 186.21 190.23 192.22 195.08 196.97 200.59 204.38 207.2 208.98 (209) (210) (222)
87 Fr 88 Ra 89 Ac 104 Rf 105Db 106 Sg 107 Bh 108 Hs 109 Mt 110 Ds 112Cn 113 Nh 114 Fl 115 Mc 116 Lv
111 Rg 112 117 Ts 118Og
francium radium act inium rutherfordium dubnium seaborgium bohrium hassium meitnerium darmstadtium roentgenium copernicium nihonium flerovium moscovium livermorium tennessine oganesson
7 (223) (226) (261) (262) (263) (262) (265) (266)
(227) (271) (272) ? ? ? ? ? ? ?
58 Ce 59 Pr 60 Nd 61 Pm 62 Sm 63 Eu 64 Gd 65 Tb 66 Dy 67 Ho 68 Er 69 Tm 70 Yb 71 Lu
cerium praseodymium neodymium promethium samarium europium gadiolinium terbium dysprosium holmium erbium thulium ytterbium Lanthanoids
lutetium
6 140.12 140.91 144.24 (145) 150.36 151.96 157.25 158.93 162.50 164.93 167.26 168.93 173.04 174.97 (lanthanides)
Numerical values of molar
masses in grams per mole (atomic 4f15d16s2 4f36s2 4f46s2 4f56s2 4f66s2 4f76s2 4f75d16s2 4f96s2 4f106s2 4f116s2 4f126s2 4f136s2 4f146s2 5d16s2
weights) are quoted to the number
of significant figures typical of 90 Th 91 Pa 92 U 93 Np 94 Pu 95 Am 96 Cm 97 Bk 98 Cf 99 Es 100Fm 101Md 102 No 103 Lr
most naturally occurring samples. thorium protactinium uranium neptunium plutonium americium curium berkelium californium einsteinium fermium mendelevium nobelium lawrencium Actinoids
7 232.04 231.04 238.03 (237) (244) (243) (247) (247) (251) (252) (257) (258) (259) (262) (actinides)
6d27s2 5f26d17s2 5f36d17s2 5f46d17s2 5f67s2 5f77s2 5f76d17s2 5f97s2 5f107s2 5f117s2 5f127s2 5f137s2 5f147s2 6d17s2
FUNDAMENTAL CONSTANTS
Constant Symbol Value

Power of 10 Units
Speed of light c 2.997 924 58* 108 m s−1
Elementary charge e 1.602 176 634* 10 −19
C
Planck’s constant h 6.626 070 15 10−34 Js
ħ = h/2π 1.054 571 817 10 −34
Js
Boltzmann’s constant k 1.380 649* 10−23 J K−1
Avogadro’s constant NA 6.022 140 76 1023 mol−1
Gas constant R = NAk 8.314 462 J K−1 mol−1
Faraday’s constant F = NAe 9.648 533 21 104 C mol−1
Mass
Electron me 9.109 383 70 10−31 kg
Proton mp 1.672 621 924 10 −27
kg
Neutron mn 1.674 927 498 10−27 kg
Atomic mass constant mu 1.660 539 067 10 −27
kg
Magnetic constant μ0 1.256 637 062 10−6 J s2 C−2 m−1
(vacuum permeability)
Electric constant 0 1/0c 2 8.854 187 813 10−12 J−1 C2 m−1
(vacuum permittivity)
4πε0 1.112 650 056 10−10 J−1 C2 m−1
Bohr magneton μB = eħ/2me 9.274 010 08 10 −24
J T−1
Nuclear magneton μN = eħ/2mp 5.050 783 75 10−27 J T−1
Proton magnetic moment μp 1.410 606 797 10 −26
J T−1
g-Value of electron ge 2.002 319 304
Magnetogyric ratio
Electron γe = gee/2me 1.760 859 630 1011 T−1 s−1
Proton γp = 2μp/ħ 2.675 221 674 10 8
T−1 s−1
Bohr radius a0 = 4πε0ħ2/e2me 5.291 772 109 10−11 m
Rydberg constant R m e 4 /8h3c 2
e 0 1.097 373 157 10 5
cm−1
hcR ∞ /e 13.605 693 12 eV
Fine-structure constant α = μ0e2c/2h 7.297 352 5693 10−3
α−1 1.370 359 999 08 102
Stefan–Boltzmann constant σ = 2π5k4/15h3c2 5.670 374 10−8 W m−2 K−4
Standard acceleration of free fall g 9.806 65* m s−2
Gravitational constant G 6.674 30 10 −11
N m2 kg−2
* Exact value. For current values of the constants, see the National Institute of Standards and Technology (NIST) website.
Atkins’
PHYSICAL CHEMISTRY
Twelfth edition
Peter Atkins
Fellow of Lincoln College,
University of Oxford,
Oxford, UK
Julio de Paula
Professor of Chemistry,
Lewis & Clark College,
Portland, Oregon, USA
James Keeler
Associate Professor of Chemistry,
University of Cambridge, and
Walters Fellow in Chemistry at Selwyn College,
Cambridge, UK
Great Clarendon Street, Oxford, OX2 6DP,
United Kingdom
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Eighth edition 2006
Ninth edition 2009
Tenth edition 2014
Eleventh edition 2018
Impression: 1
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PREFACE
Our Physical Chemistry is continuously evolving in response Behind all that are The chemist’s toolkits, which provide brief
to users’ comments, our own imagination, and technical in- reminders of the underlying mathematical techniques. There
novation. The text is mature, but it has been given a new vi- is more behind them, for the collections of Toolkits available
brancy: it has become dynamic by the creation of an e-book via the e-book take their content further and provide illustra-
version with the pedagogical features that you would expect. tions of how the material is used.
They include the ability to summon up living graphs, get The text covers a very wide area and we have sought to add
mathematical assistance in an awkward derivation, find solu- another dimension: depth. Material that we judge too detailed
tions to exercises, get feedback on a multiple-choice quiz, and for the text itself but which provides this depth of treatment,
have easy access to data and more detailed information about or simply adds material of interest springing form the intro-
a variety of subjects. These innovations are not there simply ductory material in the text, can now be found in enhanced
because it is now possible to implement them: they are there to A deeper look sections available via the e-book. These sections
help students at every stage of their course. are there for students and instructors who wish to extend their
The flexible, popular, and less daunting arrangement of the knowledge and see the details of more advanced calculations.
text into readily selectable and digestible Topics grouped to- The main text retains Examples (where we guide the reader
gether into conceptually related Focuses has been retained. through the process of answering a question) and Brief illus-
There have been various modifications of emphasis to match the trations (which simply indicate the result of using an equation,
evolving subject and to clarify arguments either in the light of giving a sense of how it and its units are used). In this edition a
readers’ comments or as a result of discussion among ourselves. few Exercises are provided at the end of each major section in a
We learn as we revise, and pass on that learning to our readers. Topic along with, in the e-book, a selection of multiple-choice
Our own teaching experience ceaselessly reminds us that questions. These questions give the student the opportunity to
mathematics is the most fearsome part of physical chemis- check their understanding, and, in the case of the e-book, re-
try, and we likewise ceaselessly wrestle with finding ways to ceive immediate feedback on their answers. Straightforward
overcome that fear. First, there is encouragement to use math- Exercises and more demanding Problems appear at the end of
ematics, for it is the language of much of physical chemistry. each Focus, as in previous editions.
The How is that done? sections are designed to show that if The text is living and evolving. As such, it depends very
you want to make progress with a concept, typically making much on input from users throughout the world. We welcome
it precise and quantitative, then you have to deploy mathemat- your advice and comments.
ics. Mathematics opens doors to progress. Then there is the PWA
fine-grained help with the manipulation of equations, with JdeP
their detailed annotations to indicate the steps being taken. JK
viii 12 The properties of gases
USING THE BOOK
TO THE STUDENT
The twelfth edition of Atkins’ Physical Chemistry has been

developed in collaboration with current students of physical
chemistry in order to meet your needs better than ever before.
Our student reviewers have helped us to revise our writing
style to retain clarity but match the way you read. We have also
introduced a new opening section, Energy: A first look, which
summarizes some key concepts that are used throughout the
text and are best kept in mind right from the beginning. They
are all revisited in greater detail later. The new edition also
brings with it a hugely expanded range of digital resources, AVAIL ABLE IN THE E-BOOK
including living graphs, where you can explore the conse-
quences of changing parameters, video interviews with prac- ‘Impact on…’ sections Group theory tables
tising scientists, video tutorials that help to bring key equations ‘Impact on’ sections show how physical chemistry is applied in A link to comprehensive group theory tables can be found at
a variety of modern contexts. They showcase physical chemis- the end of the accompanying e-book.
to life in each Focus, and a suite of self-check questions. These try as an evolving subject.
Go to this location in the accompanying e-book to view a
features are provided as part of an enhanced e-book, which is list of Impacts.
The chemist’s toolkits
accessible by using the access code included in the book. The chemist’s toolkits are reminders of the key mathematical,
You will find that the e-book offers a rich, dynamic learn- ‘A deeper look’ sections physical, and chemical concepts that you need to understand in
order to follow the text.
ing experience. The digital enhancements have been crafted to These sections take some of the material in the text further and
are there if you want to extend your knowledge and see the de-
For a consolidated and enhanced collection of the toolkits
found throughout the text, go to this location in the accompa-
tails of some of the more advanced derivations. nying e-book.
help your study and assess how well you have understood the Go to this location in the accompanying e-book to view a
list of Deeper Looks.
material. For instance, it provides assessment materials that
give you regular opportunities to test your understanding.
Innovative structure TOPIC 2A Internal energy

Short, selectable Topics are grouped into overarching Focus
sections. The former make the subject accessible; the latter
provides its intellectual integrity. Each Topic opens with
RESOURCE SEC TION ➤ Why do you need to know this material?
The First Law of thermodynamics is the foundation of the
discussion of the role of energy in chemistry. Wherever the
A closed system has a boundary through which matter
cannot be transferred.
Both open and closed systems can exchange energy with their
surroundings.
the questions that are commonly asked: why is this material generation or use of energy in physical transformations or
chemical reactions is of interest, lying in the background
An isolated system can exchange neither energy nor
matter with its surroundings.
important?, what should you look out for as a key idea?, and are the concepts introduced by the First Law.
➤ What is the key idea?

what do you need to know already? The total energy of an isolated system is constant.
2A.1 Work, heat, and energy
➤ What do you need to know already?
Although thermodynamics deals with the properties of bulk
This Topic makes use of the discussion of the properties of
systems, it is enriched by understanding the molecular origins
gases (Topic 1A), particularly the perfect gas law. It builds
of these properties. What follows are descriptions of work, heat,
Resource section
on the definition of work given in Energy: A first look.
and energy from both points of view.
2A.1(a) Definitions
The Resource section at the end of the book includes a brief In thermodynamics, the universe is divided into two parts:
the system and its surroundings. The system is the part of the
The fundamental physical property in thermodynamics is
work: work is done in order to achieve motion against an op-
world of interest. It may be a reaction vessel, an engine, an elec- posing force (Energy: A first look 2a). A simple example is the
review of two mathematical tools that are used throughout the trochemical cell, a biological cell, and so on. The surroundings
comprise the region outside the system. Measurements are
process of raising a weight against the pull of gravity. A process
does work if in principle it can be harnessed to raise a weight
text: differentiation and integration, including a table of the Contents

made in the surroundings. The type of system depends on the
characteristics of the boundary that divides it from the sur-
somewhere in the surroundings. An example is the expansion
of a gas that pushes out a piston: the motion of the piston can in
roundings (Fig. 2A.1): principle be used to raise a weight. Another example is a chemi-
integrals that are encountered in the text. There is a review of An open system has a boundary through which matter
cal reaction in a battery: the reaction generates an electric cur-
units, and how to use them, an extensive compilation of tables PART 1 Mathematical resources
can be transferred.
878
rent that can drive a motor and be used to raise a weight.
The energy of a system is its capacity to do work (Energy: A
first look 2b). When work is done on an otherwise isolated sys-
of physical and chemical data, and a set of character tables. 1.1 Integration 878
tem (for instance, by compressing a gas with a piston or wind-
ing a spring), the capacity of the system to do work is increased.
Energy
Energy
Matter
That is, the energy of the system is increased. When the system
Short extracts of most of these tables appear in the Topics 1.2 Differentiation 878
does work (when the piston moves out or the spring unwinds),
it can do less work than before. That is, its energy is decreased.
When a gas is compressed by a piston, work is done on the sys-
themselves: they are there to give you an idea of the typical 1.3Open Series expansions
Closed Isolated 881
tem and its energy is increased. When that gas is allowed to ex-
pand again the piston moves out, work is done by the system,
values of the physical quantities mentioned in the text. (a) (b) (c)
and the energy of the system is decreased.
It is very important to note that when the energy of the sys-
PART 2 Quantities and units
Figure 2A.1 (a) An open system can exchange matter and energy
with its surroundings. (b) A closed system can exchange energy
882
tem increases that of the surroundings decreases by exactly the
same amount, and vice versa. Thus, the weight raised when the
with its surroundings, but it cannot exchange matter. (c) An system does work has more energy than before the expansion,
isolated system can exchange neither energy nor matter with its because a raised weight can do more work than a lowered one.
PART 3 Data
surroundings.
884
The weight lowered when work is done on the system has less
PART 4 Character tables 910

qV CV T (2A.16)
TOPIC 2B Enthalpy
E2A.4 A sample co
This relation provides a simple way of measuring the heat ca- CV,m = 23 R , initially
pacity of a sample. A measured quantity of energy is transferred constant volume.
Using the book ix
Checklist of concepts
A checklist of key concepts is provided at the end of each Checklist of concepts
Topic, so that you can tick off the ones you have mastered.
2B.1 The definition of enthalpy
➤ Why do you need to know this material? ☐ 1. Work is the process of achieving motion against an ☐ 8. The in
The enthalpy, H, is
opposing defined as
force. raised.
The concept of enthalpy is central to many thermodynamic
discussions about processes, such as physical transforma- ☐ 2. Energy is the capacity to do work. Enthalpy
☐ 9. The Fir
H U pV
☐ 3. Heat is the process of transferring energy
(2B.1)
as a result of lated sy
tions and chemical reactions taking place under conditions [definition]
of constant pressure. a temperature difference. ☐ 10. Free ex
where p is the pressure of the system and V is its volume.
Physical chemistry: people and perspectives ☐ 4. An exothermic process is a process that releases energy
Because U, p, and V are all state functions, the enthalpy is a
no wor
➤ What is the key idea? as heat. ☐ 11. A rever
LeadingA figures state function too. As is true of any state function, the change
change in a varity of
in enthalpy is fields
equal share
to the their
energyunique and var-
transferred as ☐ 5. An endothermic process is a process in which energy is an infin
ied experiences and careers, and talk about the challenges they in enthalpy, ΔH, between any pair of initial and final states is
heat at constant pressure. acquired as heat. ☐ 12. To ach
faced and their achievements to give you a sense of where the independent of the path between them.
☐ 6. In molecular terms, work is the transfer of energy that sure is
study of➤physical
What dochemistry
you need cantolead.
know already? system
makes use of organized motion of atoms in the sur-
This Topic makes use of the discussion of internal energy Enthalpy
2B.1(a) roundings andchange and heat
heat is the transfer transfer
of energy that makes ☐ 13. The en
(Topic 2A) and draws on some aspects of perfect gases use of their disorderly motion. equal to
An important consequence of the definition of enthalpy in
(Topic 1A). ☐ 2B.1
7. Internal ☐ 14. Calorim
PRESENTING THE MATHEMATICS eqn is that itenergy,
can bethe total that
shown energy
theof=change
Probability
a system, is a state
|ψ|2dx in enthalpy is
function.
equal to the energy supplied as heat under conditions of con-
|ψ|2
stant pressure.
How
Theischange
thatindone?
internal energy is not equal to the energy trans- dx
How is that done? 2B.1 Deriving the relation between
ferred as heat when the system is free to change its volume,
You need to understand how an equation is derived from enthalpy change and heat transfer at constant pressure
such as when it is able to expand or contract under conditions
reasonable assumptions and the details of the steps involved.
of constant pressure. Under these circumstances some of the In a typical thermodynamic derivation, as here, a common way
This is one role for the How is that done? sections. Each one x x + dx
energy supplied as heat to the system is returned to the sur- to proceed is to introduce successive definitions of the quanti-
leadsroundings
from an asissue that arises
expansion work in the2B.1),
(Fig. text, so
develops thethan
dU is less nec-
dq. ties of 7B.1
Figure interest
Theand then apply ψ
wavefunction the
is appropriate
a probabilityconstraints.
amplitude in Figure 7B.2
essary equations, and arrives at a conclusion. These sections
In this case the energy supplied as heat at constant pressure is dimensional
the sense that its square modulus (ψ⋆ψ or |ψ|2) is a probability
maintain thethe
separation Step 1 Write an expression for H + dH in terms of the definition
equal to change in of the equation
another and its derivation
thermodynamic property ofso
the density. The probability of finding a particle in the region particle in th
thatsystem,
you canthe
find them easily for review, but at the same time of H proportiona
‘enthalpy’. between x and x + dx is proportional to |ψ|2dx. Here, the
emphasize that mathematics is an essential feature of physical For a general
probability infinitesimal
density change
is represented by in
thethe state of
density ofshading
the system,
in position.
chemistry. U changes
the to U +band.
superimposed dU, p changes to p + dp, and V changes to
V + dV, so from the definition in eqn 2B.1, H changes by dH to
13A The Boltzmann distribution 545
H + dH = (U + dU) + (p + dp)(V + dV) Wav
The chemist’s toolkits
Energy as
The chemist’s toolkit 7B.1 Complex
work = U + dU+ pV + pdV + Vdpnumbers
+ dpdV
The chemist’s
If, astoolkits areofreminders
a result collisions, ofthethe key mathematical,
system were to fluctuate Brief illustration 13A.1
physical,between
and chemical concepts that
the configurations you need
{N,0,0,…} and to
{Nunderstand
− 2,2,0,…}, it A complex
The last termnumber
is the productz has the z = x + iy, where
forminfinitesimally
of two i quan-
small 1.
The complex
titiesthe conjugate
andconfiguration
can be neglected. of a complex
Now recognize number z is z*
thatNU=+20, =
pV =x − iy.
theH on
in orderwould
to follow
almostthe text.beMany
always foundof these
inΔU Toolkits
the< second,
q more are rele-
likely con- For {1,0,3,5,10,1}, which has
figuration, especially if N were large. In other words, a system Complex
the right
weight numbers
(boxed), so
is calculated combine together
as (recall that 0! ≡ 1) according to the follow-
vant to more than oneEnergyTopic, and you can view a compilation
ing rules:
of them,free to enhancements
with switch between
as heat the two form
in the configurations
of more would
informa- show H + dH =20H! + dU + pdV +8 Vdp
properties characteristic almost exclusively of the second W
Addition 9.31 10
and subtraction:
tion and brief illustrations, in this section of the accompany- 1!0 !3!5!10 !1!
configuration. and hence
ing e-book. (a ib) (c id ) (a c) i(b d )
The next step is to develop an expression for the number of dH dU pdV Vdp
ways that a general configuration {N0,N1,…} can be achieved. Multiplication:
It will turn out to be more convenient to deal with the natu- Figure 7B.3
Figure 2B.1 When is a system
called is subjected to constant pressure and physical sign
Annotated This numberequations and
the equation
weight of labels
the configuration and ralStep 2 Introduceidthe
logarithm
(a ib) (cofthe
definition
) weight,
(ac bd ln)W of dU
i,(rather
bc adthan ) with the weight
is free to change its volume, some of the energy supplied as heat this wavefun
denoted W . itself:
Because dU = dq + dw this expression becomes
may escape backmany
into the surroundings as work. In suchhow
a case, Two special relations are: distribution
We have annotated equations to help you follow they
the change in internal energy is smaller than the energy supplied dU 2 (x/y) =2 ln x −2ln1/y2
1/ln
the density
are developed. An annotation can help you travel across the dH dq d| w
Modulus: z | p(zdVz ) Vd(px y )
as heat. N ! i
equals sign:Howit isis athat
reminder of the substitution used, an approxi-
done? 13A.1 = ln relation: e =cos lnN! i−sin ln(N
Evaluating the weight of a ln WEuler’s 0 ! N1 ! N
, which 2!
implies ) that ei 1, The Born
mation made, the terms that have been assumed constant, an N0 i!N1 ! N
i2
!
configuration cos 2 (e e ), and sin 2 i(e e ).
1 1 i i
significance
integral used, and so on. An annotation can also be a reminder because |ψ
Consider the N balls into We ln xy = ln x + ln y
of the significance of number of ways of
an individual distributing
term in an expression. bins
direct signifi
labelled 0, 1, 2 … such that there are N0 balls in bin 0, N1 in bin
sometimes collect into a small box a collection of numbers or = ln N ! − ln N 0 ! − ln N1 ! − ln N 2 ! − = ln N ! − ∑ ln N i ! function: o
1, and so on. The first ball can be selected in N different ways,
symbols totheshow
next how they
ball in N −carry fromways
1 different one from
line to
thethe next.
balls Many
remaining, Because |ψ| dx is a (dimensionless) probability, |ψ|2 is the
2 i
cant, and b
of the equations are labelled to highlight their significance.
and so on. Therefore, there are N(N − 1) … 1 = N! ways of probability
One reason fordensity, withlnW
introducing the isdimensions of 1/length
that it is easier to make ap-(for a may corres
selecting the balls. one-dimensional
proximations. system). the
In particular, Thefactorials can be ψ
wavefunction itself is called
simplified by region (Fig
There are N0! different ways in which the balls could have the probability
using amplitude.1For a particle free to move in three
Stirling’s approximation tive regions
been chosen to fill bin 0 (Fig. 13A.1). Similarly, there are N1! dimensions (for example, an electron near a nucleus in an because it g
ways in which the balls in bin 1 could have been chosen, and ln x ! x ln x x Stirling’s approximation [ x >> 1] (13A.2)
atom), the wavefunction depends on the coordinates x, y, and structive in
so on. Therefore, the total number of distinguishable ways of Then
z andtheisapproximate
denoted ψ(r). expression
In this forcasethethe
weight
Bornisinterpretation is A wavefu
☐ 1. Energy transferred as heat at constant pressure is equal ☐ 3. The heat capa
to the change in enthalpy of a system. heat capacity)
☐ 2. Enthalpy changes can be measured in an isobaric temperature.
x Using the book
calorimeter.
Checklists of equations
A handy checklist at the end of each topic summarizes the Checklist of equations
most important equations and the conditions under which
Property Equation Comment
they apply. Don’t think, however, that you have to memorize
every equation in these checklists: they are collected there for Enthalpy H = U + pV Definition
ready reference. Heat transfer at constant pressure dH = dqp, ΔH = qp No additional work

Relation between ΔH and ΔU ΔH = ΔU + ΔngRT Isothermal process, per
Heat capacity at constant pressure Cp = (∂H/∂T)p Definition
Relation between heat capacities Cp − CV = nR Perfect gas
Video tutorials on key equations
Video tutorials to accompany each Focus dig deeper into some
of the key equations used throughout that Focus, emphasizing
the significance of an equation, and highlighting connections
with material elsewhere in the book.
48 2 The First Law
Living graphs
The educational value of many graphs can be heightened by Table 2B.1 Temperature variation of molar heat capacities,
seeing—in a very direct way—how relevant parameters, such C p ,m /(JK −1 mol−1 ) = a + bT + c /T 2 *
as temperature or pressure, affect the plot. You Bcan now in- a b/(10−3 K−1) c/(105 K2)
teract with key graphs throughout the text in order to ex-
Enthalpy, H
C(s, graphite) 16.86 4.77 −8.54

plore how they respond as the parameters are changed. These
CO2(g) 44.22 8.79 −8.62
graphs are clearly flagged throughout theInternal
book, and you can
H2O(l) 75.29 0 0
find links to the dynamic
A versions in the corresponding
energy, U loca-
N2(g) 28.58 3.77 −0.50
tion in the e-book.
* More values are given in the Resource section.
Temperature, T
Atkins-Chap02_033-074.indd 49
The empirical parameters a, b, and c are independent of tempera-
Figure 2B.3 The constant-pressure heat capacity at a particular ture. Their values are found by fitting this expression to experi-
SET TING UP AND SOLVING PROBLEMS
temperature is the slope of the tangent to a curve of the enthalpy mental data on many substances, as shown in Table 2B.1.
of a system plotted against temperature (at constant pressure). If eqn 2B.8 is used to calculate the change 2BinEnthalpy 47
enthalpy be-
For gases, at a given temperature the slope of enthalpy versus
tween two temperatures T1 and T2, integration by using Integral
temperature is steeper than that of internal energy versus
A.1 in the Resource section gives
Brief illustrations
temperature, and Cp,m is larger than CV,m.
mass densities of the polymorphs are 2.71 g cm −3
(calcite) and Brief illustration
T2 2B.2
T2 c
H C p dT a bT 2 dT
−3
2.93 g cm (aragonite).
A Brief illustration shows you how to use an equation or con- T1 T1
T
In the reaction 3 H2(g) + N2(g) → 2 NH3(g), 4 mol of gas-
cept thatishas
thejust
Collect been
your
heat introduced
perThe
thoughts
capacity mole in of
thesubstance;
starting text.
pointItfor
shows
itthe youintensive
how
calculation
is an T2
phase molecules
c by 2 mol of gas-phase molecules,
is replaced
to use data is and
the manipulate
property. units
relation between thecorrectly.
enthalpy ofIt also helps you
a substance andtoits aT 12 bT 2 (2B.9)
−1
internal energy (eqn so Δng = −2 mol. Therefore,T T at 298 K, when RT = 2.5 kJ mol ,
become familiar
The heat with the
capacity at2B.1).
magnitudesYou need
constant to express
relatesthe
of quantities.
pressure thedifference
change in the molar enthalpy and molar internal energy changes taking
1
betweentothe
enthalpy two quantities
a change in termsFor
in temperature. of the pressure and
infinitesimal the
changes 2 by 1 1
difference of their molar volumes.
of temperature, eqn 2B.5 implies thatThe latter can be calculated place inathe
(T2 system
T1 ) 12are
b Trelated
2
2 T1
c
T T
Examples
from their molar masses, M, and their mass densities, ρ, by H U (2) RT 5.0 kJmo 2 11 1
m m
dH =ρC=p dM/V
using T (at
m.
constant pressure) (2B.6a)
Worked Examples are more detailed illustrations of the appli- Note that the difference is expressed in kilojoules, not joules
If The
the heat capacity
solution The ischange
constant inover the range
enthalpy when of temperatures
the transitionof as in Example
Example 2B.2 2B.1. The enthalpy change is more negative than
cation ofinterest,
the material,
occurs then
and typically require you to assemble
is for a measurable increase in temperature
Evaluating an increase in enthalpy with
the change in internal energy because, although energy escapes
and deploy several relevant concepts and equations. temperature
T
from the system as heat when the reaction occurs, the system
Everyone has
H madifferent
T
H
C pmd(T Cway
aragonite T to approach solving a problem,
)dTHm (Ccalcite )
p T
H
2 2
What is the
contracts change
as the in molar
product enthalpy
is formed, of N2 is
so energy when it is heated
restored to it as
p (T 2 T1 )
and it changes with {U mexperience.
T 1
(a ) pVm (a )} To
1
{U mhelp
(c) in
pVmthis
(c)}} process, we from 25 °Cthe
work from 100 °C? Use the heat capacity information in
to surroundings.
suggest how
whichyou canshould
be collect
summarized your
as
U m p{Vm (a ) Vm (c)} thoughts and then pro- Table 2B.1.
ceed to a solution. All the worked
H C p T (at constant pressure) Examples are accompanied
(2B.6b) Collect your thoughts The heat capacity of N2 changes with
by closely related self-tests to enable you to test yourbygrasp
where a denotes aragonite and c calcite. It follows of
substitut- temperature significantly in this range, so use eqn 2B.9.
Because
the material Vm =a M/ρ
ing after change
that in through
working enthalpy can be equated to the energy
our solution as set out in Exercises
supplied as heat at constant pressure, the practical form of this The solution Using a = 28.58 J K−1 mol−1, b = 3.77 × 10−3 J K−2 mol−1,
the Example. 5 of ΔHm−1− ΔUm for the reaction N2(g) + 3 H2(g) →
E2B.1 Calculate the value
c =at−0.50
equation is 1 1 2 and
NH3(g) 473 K.× 10 J K mol , T1 = 298 K, and T2 = 373 K, eqn 2B.9
H m U m pM is written as
q p C p T (a ) (c) (2B.7) E2B.2 When 0.100 mol of H2(g) is burnt in a flame calorimeter it is
H that
observed H mthe water
(373 H min(298
K )bath which
K )the apparatus is immersed increases
This expression
Substitution of shows
the data, how to measure
using M = 100.09 the constant-pressure
g mol −1
, gives in temperature by 13.64 1 K. When
1 0.100 mol C4H10(g), butane, is burnt in
heat capacity of a sample: a measured quantity of energy is the same (28.58 JK
apparatus mol ) (373
the temperature riseKis6.03
298K.KThe
) molar enthalpy of
combustionof1 (H32.(g) −285
is 10 2 −1. Calculate
3 kJ mol the
supplied
H m as
heat
U m under
(1.0 conditions
105 Pa ) (100of.09
constant
g mo11pressure
) (as in a 2
77 JK mol 1
) {(373 K )molar
2 enthalpy
(298 K )2 }
of combustion of butane.
sample exposed to the atmosphere and free to expand), and the 1 1
1 1 (0.50 105 JKmol 1 )
temperature rise ismonitored.
3
373 K 298 K
2.93capacity
g cm 3
2.71 g cm
The variation of heat
5
with temperature
3 1
can some-
Pa m 3this 1
2B.2 The variation of enthalpy with
times be ignored if 2the.8 10 Pa cm molrange
temperature 0is.28
small; molis an The final result is
Using the book xi
Self-check questions
This edition introduces self-check questions throughout the
text, which can be found at the end of most sections in the
e-book. They test your comprehension of the concepts dis-
cussed in each section, and provide instant feedback to help
you monitor your progress and reinforce your learning. Some
of the questions are multiple choice; for them the ‘wrong’ an-
swers are not simply random numbers but the result of errors
that, in our experience, students often make. The feedback
from the multiple choice questions not only explains the cor-
rect method, but also points out the mistakes that led to the
incorrect answer. By working through the multiple-choice Exercises and problems 27
questions you will be well prepared to tackle more challenging

FOCUS 1 The properties of gases
exercises and problems.
To test your understanding of this material, work through Selected solutions can be found at the end of this Focus in
the Exercises, Additional exercises, Discussion questions, and the e-book. Solutions to even-numbered questions are available
Discussion questions Problems found throughout this Focus. online only to lecturers.
Discussion questions appear at the end of each Focus, and are TOPIC 1A The perfect gas
organized by Topic. They are designed to encourage you to Discussion questions
D1A.1 Explain how the perfect gas equation of state arises by combination of D1A.2 Explain the term ‘partial pressure’ and explain why Dalton’s law is a
reflect on the material you have just read, to review the key Boyle’s law, Charles’s law, and Avogadro’s principle. limiting law.
concepts, and sometimes to think about its implications and Additional exercises
limitations.
E1A.8 Express (i) 22.5 kPa in atmospheres and (ii) 770 Torr in pascals. 60 per cent. Hint: Relative humidity is the prevailing partial pressure of water
3 vapour expressed as a percentage of the vapour pressure of water vapour at
E1A.9 Could 25 g of argon gas in a vessel of volume 1.5 dm exert a pressure
the same temperature (in this case, 35.6 mbar).
of 2.0 bar at 30 °C if it behaved as a perfect gas? If not, what pressure would
it exert? E1A.18 Calculate the mass of water vapour present in a room of volume
250 m3 that contains air at 23 °C on a day when the relative humidity is
E1A.10 A perfect gas undergoes isothermal expansion, which increases its
53 per cent (in this case, 28.1 mbar).
volume by 2.20 dm3. The final pressure and volume of the gas are 5.04 bar
and 4.65 dm3, respectively. Calculate the original pressure of the gas in E1A.19 Given that the mass density of air at 0.987 bar and 27 °C is
(i) bar, (ii) atm. 1.146 kg m−3, calculate the mole fraction and partial pressure of nitrogen
and oxygen assuming that (i) air consists only of these two gases, (ii) air also
Exercises and problems

E1A.11 A perfect gas undergoes isothermal compression, which reduces its
3 contains 1.0 mole per cent Ar.
volume by 1.80 dm . The final pressure and volume of the gas are 1.97 bar
and 2.14 dm3, respectively. Calculate the original pressure of the gas in E1A.20 A gas mixture consists of 320 mg of methane, 175 mg of argon, and
(i) bar, (ii) torr. 225 mg of neon. The partial pressure of neon at 300 K is 8.87 kPa. Calculate
−2 (i) the volume and (ii) the total pressure of the mixture.
E1A.12 A car tyre (an automobile tire) was inflated to a pressure of 24 lb in
Exercises are provided throughout the main text and, along −2

(1.00 atm = 14.7 lb in ) on a winter’s day when the temperature was −5 °C.
What pressure will be found, assuming no leaks have occurred and that the
E1A.21 The mass density of a gaseous compound was found to be 1.23 kg m
at 330 K and 20 kPa. What is the molar mass of the compound?
−3
volume is constant, on a subsequent summer’s day when the temperature is

with Problems, at the end of every Focus. They are all organ-
3
E1A.22 In an experiment to measure the molar mass of a gas, 250 cm of the
35 °C? What complications should be taken into account in practice?
gas was confined in a glass vessel. The pressure was 152 Torr at 298 K, and
E1A.13 A sample of hydrogen gas was found to have a pressure of 125 kPa after correcting for buoyancy effects, the mass of the gas was 33.5 mg. What is
ised by Topic. Exercises are designed as relatively straightfor- when the temperature was 23 °C. What can its pressure be expected to be
when the temperature is 11 °C?
the molar mass of the gas?
−3
E1A.23 The densities of air at −85 °C, 0 °C, and 100 °C are 1.877 g dm ,
ward numerical tests; the Problems are more challenging and 1.294 g dm−3, and 0.946 g dm−3, respectively. From these data, and assuming
3
E1A.14 A sample of 255 mg of neon occupies 3.00 dm at 122 K. Use the
perfect gas law to calculate the pressure of the gas. that air obeys Charles’s law, determine a value for the absolute zero of
temperature in degrees Celsius.
typically involve constructing a more detailed answer. For this

3 3
E1A.15 A homeowner uses 4.00 × 10 m of natural gas in a year to heat a
3
home. Assume that natural gas is all methane, CH4, and that methane is a E1A.24 A certain sample of a gas has a volume of 20.00 dm at 0 °C and
perfect gas for the conditions of this problem, which are 1.00 atm and 20 °C. 1.000 atm. A plot of the experimental data of its volume against the Celsius
new edition, detailed solutions are provided in the e-book in What is the mass of gas used?
E1A.16 At 100 °C and 16.0 kPa, the mass density of phosphorus vapour is
temperature, θ, at constant p, gives a straight line of slope 0.0741 dm3 °C−1.
From these data alone (without making use of the perfect gas law), determine
the absolute zero of temperature in degrees Celsius.
the same location as they appear in print.

0.6388 kg m−3. What is the molecular formula of phosphorus under these
3
conditions? E1A.25 A vessel of volume 22.4 dm contains 1.5 mol H2(g) and 2.5 mol N2(g)
at 273.15 K. Calculate (i) the mole fractions of each component, (ii) their
E1A.17 Calculate the mass of water vapour present in a room of volume
For the Examples and Problems at the end of each Focus de- 3
400 m that contains air at 27 °C on a day when the relative humidity is
partial pressures, and (iii) their total pressure.
tailed solutions to the odd-numbered questions are provided Problems

P1A.1 A manometer consists of a U-shaped tube containing a liquid. One side pressure, ρ is the mass density of the liquid in the tube, g = 9.806 m s−2 is the
in the e-book; solutions to the even-numbered questions are is connected to the apparatus and the other is open to the atmosphere. The
pressure p inside the apparatus is given p = pex + ρgh, where pex is the external
acceleration of free fall, and h is the difference in heights of the liquid in the
two sides of the tube. (The quantity ρgh is the hydrostatic pressure exerted by
available only to lecturers. Exercises and problems 139
P4B.16 Figure 4B.1 gives a schematic representation of how the chemical temperature, of these lines. Is there any restriction on the value this curvature
potentials of the solid, liquid, and gaseous phases of a substance vary with can take? For water, compare the curvature of the liquid line with that for the
Atkins-Chap01_003-032.indd 27 01-10-2022 12:40:10
temperature. All have a negative slope, but it is unlikely that they are straight gas in the region of the normal boiling point. The molar heat capacities at
lines as indicated in the illustration. Derive an expression for the curvatures, constant pressure of the liquid and gas are 75.3 J K−1 mol−1 and 33.6 J K−1 mol−1,
that is, the second derivative of the chemical potential with respect to respectively.
Integrated activities
FOCUS 4 Physical transformations of pure substances
At the end of every Focus you will find questions that span Integrated activities
several Topics. They are designed to help you use your I4.1 Construct the phase diagram for benzene near its triple point at
36 Torr and 5.50 °C from the following data: ∆fusH = 10.6 kJ mol−1,
∆vapH = 30.8 kJ mol−1, ρ(s) = 0.891 g cm−3, ρ(l) = 0.879 g cm−3.
(c) Plot Tm/(ΔhbHm/ΔhbSm) for 5 ≤ N ≤ 20. At what value of N does Tm change
by less than 1 per cent when N increases by 1?
knowledge creatively in a variety of ways.

‡
I4.4 A substance as well-known as methane still receives research attention
‡
I4.2 In an investigation of thermophysical properties of methylbenzene because it is an important component of natural gas, a commonly used fossil
R.D. Goodwin (J. Phys. Chem. Ref. Data 18, 1565 (1989)) presented fuel. Friend et al. have published a review of thermophysical properties of
expressions for two coexistence curves. The solid–liquid curve is given by methane (D.G. Friend, J.F. Ely, and H. Ingham, J. Phys. Chem. Ref. Data 18,
583 (1989)), which included the following vapour pressure data describing the
p/bar p3 /bar 1000(5.60 11.727 x )x liquid–vapour coexistence curve.
where x = T/T3 − 1 and the triple point pressure and temperature are T/K 100 108 110 112 114 120 130 140 150 160 170 190
p/MPa 0.034 0.074 0.088 0.104 0.122 0.192 0.368 0.642 1.041 1.593 2.329 4.521
p3 = 0.4362 μbar and T3 = 178.15 K. The liquid–vapour curve is given by
(a) Plot the liquid–vapour coexistence curve. (b) Estimate the standard
ln( p/bar ) 10.418/y 21.157 15.996 y 14.015 y 2 boiling point of methane. (c) Compute the standard enthalpy of vaporization
5.0120 y 3 4.7334(1 y )1.70 of methane (at the standard boiling point), given that the molar volumes of
the liquid and vapour at the standard boiling point are 3.80 × 10−2 dm3 mol−1
and 8.89 dm3 mol−1, respectively.
where y = T/Tc = T/(593.95 K). (a) Plot the solid–liquid and liquid–vapour
coexistence curves. (b) Estimate the standard melting point of methylbenzene. ‡
I4.5 Diamond is the hardest substance and the best conductor of heat yet
(c) Estimate the standard boiling point of methylbenzene. (The equation you characterized. For these reasons, it is used widely in industrial applications
will need to solve to find this quantity cannot be solved by hand, so you should that require a strong abrasive. Unfortunately, it is difficult to synthesize
use a numerical approach, e.g. by using mathematical software.) (d) Calculate diamond from the more readily available allotropes of carbon, such as
the standard enthalpy of vaporization of methylbenzene at the standard boiling graphite. To illustrate this point, the following approach can be used to
point, given that the molar volumes of the liquid and vapour at the standard estimate the pressure required to convert graphite into diamond at 25 °C (i.e.
boiling point are 0.12 dm3 mol−1 and 30.3 dm3 mol−1, respectively. the pressure at which the conversion becomes spontaneous). The aim is to
find an expression for ∆rG for the process graphite → diamond as a function
I4.3 Proteins are polymers of amino acids that can exist in ordered structures
of the applied pressure, and then to determine the pressure at which the Gibbs
stabilized by a variety of molecular interactions. However, when certain
energy change becomes negative. (a) Derive the following expression for the
conditions are changed, the compact structure of a polypeptide chain may
pressure variation of ∆rG at constant temperature
collapse into a random coil. This structural change may be regarded as a phase
transition occurring at a characteristic transition temperature, the melting
r G
temperature, Tm, which increases with the strength and number of intermolecular Vm,d Vm,gr
interactions in the chain. A thermodynamic treatment allows predictions to p T
be made of the temperature Tm for the unfolding of a helical polypeptide held
together by hydrogen bonds into a random coil. If a polypeptide has N amino where Vm,gr is the molar volume of graphite and Vm,d that of diamond. (b) The
acid residues, N − 4 hydrogen bonds are formed to form an α-helix, the most difficulty with dealing with the previous expression is that the Vm depends
common type of helix in naturally occurring proteins (see Topic 14D). Because on the pressure. This dependence is handled as follows. Consider ∆rG to be a
the first and last residues in the chain are free to move, N − 2 residues form the function of pressure and form a Taylor expansion about p = p⦵:
compact helix and have restricted motion. Based on these ideas, the molar Gibbs A
B
energy of unfolding of a polypeptide with N ≥ 5 may be written as ⦵ G ⦵ 2 rG ⦵
r G( p) r G( p ) r ( p p ) 12 2
( p p )2
p p p⦵ p p p⦵
unfoldG (N 4) hb H (N 2)T hb S where the derivatives are evaluated at p = p⦵ and the series is truncated after
the second-order term. Term A can be found from the expression in part (a)
xii Using the book
TAKING YOUR LEARNING FURTHER
‘Impact’ sections details of some of the more advanced derivations. They are
listed at the beginning of the text and are referred to where
‘Impact’ sections show you how physical chemistry is ap- they are relevant. You can find a compilation of Deeper Looks
plied in a variety of modern contexts. They showcase physical at the end of the e-book.
chemistry as an evolving subject. These sections are listed at
the beginning of the text, and are referred to at appropriate
places elsewhere. You can find a compilation of ‘Impact’ sec- Group theory tables
tions at the end of the e-book. If you need character tables, you can find them at the end of
the Resource section.
A deeper look
These sections take some of the material in the text further.
Read them if you want to extend your knowledge and see the
TO THE INSTRUC TOR

We have designed the text to give you maximum flexibility in Key equations
the selection and sequence of Topics, while the grouping of
Topics into Focuses helps to maintain the unity of the subject. Supplied in Word format so you can download and edit them.
Additional resources are:
Solutions to exercises, problems, and
Figures and tables from the book integrated activities
Lecturers can find the artwork and tables from the book in For the discussion questions, examples, problems, and in-
ready-to-download format. They may be used for lectures tegrated activities detailed solutions to the even-numbered
without charge (but not for commercial purposes without spe- questions are available to lecturers online, so they can be set as
cific permission). homework or used as discussion points in class.
Lecturer resources are available only to registered adopters
of the textbook. To register, simply visit www.oup.com/he/
pchem12e and follow the appropriate links.
ABOUT THE AUTHORS
Peter Atkins is a fellow of Lincoln College, Oxford, and emeritus professor of physical chemistry in
the University of Oxford. He is the author of over seventy books for students and a general audience.
His texts are market leaders around the globe. A frequent lecturer throughout the world, he has held
visiting professorships in France, Israel, Japan, China, Russia, and New Zealand. He was the founding
chairman of the Committee on Chemistry Education of the International Union of Pure and Applied
Chemistry and was a member of IUPAC’s Physical and Biophysical Chemistry Division.
Photograph by Natasha
Ellis-Knight.
Julio de Paula is Professor of Chemistry at Lewis & Clark College. A native of Brazil, he received a
B.A. degree in chemistry from Rutgers, The State University of New Jersey, and a Ph.D. in biophysical
chemistry from Yale University. His research activities encompass the areas of molecular spectroscopy,
photochemistry, and nanoscience. He has taught courses in general chemistry, physical chemistry, bio-
chemistry, inorganic chemistry, instrumental analysis, environmental chemistry, and writing. Among
his professional honours are a Christian and Mary Lindback Award for Distinguished Teaching, a
Henry Dreyfus Teacher-Scholar Award, and a STAR Award from the Research Corporation for Science
Advancement.
James Keeler is Associate Professor of Chemistry, University of Cambridge, and Walters Fellow in
Chemistry at Selwyn College. He received his first degree and doctorate from the University of Oxford,
specializing in nuclear magnetic resonance spectroscopy. He is presently Head of Department, and be-
fore that was Director of Teaching in the department and also Senior Tutor at Selwyn College.
Photograph by Nathan Pitt,

© University of Cambridge.
ACKNOWLEDGEMENTS
A book as extensive as this could not have been written with- Rosalind Baverstock, Durham University
out significant input from many individuals. We would like to Grace Butler, Trinity College Dublin
thank the hundreds of instructors and students who contrib- Kaylyn Cater, Cardiff University
uted to this and the previous eleven editions: Ruth Comerford, University College Dublin
Orlagh Fraser, University of Aberdeen
Scott Anderson, University of Utah
Dexin Gao, University College London
Milan Antonijevic, University of Greenwich
Suruthi Gnanenthiran, University of Bath
Elena Besley, University of Greenwich
Milena Gonakova, University of the West of England Bristol
Merete Bilde, Aarhus University
Joseph Ingle, University of Lincoln
Matthew Blunt, University College London
Jeremy Lee, University of Durham
Simon Bott, Swansea University
Luize Luse, Heriot-Watt University
Klaus Braagaard Møller, Technical University of Denmark
Zoe Macpherson, University of Strathclyde
Wesley Browne, University of Groningen
Sukhbir Mann, University College London
Sean Decatur, Kenyon College
Declan Meehan, Trinity College Dublin
Anthony Harriman, Newcastle University
Eva Pogacar, Heriot-Watt University
Rigoberto Hernandez, Johns Hopkins University
Pawel Pokorski, Heriot-Watt University
J. Grant Hill, University of Sheffield
Fintan Reid, University of Strathclyde
Kayla Keller, Kentucky Wesleyan College
Gabrielle Rennie, University of Strathclyde
Kathleen Knierim, University of Louisiana Lafayette
Annabel Savage, Manchester Metropolitan University
Tim Kowalczyk, Western Washington University
Sophie Shearlaw, University of Strathclyde
Kristin Dawn Krantzman, College of Charleston
Yutong Shen, University College London
Hai Lin, University of Colorado Denver
Saleh Soomro, University College London
Mikko Linnolahti, University of Eastern Finland
Matthew Tully, Bangor University
Mike Lyons, Trinity College Dublin
Richard Vesely, University of Cambridge
Jason McAfee, University of North Texas
Phoebe Williams, Nottingham Trent University
Joseph McDouall, University of Manchester
Hugo Meekes, Radboud University We would also like to thank Michael Clugston for proofread-
Gareth Morris, University of Manchester ing the entire book, and Peter Bolgar, Haydn Lloyd, Aimee
David Rowley, University College London North, Vladimiras Oleinikovas, and Stephanie Smith who all
Nessima Salhi, Uppsala University worked alongside James Keeler in the writing of the solutions
Andy S. Sardjan, University of Groningen to the exercises and problems. The multiple-choice questions
Trevor Sears, Stony Brook University were developed in large part by Dr Stephanie Smith (Yusuf
Gemma Shearman, Kingston University Hamied Department of Chemistry and Pembroke College,
John Slattery, University of York University of Cambridge). These questions and further exer-
Catherine Southern, DePaul University cises were integrated into the text by Chloe Balhatchet (Yusuf
Michael Staniforth, University of Warwick Hamied Department of Chemistry and Selwyn College,
Stefan Stoll, University of Washington University of Cambridge), who also worked on the living
Mahamud Subir, Ball State University graphs. The solutions to the exercises and problems are taken
Enrico Tapavicza, CSU Long Beach from the solutions manual for the eleventh edition prepared
Jeroen van Duifneveldt, University of Bristol by Peter Bolgar, Haydn Lloyd, Aimee North, Vladimiras
Darren Walsh, University of Nottingham Oleinikovas, Stephanie Smith, and James Keeler, with addi-
Graeme Watson, Trinity College Dublin tional contributions from Chloe Balhatchet.
Darren L. Williams, Sam Houston State University Last, but by no means least, we acknowledge our two com-
Elisabeth R. Young, Lehigh University missioning editors, Jonathan Crowe of Oxford University
Press and Jason Noe of OUP USA, and their teams for their
Our special thanks also go to the many student reviewers who
assistance, advice, encouragement, and patience. We owe
helped to shape this twelfth edition:
special thanks to Katy Underhill, Maria Bajo Gutiérrez, and
Katherine Ailles, University of York Keith Faivre from OUP, who skillfully shepherded this com-
Mohammad Usman Ali, University of Manchester plex project to completion.
BRIEF CONTENTS
ENERGY A First Look xxxiii FOCUS 12 Magnetic resonance 499
FOCUS 1 The properties of gases 3 FOCUS 13 Statistical thermodynamics 543
FOCUS 2 The First Law 33 FOCUS 14 Molecular interactions 597
FOCUS 3 The Second and Third Laws 75 FOCUS 15 Solids 655
FOCUS 4 hysical transformations of pure

P FOCUS 16 Molecules in motion 707
substances 119
FOCUS 17 Chemical kinetics 737
FOCUS 5 Simple mixtures 141
FOCUS 18 Reaction dynamics 793
FOCUS 6 Chemical equilibrium 205
FOCUS 19 Processes at solid surfaces 835
FOCUS 7 Quantum theory 237
Resource section
FOCUS 8 Atomic structure and spectra 305 1 Mathematical resources 878
2 Quantities and units 882
FOCUS 9 Molecular structure 343 3 Data 884
4 Character tables 910
FOCUS 10 Molecular symmetry 397
Index 915
FOCUS 11 Molecular spectroscopy 427
FULL CONTENTS
Conventions xxvii FOCUS 2 The First Law 33

Physical chemistry: people and perspectives xxvii TOPIC 2A Internal energy 34
List of tables xxviii 2A.1 Work, heat, and energy 34
List of The chemist’s toolkits xxx (a) Definitions 34
List of material provided as A deeper look xxxi (b) The molecular interpretation of heat and work 35
List of Impacts xxxii 2A.2 The definition of internal energy 36

(a) Molecular interpretation of internal energy 36
(b) The formulation of the First Law 37
ENERGY A First Look xxxiii 2A.3 Expansion work 37
(a) The general expression for work 37
FOCUS 1 The properties of gases 3 (b) Expansion against constant pressure 38

(c) Reversible expansion 39
TOPIC 1A The perfect gas 4 (d) Isothermal reversible expansion of a perfect gas 39
1A.1 Variables of state 4 2A.4 Heat transactions 40
(a) Pressure and volume 4 (a) Calorimetry 40
(b) Temperature 5 (b) Heat capacity 41
(c) Amount 5 Checklist of concepts 43
(d) Intensive and extensive properties 5
Checklist of equations 44
1A.2 Equations of state 6
(a) The empirical basis of the perfect gas law 6 TOPIC 2B Enthalpy 45
(b) The value of the gas constant 8 2B.1 The definition of enthalpy 45
(c) Mixtures of gases 9 (a) Enthalpy change and heat transfer 45
Checklist of concepts 10 (b) Calorimetry 46
Checklist of equations 10 2B.2 The variation of enthalpy with temperature 47
(a) Heat capacity at constant pressure 47
TOPIC 1B The kinetic model 11 (b) The relation between heat capacities 49
1B.1 The model 11 Checklist of concepts 49
(a) Pressure and molecular speeds 11
(b) The Maxwell–Boltzmann distribution of speeds 12
(c) Mean values 14 TOPIC 2C Thermochemistry 50
1B.2 Collisions 16 2C.1 Standard enthalpy changes 50
(a) The collision frequency 16 (a) Enthalpies of physical change 50
(b) The mean free path 16 (b) Enthalpies of chemical change 51
Checklist of concepts 17 (c) Hess’s law 52
Checklist of equations 17 2C.2 Standard enthalpies of formation 53
2C.3 The temperature dependence of reaction enthalpies 54
TOPIC 1C Real gases 18 2C.4 Experimental techniques 55
1C.1 Deviations from perfect behaviour 18 (a) Differential scanning calorimetry 55
(a) The compression factor 19
(b) Isothermal titration calorimetry 56
(b) Virial coefficients 20
Checklist of concepts 56
(c) Critical constants 21
1C.2 The van der Waals equation 22
(a) Formulation of the equation 22 TOPIC 2D State functions and exact differentials 58
(b) The features of the equation 23
2D.1 Exact and inexact differentials 58
(c) The principle of corresponding states 24
2D.2 Changes in internal energy 59
(a) General considerations 59
Checklist of equations 26 (b) Changes in internal energy at constant pressure 60
xviii Full Contents
2D.3 Changes in enthalpy 62 TOPIC 3E Combining the First and Second Laws 104
2D.4 The Joule–Thomson effect 63 3E.1 Properties of the internal energy 104
Checklist of concepts 64 (a) The Maxwell relations 105
Checklist of equations 65 (b) The variation of internal energy with volume 106
3E.2 Properties of the Gibbs energy 107
TOPIC 2E Adiabatic changes 66 (a) General considerations 107
2E.1 The change in temperature 66 (b) The variation of the Gibbs energy with temperature 108
2E.2 The change in pressure 67 (c) The variation of the Gibbs energy of condensed phases
Checklist of concepts 68 with pressure 109
(d) The variation of the Gibbs energy of gases with pressure 109
FOCUS 3 The Second and Third Laws 75
TOPIC 3A Entropy 76 FOCUS 4 Physical transformations
3A.1 The Second Law 76 of pure substances 119
3A.2 The definition of entropy 78
(a) The thermodynamic definition of entropy 78
TOPIC 4A Phase diagrams of pure substances 120
(b) The statistical definition of entropy 79
4A.1 The stabilities of phases 120
(a) The number of phases 120
3A.3 The entropy as a state function 80
(b) Phase transitions 121
(a) The Carnot cycle 81
(c) Thermodynamic criteria of phase stability 121
(b) The thermodynamic temperature 83
(c) The Clausius inequality 84
4A.2 Coexistence curves 122
(a) Characteristic properties related to phase transitions 122
(b) The phase rule 123
4A.3 Three representative phase diagrams 125
TOPIC 3B Entropy changes accompanying (a) Carbon dioxide 125
(b) Water 125
specific processes 86
(c) Helium 126
3B.1 Expansion 86
3B.2 Phase transitions 87
3B.3 Heating 88
3B.4 Composite processes 89 TOPIC 4B Thermodynamic aspects of phase
Checklist of concepts 90 transitions 128
Checklist of equations 90 4B.1 The dependence of stability on the conditions 128
(a) The temperature dependence of phase stability 128
TOPIC 3C The measurement of entropy 91 (b) The response of melting to applied pressure 129
3C.1 The calorimetric measurement of entropy 91 (c) The vapour pressure of a liquid subjected to pressure 130
3C.2 The Third Law 92 4B.2 The location of coexistence curves 131
(a) The Nernst heat theorem 92 (a) The slopes of the coexistence curves 131
(b) Third-Law entropies 93 (b) The solid–liquid coexistence curve 132
(c) The temperature dependence of reaction entropy 94 (c) The liquid–vapour coexistence curve 132
Checklist of concepts 95 (d) The solid–vapour coexistence curve 134
Checklist of equations 95 Checklist of concepts 134
TOPIC 3D Concentrating on the system 96
3D.1 The Helmholtz and Gibbs energies 96
(a) Criteria of spontaneity 96
FOCUS 5 Simple mixtures 141
(b) Some remarks on the Helmholtz energy 97 TOPIC 5A The thermodynamic description
(c) Maximum work 97 of mixtures 143
(d) Some remarks on the Gibbs energy 99 5A.1 Partial molar quantities 143
(e) Maximum non-expansion work 99 (a) Partial molar volume 143
3D.2 Standard molar Gibbs energies 100 (b) Partial molar Gibbs energies 145
(a) Gibbs energies of formation 100 (c) The Gibbs–Duhem equation 146
(b) The Born equation 101 5A.2 The thermodynamics of mixing 147
Checklist of concepts 102 (a) The Gibbs energy of mixing of perfect gases 148
Checklist of equations 103 (b) Other thermodynamic mixing functions 149
Full Contents xix
5A.3 The chemical potentials of liquids 150 (c) Activities in terms of molalities 188
(a) Ideal solutions 150 5F.3 The activities of regular solutions 189
(b) Ideal–dilute solutions 151 5F.4 The activities of ions 190
Checklist of concepts 153 (a) Mean activity coefficients 190
Checklist of equations 154 (b) The Debye–Hückel limiting law 191
(c) Extensions of the limiting law 192
TOPIC 5B The properties of solutions 155 Checklist of concepts 193
5B.1 Liquid mixtures 155 Checklist of equations 193
(a) Ideal solutions 155
(b) Excess functions and regular solutions 156
5B.2 Colligative properties 158 FOCUS 6 Chemical equilibrium 205
(a) The common features of colligative properties 158
(b) The elevation of boiling point 159
TOPIC 6A The equilibrium constant 206
(c) The depression of freezing point 161 6A.1 The Gibbs energy minimum 206
(d) Solubility 161 (a) The reaction Gibbs energy 206
(e) Osmosis 162 (b) Exergonic and endergonic reactions 207
Checklist of concepts 165 6A.2 The description of equilibrium 207

(a) Perfect gas equilibria 208
(b) The general case of a reaction 209
TOPIC 5C Phase diagrams of binary (c) The relation between equilibrium constants 211
(d) Molecular interpretation of the equilibrium constant 212
systems: liquids 166
5C.1 Vapour pressure diagrams 166 Checklist of concepts 213
5C.2 Temperature–composition diagrams 168 Checklist of equations 213

(a) The construction of the diagrams 168
(b) The interpretation of the diagrams 169 TOPIC 6B The response of equilibria to
5C.3 Distillation 170 the conditions 214
(a) Fractional distillation 170 6B.1 The response to pressure 214
(b) Azeotropes 171 6B.2 The response to temperature 216
(c) Immiscible liquids 172 (a) The van ’t Hoff equation 216
5C.4 Liquid–liquid phase diagrams 172 (b) The value of K at different temperatures 217
(a) Phase separation 173 Checklist of concepts 218
(b) Critical solution temperatures 174 Checklist of equations 218
(c) The distillation of partially miscible liquids 175
Checklist of concepts 177 TOPIC 6C Electrochemical cells 219
Checklist of equations 177 6C.1 Half-reactions and electrodes 219
6C.2 Varieties of cell 220
TOPIC 5D Phase diagrams of binary (a) Liquid junction potentials 220
systems: solids 178 (b) Notation 221
5D.1 Eutectics 178 6C.3 The cell potential 221
5D.2 Reacting systems 180 (a) The Nernst equation 222
5D.3 Incongruent melting 180 (b) Cells at equilibrium 223
Checklist of concepts 181 6C.4 The determination of thermodynamic functions 224

TOPIC 5E Phase diagrams of ternary systems 182 Checklist of equations 225
5E.1 Triangular phase diagrams 182
5E.2 Ternary systems 183 TOPIC 6D Electrode potentials 226
(a) Partially miscible liquids 183 6D.1 Standard potentials 226
(b) Ternary solids 184 (a) The measurement procedure 227
Checklist of concepts 185 (b) Combining measured values 228
6D.2 Applications of standard electrode potentials 228
TOPIC 5F Activities 186 (a) The electrochemical series 228
5F.1 The solvent activity 186 (b) The determination of activity coefficients 229
5F.2 The solute activity 187 (c) The determination of equilibrium constants 229
(a) Ideal–dilute solutions 187 Checklist of concepts 230

(b) Real solutes 187 Checklist of equations 230
xx Full Contents
FOCUS 7 Quantum theory 237 Checklist of concepts 282

TOPIC 7A The origins of quantum mechanics 239
7A.1 Energy quantization 239 TOPIC 7F Rotational motion 283
(a) Black-body radiation 239 7F.1 Rotation in two dimensions 283
(b) Heat capacity 242 (a) The solutions of the Schrödinger equation 284
(c) Atomic and molecular spectra 243 (b) Quantization of angular momentum 285
7A.2 Wave–particle duality 244 7F.2 Rotation in three dimensions 286
(a) The particle character of electromagnetic radiation 244 (a) The wavefunctions and energy levels 286
(b) The wave character of particles 246 (b) Angular momentum 289
Checklist of concepts 247 (c) The vector model 290
TOPIC 7B Wavefunctions 248
7B.1 The Schrödinger equation 248
7B.2 The Born interpretation 248 FOCUS 8 Atomic structure and spectra 305
(a) Normalization 250 TOPIC 8A Hydrogenic atoms 306
(b) Constraints on the wavefunction 251
8A.1 The structure of hydrogenic atoms 306
(c) Quantization 251
(a) The separation of variables 306
Checklist of concepts 252 (b) The radial solutions 307
Checklist of equations 252 8A.2 Atomic orbitals and their energies 309
(a) The specification of orbitals 310
TOPIC 7C Operators and observables 253
(b) The energy levels 310
7C.1 Operators 253
(c) Ionization energies 311
(a) Eigenvalue equations 253
(d) Shells and subshells 311
(b) The construction of operators 254
(e) s Orbitals 312
(c) Hermitian operators 255
(f) Radial distribution functions 313
(d) Orthogonality 256
(g) p Orbitals 315
7C.2 Superpositions and expectation values 257 (h) d Orbitals 316
7C.3 The uncertainty principle 259 Checklist of concepts 316
7C.4 The postulates of quantum mechanics 261 Checklist of equations 317
Checklist of equations 262 TOPIC 8B Many-electron atoms 318
8B.1 The orbital approximation 318
TOPIC 7D Translational motion 263 8B.2 The Pauli exclusion principle 319
7D.1 Free motion in one dimension 263 (a) Spin 319
7D.2 Confined motion in one dimension 264 (b) The Pauli principle 320
(a) The acceptable solutions 264 8B.3 The building-up principle 322
(b) The properties of the wavefunctions 265 (a) Penetration and shielding 322
(c) The properties of the energy 266 (b) Hund’s rules 323
7D.3 Confined motion in two and more dimensions 268 (c) Atomic and ionic radii 325
(a) Energy levels and wavefunctions 268 (d) Ionization energies and electron affinities 326
(b) Degeneracy 270 8B.4 Self-consistent field orbitals 327
7D.4 Tunnelling 271 Checklist of concepts 328
Checklist of concepts 273 Checklist of equations 328
TOPIC 8C Atomic spectra 329
TOPIC 7E Vibrational motion 275 8C.1 The spectra of hydrogenic atoms 329
7E.1 The harmonic oscillator 275 8C.2 The spectra of many-electron atoms 330
(a) The energy levels 276 (a) Singlet and triplet terms 330
(b) The wavefunctions 276 (b) Spin–orbit coupling 331
7E.2 Properties of the harmonic oscillator 279 (c) Term symbols 334
(a) Mean values 279 (d) Hund’s rules and term symbols 337
(b) Tunnelling 280 (e) Selection rules 337
Full Contents xxi
Checklist of concepts 338 (c) Density functional theory 381

Checklist of equations 338 (d) Practical calculations 381
(e) Graphical representations 381
FOCUS 9 Molecular structure 343 Checklist of equations 382
PROLOGUE The Born–Oppenheimer approximation 345
TOPIC 9F Computational chemistry 383
TOPIC 9A Valence-bond theory 346 9F.1 The central challenge 383
9A.1 Diatomic molecules 346 9F.2 The Hartree−Fock formalism 384
9A.2 Resonance 348 9F.3 The Roothaan equations 385
9A.3 Polyatomic molecules 348 9F.4 Evaluation and approximation of the integrals 386
(a) Promotion 349
9F.5 Density functional theory 388
(b) Hybridization 350
TOPIC 9B Molecular orbital theory: the hydrogen FOCUS 10 Molecular symmetry 397
molecule-ion 353
9B.1 Linear combinations of atomic orbitals 353
TOPIC 10A Shape and symmetry 398
(a) The construction of linear combinations 353 10A.1 Symmetry operations and symmetry elements 398
(b) Bonding orbitals 354 10A.2 The symmetry classification of molecules 400
(c) Antibonding orbitals 356 (a) The groups C1, Ci, and Cs 402
9B.2 Orbital notation 358 (b) The groups Cn, Cnv, and Cnh 402
(c) The groups Dn, Dnh, and Dnd 403
(d) The groups Sn 403
(e) The cubic groups 403
TOPIC 9C Molecular orbital theory: homonuclear (f) The full rotation group 404
diatomic molecules 359 10A.3 Some immediate consequences of symmetry 404

(a) Polarity 404
9C.1 Electron configurations 359
(b) Chirality 405
(a) MO energy level diagrams 359
(b) σ Orbitals and π orbitals 360 Checklist of concepts 406
(c) The overlap integral 361 Checklist of symmetry operations and elements 406
(d) Period 2 diatomic molecules 362
9C.2 Photoelectron spectroscopy 364
TOPIC 10B Group theory 407
10B.1 The elements of group theory 407
10B.2 Matrix representations 409
(a) Representatives of operations 409
TOPIC 9D Molecular orbital theory: heteronuclear (b) The representation of a group 409
diatomic molecules 366 (c) Irreducible representations 410
9D.1 Polar bonds and electronegativity 366 (d) Characters 411
9D.2 The variation principle 367 10B.3 Character tables 412

(a) The procedure 368 (a) The symmetry species of atomic orbitals 413
(b) The features of the solutions 370 (b) The symmetry species of linear combinations of orbitals 413
(c) Character tables and degeneracy 414
TOPIC 9E Molecular orbital theory: polyatomic
molecules 373 TOPIC 10C Applications of symmetry 417
9E.1 The Hückel approximation 373 10C.1 Vanishing integrals 417
(a) An introduction to the method 373 (a) Integrals of the product of functions 418
(b) The matrix formulation of the method 374 (b) Decomposition of a representation 419
9E.2 Applications 376 10C.2 Applications to molecular orbital theory 420

(a) π-Electron binding energy 376 (a) Orbital overlap 420
(b) Aromatic stability 378 (b) Symmetry-adapted linear combinations 421
9E.3 Computational chemistry 379 10C.3 Selection rules 422

(a) Basis functions and basis sets 379 Checklist of concepts 423
(b) Electron correlation 380 Checklist of equations 423
xxii Full Contents
FOCUS 11 Molecular spectroscopy 427 TOPIC 11E Symmetry analysis of vibrational spectra 466
11E.1 Classification of normal modes according to symmetry 466
TOPIC 11A General features of molecular
11E.2 Symmetry of vibrational wavefunctions 468
spectroscopy 429
(a) Infrared activity of normal modes 468
11A.1 The absorption and emission of radiation 430
(b) Raman activity of normal modes 469
(a) Stimulated and spontaneous radiative processes 430
(c) The symmetry basis of the exclusion rule 469
(b) Selection rules and transition moments 431
(c) The Beer–Lambert law 431
11A.2 Spectral linewidths 433 TOPIC 11F Electronic spectra 470
(a) Doppler broadening 433
11F.1 Diatomic molecules 470
(b) Lifetime broadening 435
(a) Term symbols 470
11A.3 Experimental techniques 435 (b) Selection rules 473
(a) Sources of radiation 436 (c) Vibrational fine structure 473
(b) Spectral analysis 436 (d) Rotational fine structure 476
(c) Detectors 438
11F.2 Polyatomic molecules 477
(d) Examples of spectrometers 438
(a) d-Metal complexes 478
Checklist of concepts 439 (b) π* ← π and π* ← n transitions 479
TOPIC 11B Rotational spectroscopy 440
11B.1 Rotational energy levels 440 TOPIC 11G Decay of excited states 481
(a) Spherical rotors 441 11G.1 Fluorescence and phosphorescence 481
(b) Symmetric rotors 442 11G.2 Dissociation and predissociation 483
(c) Linear rotors 444
11G.3 Lasers 484
(d) Centrifugal distortion 444
11B.2 Microwave spectroscopy 444
(a) Selection rules 445
(b) The appearance of microwave spectra 446 FOCUS 12 Magnetic resonance 499
11B.3 Rotational Raman spectroscopy 447
TOPIC 12A General principles 500
11B.4 Nuclear statistics and rotational states 449
12A.1 Nuclear magnetic resonance 500
Checklist of concepts 451 (a) The energies of nuclei in magnetic fields 500
Checklist of equations 451 (b) The NMR spectrometer 502
12A.2 Electron paramagnetic resonance 503
TOPIC 11C Vibrational spectroscopy of (a) The energies of electrons in magnetic fields 503
diatomic molecules 452 (b) The EPR spectrometer 504
11C.1 Vibrational motion 452
11C.2 Infrared spectroscopy 453
11C.3 Anharmonicity 454
(a) The convergence of energy levels 454 TOPIC 12B Features of NMR spectra 506
(b) The Birge–Sponer plot 455 12B.1 The chemical shift 506
11C.4 Vibration–rotation spectra 456 12B.2 The origin of shielding constants 508
(a) Spectral branches 457 (a) The local contribution 508
(b) Combination differences 458 (b) Neighbouring group contributions 509
11C.5 Vibrational Raman spectra 458 (c) The solvent contribution 510
Checklist of concepts 460 12B.3 The fine structure 511
Checklist of equations 460 (a) The appearance of the spectrum 511
(b) The magnitudes of coupling constants 513
TOPIC 11D Vibrational spectroscopy of 12B.4 The origin of spin-spin coupling 514
polyatomic molecules 461 (a) Equivalent nuclei 516
11D.1 Normal modes 461 (b) Strongly coupled nuclei 517
11D.2 Infrared absorption spectra 462 12B.5 Exchange processes 517
11D.3 Vibrational Raman spectra 464 12B.6 Solid-state NMR 518
Checklist of concepts 464 Checklist of concepts 519
Checklist of equations 465 Checklist of equations 520
Full Contents xxiii
TOPIC 12C Pulse techniques in NMR 521 TOPIC 13D The canonical ensemble 567
12C.1 The magnetization vector 521 13D.1 The concept of ensemble 567
(a) The effect of the radiofrequency field 522 (a) Dominating configurations 568
(b) Time- and frequency-domain signals 523 (b) Fluctuations from the most probable distribution 568
12C.2 Spin relaxation 525 13D.2 The mean energy of a system 569
(a) The mechanism of relaxation 525 13D.3 Independent molecules revisited 569
(b) The measurement of T1 and T2 526 13D.4 The variation of the energy with volume 570
12C.3 Spin decoupling 527 Checklist of concepts 572
12C.4 The nuclear Overhauser effect 528 Checklist of equations 572
Checklist of equations 530 TOPIC 13E The internal energy and the entropy 573
13E.1 The internal energy 573
TOPIC 12D Electron paramagnetic resonance 531 (a) The calculation of internal energy 573
(b) Heat capacity 574
12D.1 The g-value 531
13E.2 The entropy 575
12D.2 Hyperfine structure 532
(a) Entropy and the partition function 576
(a) The effects of nuclear spin 532
(b) The translational contribution 577
12D.3 The McConnell equation 533
(c) The rotational contribution 578
(a) The origin of the hyperfine interaction 534
(d) The vibrational contribution 579
Checklist of concepts 535 (e) Residual entropies 579
FOCUS 13 Statistical thermodynamics 543 TOPIC 13F Derived functions 582

13F.1 The derivations 582
TOPIC 13A The Boltzmann distribution 544 13F.2 Equilibrium constants 585
13A.1 Configurations and weights 544 (a) The relation between K and the partition function 585
(a) Instantaneous configurations 544
(b) A dissociation equilibrium 586
(b) The most probable distribution 545
(c) Contributions to the equilibrium constant 586
13A.2 The relative populations of states 548 Checklist of concepts 588
FOCUS 14 Molecular interactions 597

TOPIC 13B Molecular partition functions 550
13B.1 The significance of the partition function 550 TOPIC 14A The electric properties of molecules 599
13B.2 Contributions to the partition function 552 14A.1 Electric dipole moments 599
(a) The translational contribution 552 14A.2 Polarizabilities 601
(b) The rotational contribution 554 14A.3 Polarization 603
(c) The vibrational contribution 558 (a) The mean dipole moment 603
(d) The electronic contribution 559 (b) The frequency dependence of the polarization 604
Checklist of concepts 560 (c) Molar polarization 604
TOPIC 13C Molecular energies 561
13C.1 The basic equations 561
TOPIC 14B Interactions between molecules 608
14B.1 The interactions of dipoles 608
13C.2 Contributions of the fundamental modes
of motion 562 (a) Charge–dipole interactions 608
(a) The translational contribution 562 (b) Dipole–dipole interactions 609
(b) The rotational contribution 562 (c) Dipole–induced dipole interactions 612
(c) The vibrational contribution 563 (d) Induced dipole–induced dipole interactions 612
(d) The electronic contribution 564 14B.2 Hydrogen bonding 613

(e) The spin contribution 565 14B.3 The total interaction 614
xxiv Full Contents
TOPIC 14C Liquids 618 (c) Scattering factors 666

(d) The electron density 666
14C.1 Molecular interactions in liquids 618
(e) The determination of structure 669
(a) The radial distribution function 618
(b) The calculation of g(r) 619 15B.2 Neutron and electron diffraction 671
(c) The thermodynamic properties of liquids 620 Checklist of concepts 672
14C.2 The liquid–vapour interface 621 Checklist of equations 672
(a) Surface tension 621
(b) Curved surfaces 622 TOPIC 15C Bonding in solids 673
(c) Capillary action 623 15C.1 Metals 673
14C.3 Surface films 624 (a) Close packing 673
(a) Surface pressure 624 (b) Electronic structure of metals 675
(b) The thermodynamics of surface layers 626 15C.2 Ionic solids 677
14C.4 Condensation 627 (a) Structure 677
(b) Energetics 678
15C.3 Covalent and molecular solids 681
TOPIC 14D Macromolecules 629 Checklist of equations 682
14D.1 Average molar masses 629
14D.2 The different levels of structure 630 TOPIC 15D The mechanical properties of solids 683
14D.3 Random coils 631 Checklist of concepts 685
(a) Measures of size 631 Checklist of equations 685
(b) Constrained chains 634
(c) Partly rigid coils 634 TOPIC 15E The electrical properties of solids 686
14D.4 Mechanical properties 635 15E.1 Metallic conductors 686
(a) Conformational entropy 635 15E.2 Insulators and semiconductors 687
(b) Elastomers 636 15E.3 Superconductors 689
14D.5 Thermal properties 637 Checklist of concepts 690
TOPIC 15F The magnetic properties of solids 691
TOPIC 14E Self-assembly 640 15F.1 Magnetic susceptibility 691
14E.1 Colloids 640 15F.2 Permanent and induced magnetic moments 692
(a) Classification and preparation 640
15F.3 Magnetic properties of superconductors 693
(b) Structure and stability 641
(c) The electrical double layer 641
14E.2 Micelles and biological membranes 643
(a) The hydrophobic interaction 643
TOPIC 15G The optical properties of solids 695
(b) Micelle formation 644
15G.1 Excitons 695
(c) Bilayers, vesicles, and membranes 646
15G.2 Metals and semiconductors 696
(a) Light absorption 696
(b) Light-emitting diodes and diode lasers 697
15G.3 Nonlinear optical phenomena 697
FOCUS 15 Solids 655 Checklist of concepts 698
TOPIC 15A Crystal structure 657
15A.1 Periodic crystal lattices 657 FOCUS 16 Molecules in motion 707
15A.2 The identification of lattice planes 659
(a) The Miller indices 659
TOPIC 16A Transport properties of a perfect gas 708
(b) The separation of neighbouring planes 660
16A.1 The phenomenological equations 708
Checklist of concepts 661 16A.2 The transport parameters 710

(a) The diffusion coefficient 711
(b) Thermal conductivity 712
TOPIC 15B Diffraction techniques 663 (c) Viscosity 714
(d) Effusion 715
15B.1 X-ray crystallography 663
(a) X-ray diffraction 663 Checklist of concepts 716
(b) Bragg’s law 665 Checklist of equations 716
Full Contents xxv
TOPIC 16B Motion in liquids 717 TOPIC 17E Reaction mechanisms 762
16B.1 Experimental results 717 17E.1 Elementary reactions 762
(a) Liquid viscosity 717 17E.2 Consecutive elementary reactions 763
(b) Electrolyte solutions 718 17E.3 The steady-state approximation 764
16B.2 The mobilities of ions 719 17E.4 The rate-determining step 766
(a) The drift speed 719
17E.5 Pre-equilibria 767
(b) Mobility and conductivity 721
17E.6 Kinetic and thermodynamic control of reactions 768
(c) The Einstein relations 722
TOPIC 17F Examples of reaction mechanisms 769
TOPIC 16C Diffusion 724
17F.1 Unimolecular reactions 769
16C.1 The thermodynamic view 724
17F.2 Polymerization kinetics 771
16C.2 The diffusion equation 726
(a) Stepwise polymerization 771
(a) Simple diffusion 726
(b) Chain polymerization 772
(b) Diffusion with convection 728
17F.3 Enzyme-catalysed reactions 774
(c) Solutions of the diffusion equation 728
16C.3 The statistical view 730
Checklist of equations 732 TOPIC 17G Photochemistry 778
17G.1 Photochemical processes 778
FOCUS 17 Chemical kinetics 737 17G.2 The primary quantum yield 779
17G.3 Mechanism of decay of excited singlet states 780
TOPIC 17A The rates of chemical reactions 739
17G.4 Quenching 781
17A.1 Monitoring the progress of a reaction 739
17G.5 Resonance energy transfer 783
(a) General considerations 739
(b) Special techniques 740
17A.2 The rates of reactions 741
(a) The definition of rate 741
(b) Rate laws and rate constants 742 FOCUS 18 Reaction dynamics 793
(c) Reaction order 743
(d) The determination of the rate law 744 TOPIC 18A Collision theory 794
Checklist of concepts 746 18A.1 Reactive encounters 794
(a) Collision rates in gases 795
(b) The energy requirement 795
TOPIC 17B Integrated rate laws 747 (c) The steric requirement 798
17B.1 Zeroth-order reactions 747 18A.2 The RRK model 799

17B.2 First-order reactions 747 Checklist of concepts 800
17B.3 Second-order reactions 749 Checklist of equations 800
TOPIC 18B Diffusion-controlled reactions 801
18B.1 Reactions in solution 801
(a) Classes of reaction 801
TOPIC 17C Reactions approaching equilibrium 753
(b) Diffusion and reaction 802
17C.1 First-order reactions approaching equilibrium 753
18B.2 The material-balance equation 803
17C.2 Relaxation methods 754
(a) The formulation of the equation 803
Checklist of concepts 756 (b) Solutions of the equation 804
TOPIC 17D The Arrhenius equation 757
17D.1 The temperature dependence of rate constants 757 TOPIC 18C Transition-state theory 806
17D.2 The interpretation of the Arrhenius parameters 759 18C.1 The Eyring equation 806
(a) A first look at the energy requirements of reactions 759 (a) The formulation of the equation 806
(b) The effect of a catalyst on the activation energy 760 (b) The rate of decay of the activated complex 807
Checklist of concepts 761 (c) The concentration of the activated complex 807
Checklist of equations 761 (d) The rate constant 808
xxvi Full Contents
18C.2 Thermodynamic aspects 809 TOPIC 19B Adsorption and desorption 844
(a) Activation parameters 809
19B.1 Adsorption isotherms 844
(b) Reactions between ions 811
(a) The Langmuir isotherm 844
18C.3 The kinetic isotope effect 812 (b) The isosteric enthalpy of adsorption 845
Checklist of concepts 814 (c) The BET isotherm 847
Checklist of equations 814 (d) The Temkin and Freundlich isotherms 849
19B.2 The rates of adsorption and desorption 850
TOPIC 18D The dynamics of molecular collisions 815 (a) The precursor state 850
18D.1 Molecular beams 815 (b) Adsorption and desorption at the molecular level 850
(a) Techniques 815 (c) Mobility on surfaces 852
(b) Experimental results 816 Checklist of concepts 852
18D.2 Reactive collisions 818 Checklist of equations 852
(a) Probes of reactive collisions 818
(b) State-to-state reaction dynamics 818 TOPIC 19C Heterogeneous catalysis 853
18D.3 Potential energy surfaces 819 19C.1 Mechanisms of heterogeneous catalysis 853
18D.4 Some results from experiments and calculations 820 (a) Unimolecular reactions 853
(a) The direction of attack and separation 821 (b) The Langmuir–Hinshelwood mechanism 854
(b) Attractive and repulsive surfaces 821 (c) The Eley-Rideal mechanism 855
(c) Quantum mechanical scattering theory 822 19C.2 Catalytic activity at surfaces 855
TOPIC 18E Electron transfer in homogeneous TOPIC 19D Processes at electrodes 857
systems 824 19D.1 The electrode–solution interface 857
18E.1 The rate law 824 19D.2 The current density at an electrode 858
18E.2 The role of electron tunnelling 825 (a) The Butler–Volmer equation 858
18E.3 The rate constant 826 (b) Tafel plots 862
18E.4 Experimental tests of the theory 828 19D.3 Voltammetry 862

Checklist of concepts 829 19D.4 Electrolysis 865
Checklist of equations 829 19D.5 Working galvanic cells 865
FOCUS 19 Processes at solid surfaces 835
TOPIC 19A An introduction to solid surfaces 836
Resource section 877
19A.1 Surface growth 836
1 Mathematical resources 878
19A.2 Physisorption and chemisorption 837
19A.3 Experimental techniques 838
1.1 Integration 878
(a) Microscopy 839 1.2 Differentiation 878
(b) Ionization techniques 840 1.3 Series expansions 881
(c) Diffraction techniques 841 2 Quantities and units 882
(d) Determination of the extent and rates of adsorption
and desorption 842
3 Data 884
Checklist of concepts 843 4 Character tables 910
Index 915
CONVENTIONS
To avoid intermediate rounding errors, but to keep track of e rrors, we display intermediate results as n.nnn. . . and round
values in order to be aware of values and to spot numerical the calculation only at the final step.
PHYSIC AL CHEMISTRY: PEOPLE AND PERSPEC TIVES

To watch these interviews, go to this section of the e-book.
LIST OF TABLES
Table 1A.1 Pressure units 4 Table 5F.1 Ionic strength and molality, I = kb/b⦵ 191
Table 1A.2 The (molar) gas constant 9 Table 5F.2 Mean activity coefficients in water at 298 K 192
Table 1B.1 Collision cross-sections 16 Table 5F.3 Activities and standard states: a summary 193
Table 1C.1 Second virial coefficients, B/(cm mol )
3 −1
20 Table 6C.1 Varieties of electrode 219
Table 1C.2 Critical constants of gases 21 Table 6D.1 Standard potentials at 298 K 226
Table 1C.3 van der Waals coefficients 22 Table 6D.2 The electrochemical series 229
Table 1C.4 Selected equations of state 23 Table 7E.1 The Hermite polynomials 277
Table 2A.1 Varieties of work 38 Table 7F.1 The spherical harmonics 288
Table 2B.1 Temperature variation of molar heat capacities, 48 Table 8A.1 Hydrogenic radial wavefunctions 308
Cp,m/(J K−1 mol−1) = a + bT + c/T2
Table 8B.1 Effective nuclear charge 322
Table 2C.1 Standard enthalpies of fusion and 51
Table 8B.2 Atomic radii of main-group elements, r/pm 325
vaporization at the transition temperature,
⦵
ΔtrsH /(kJ mol−1) Table 8B.3 Ionic radii, r/pm 326
Table 2C.2 Enthalpies of reaction and transition 51 Table 8B.4 First and second ionization energies 326
Table 2C.3 Standard enthalpies of formation and 52 Table 8B.5 Electron affinities, Ea/(kJ mol−1) 327
combustion of organic compounds at 298 K
Table 9A.1 Some hybridization schemes 352
Table 2C.4 Standard enthalpies of formation of inorganic 53
Table 9C.1 Overlap integrals between hydrogenic orbitals 361
compounds at 298 K
Standard enthalpies of formation of organic 53 Table 9C.2 Bond lengths 364
Table 2C.5
compounds at 298 K Table 9C.3 Bond dissociation energies, N AhcD 0 364
Table 2D.1 Expansion coefficients (α) and isothermal 61 Table 9D.1 Pauling electronegativities 367
compressibilities (κT) at 298 K
Table 10A.1 The notations for point groups 400
Table 2D.2 Inversion temperatures (TI), normal freezing 63
Table 10B.1 The C2v character table 412
(Tf ) and boiling (T b) points, and Joule–
Thomson coefficients (μ) at 1 bar and 298 K Table 10B.2 The C3v character table 412
Table 3B.1 Entropies of phase transitions, ΔtrsS/(J K−1 mol−1), 87 Table 10B.3 The C 4 character table 415
at the corresponding normal transition
Table 11B.1 Moments of inertia 440
temperatures (at 1 atm)
Table 11C.1 Properties of diatomic molecules 458
Table 3B.2 The standard enthalpies and entropies of 87
vaporization of liquids at their boiling Table 11F.1 Colour, frequency, and energy of light 470
temperatures
Table 11F.2 Absorption characteristics of some groups and 477
Table 3C.1 Standard Third-Law entropies at 298 K 93 molecules
Table 3D.1 Standard Gibbs energies of formation at 298 K 100 Table 11G.1 Characteristics of laser radiation and their 484
chemical applications
Table 3E.1 The Maxwell relations 105
Table 12A.1 Nuclear constitution and the nuclear spin 500
Table 5A.1 Henry’s law constants for gases in water at 153
quantum number
298 K
Table 12A.2 Nuclear spin properties 501
Table 5B.1 Freezing-point (K f ) and boiling-point (K b) 160
constants Table 12D.1 Hyperfine coupling constants for atoms, a/mT 534
List of Tables xxix
Table 13B.1 Rotational temperatures of diatomic molecules 556 Table 17D.1 Arrhenius parameters 757
Table 13B.2 Symmetry numbers of molecules 557 Table 17G.1 Examples of photochemical processes 778
Table 13B.3 Vibrational temperatures of diatomic molecules 559 Table 17G.2 Common photophysical processes 779
Table 14A.1 Dipole moments and polarizability volumes 599 Table 17G.3 Values of R0 for some donor–acceptor pairs 783
Table 14B.1 Interaction potential energies 612 Table 18A.1 Arrhenius parameters for gas-phase reactions 798
Table 14B.2 Lennard-Jones-(12,6) potential energy 616 Table 18B.1 Arrhenius parameters for solvolysis reactions 802
parameters in solution
Table 14C.1 Surface tensions of liquids at 293 K 621 Table 19A.1 Maximum observed standard enthalpies of 837
physisorption at 298 K
Table 14E.1 Micelle shape and the surfactant parameter 645
Table 19A.2 Standard enthalpies of chemisorption, 837
Table 15A.1 The seven crystal systems 658 ⦵
Δad H /(kJ mol−1), at 298 K
Table 15C.1 The crystal structures of some elements 674
Table 19C.1 Chemisorption abilities 856
Table 15C.2 Ionic radii, r/pm 678
Table 19D.1 Exchange-current densities and transfer 861
Table 15C.3 Madelung constants 679 coefficients at 298 K
Table 15C.4 Lattice enthalpies at 298 K, ΔHL/(kJ mol−1) 681 RESOURCE SECTION TABLES
Table 15F.1 Magnetic susceptibilities at 298 K 692 Table 1.1 Common integrals 879
Table 16A.1 Transport properties of gases at 1 atm 709 Table 2.1 Some common units 882
Table 16B.1 Viscosities of liquids at 298 K 717 Table 2.2 Common SI prefixes 882
Table 16B.2 Ionic mobilities in water at 298 K 720 Table 2.3 The SI base units 882
Table 16B.3 Diffusion coefficients at 298 K, D/(10 m s )

−9 2 −1
722 Table 2.4 A selection of derived units 883
Table 17B.1 Kinetic data for first-order reactions 748 Table 0.1 Physical properties of selected materials 885
Table 17B.2 Kinetic data for second-order reactions 749 Table 0.2 Masses and natural abundances of selected 886
nuclides
Table 17B.3 Integrated rate laws 751
LIST OF THE CHEMIST’S TOOLKITS
Number Title
2A.1 Electrical charge, current, power, and energy 41
2A.2 Partial derivatives 42
3E.1 Exact differentials 105
5B.1 Molality and mole fraction 160
7A.1 Electromagnetic radiation 239
7A.2 Diffraction of waves 246
7B.1 Complex numbers 249
7F.1 Cylindrical coordinates 283
7F.2 Spherical polar coordinates 287
8C.1 Combining vectors 332
9D.1 Determinants 369
9E.1 Matrices 375
11A.1 Exponential and Gaussian functions 434
12B.1 Dipolar magnetic fields 509
12C.1 The Fourier transform 524
16B.1 Electrostatics 720
17B.1 Integration by the method 751
LIST OF MATERIAL PROVIDED
AS A DEEPER LOOK
The list of A deeper look material that can be found via the e-book. You will also find references to this material where relevant
throughout the book.
Number Title
2D.1 The Joule–Thomson effect and isenthalpic change
3D.1 The Born equation
5F.1 The Debye–Hückel theory
5F.2 The fugacity
7D.1 Particle in a triangle
7F.1 Separation of variables
9B.1 The energies of the molecular orbitals of H2+
9F.1 The equations of computational chemistry
9F.2 The Roothaan equations
11A.1 Origins of spectroscopic transitions
11B.1 Rotational selection rules
11C.1 Vibrational selection rules
13D.1 The van der Waals equation of state
14B.1 The electric dipole–dipole interaction
14C.1 The virial and the virial equation of state
15D.1 Establishing the relation between bulk and molecular properties
16C.1 Diffusion in three dimensions
16C.2 The random walk
18A.1 The RRK model
19B.1 The BET isotherm
LIST OF IMPAC TS
The list of Impacts that can be found via the e-book. You will also find references to this material where relevant throughout the
book.
Number Focus Title

1 1 . . .on environmental science: The gas laws and the weather
2 1 . . .on astrophysics: The Sun as a ball of perfect gas
3 2 . . .on technology: Thermochemical aspects of fuels and foods
4 3 . . .on engineering: Refrigeration
5 3 . . .on materials science: Crystal defects
6 4 . . .on technology: Supercritical fluids
7 5 . . .on biology: Osmosis in physiology and biochemistry
8 5 . . .on materials science: Liquid crystals
9 6 . . .on biochemistry: Energy conversion in biological cells
10 6 . . .on chemical analysis: Species-selective electrodes
11 7 . . .on technology: Quantum computing
12 7 . . .on nanoscience: Quantum dots
13 8 . . .on astrophysics: The spectroscopy of stars
14 9 . . .on biochemistry: The reactivity of O2, N2, and NO
15 9 . . .on biochemistry: Computational studies of biomolecules
16 11 . . .on astrophysics: Rotational and vibrational spectroscopy of interstellar species
17 11 . . .on environmental science: Climate change
18 12 . . .on medicine: Magnetic resonance imaging
19 12 . . .on biochemistry and nanoscience: Spin probes
20 13 . . .on biochemistry: The helix–coil transition in polypeptides
21 14 . . .on biology: Biological macromolecules
22 14 . . .on medicine: Molecular recognition and drug design
23 15 . . .on biochemistry: Analysis of X-ray diffraction by DNA
24 15 . . .on nanoscience: Nanowires
25 16 . . .on biochemistry: Ion channels
26 17 . . .on biochemistry: Harvesting of light during plant photosynthesis
27 19 . . .on technology: Catalysis in the chemical industry
28 19 . . .on technology: Fuel cells
ENERGY A First Look
Much of chemistry is concerned with the transfer and trans- For example, the x-component, vx, is the particle’s rate of
formation of energy, so right from the outset it is important to change of position along the x-axis:
become familiar with this concept. The first ideas about energy dx Component of velocity
emerged from classical mechanics, the theory of motion for- vx = [definition]
(1a)
dt
mulated by Isaac Newton in the seventeenth century. In the
twentieth century classical mechanics gave way to quantum Similar expressions may be written for the y- and z-components.
mechanics, the theory of motion formulated for the descrip- The magnitude of the velocity, as represented by the length of
tion of small particles, such as electrons, atoms, and molecules. the velocity vector, is the speed, v. Speed is related to the com-
In quantum mechanics the concept of energy not only survived ponents of velocity by
but was greatly enriched, and has come to underlie the whole of Speed
physical chemistry. v (vx2 v 2y vz2 )1/ 2 [definition]
(1b)
The linear momentum, p, of a particle, like the velocity, is

a vector quantity, but takes into account the mass of the parti-
1 Force cle as well as its speed and direction. Its components are px, py,
and pz along each axis (Fig. 1b) and its magnitude is p. A heavy
Classical mechanics is formulated in terms of the forces acting particle travelling at a certain speed has a greater linear mo-
on particles, and shows how the paths of particles respond to mentum than a light particle travelling at the same speed. For a
them by accelerating or changing direction. Much of the dis- particle of mass m, the x-component of the linear momentum
cussion focuses on a quantity called the ‘momentum’ of the is given by
particle. Component of linear momentum
px = mvx [definition]
(2)
(a) Linear momentum and similarly for the y- and z-components.

‘Translation’ is the motion of a particle through space. The
velocity, v, of a particle is the rate of change of its position. Brief illustration 1
Velocity is a ‘vector quantity’, meaning that it has both a direc-
tion and a magnitude, and is expressed in terms of how fast Imagine a particle of mass m attached to a spring. When the
the particle travels with respect to x-, y-, and z-axes (Fig. 1). particle is displaced from its equilibrium position and then
released, it oscillates back and forth about this equilibrium
position. This model can be used to describe many features
of a chemical bond. In an idealized case, known as the simple
pz harmonic oscillator, the displacement from equilibrium x(t)
vz p varies with time as
Heavy
v particle
p x(t ) A sin 2 t
th
v Light
eng In this expression, ν (nu) is the frequency of the oscillation
th L
ng
particle
e and A is its amplitude, the maximum value of the displace-
vx L px ment along the x-axis. The x-component of the velocity of the
vy
py particle is therefore
(a) (b)
dx d( A sin 2 t )
vx 2 A cos 2 t
Figure 1 (a) The velocity v is denoted by a vector of magnitude dt dt
v (the speed) and an orientation that indicates the direction The x-component of the linear momentum of the particle is
of translational motion. (b) Similarly, the linear momentum p
is denoted by a vector of magnitude p and an orientation that px mvx 2 Am cos 2 t
corresponds to the direction of motion.
xxxiv ENERGY A First Look
(b) Angular momentum z

Jz
‘Rotation’ is the change of orientation in space around a cen- J
tral point (the ‘centre of mass’). Its description is very similar to
that of translation but with ‘angular velocity’ taking the place of
velocity and ‘moment of inertia’ taking the place of mass. The
angular velocity, ω (omega) is the rate of change of orientation
(for example, in radians per second); it is a vector with magni- Jx Jy
tude ω. The moment of inertia, I, is a measure of the mass that y
is being swung round by the rotational motion. For a particle x

of mass m moving in a circular path of radius r, the moment of
inertia is Figure 2 The angular momentum J of a particle is represented by
a vector along the axis of rotation and perpendicular to the plane
I = mr 2
Moment of inertia
(3a) of rotation. The length of the vector denotes the magnitude J of
[definition] the angular momentum. The direction of motion is clockwise to
an observer looking in the direction of the vector.
For a molecule composed of several atoms, each atom i gives a
contribution of this kind, and the moment of inertia around a
given axis is
the velocity and momentum, the force, F, is a vector quantity
I mi ri (3b)
2 with a direction and a magnitude (the ‘strength’ of the force).
i Force is reported in newtons, with 1 N = 1 kg m s−2. For motion
along the x-axis Newton’s second law states that
where ri is the perpendicular distance from the mass mi to
the axis. The rotation of a particle is described by its angular dp x Newton’s second law
momentum, J, a vector with a length that indicates the rate at = Fx [in one dimension]
(5a)
dt
which the particle circulates and a direction that indicates the
axis of rotation (Fig. 2). The components of angular momen- where Fx is the component of the force acting along the x-axis.
tum, Jx, Jy, and Jz, on three perpendicular axes show how much Each component of linear momentum obeys the same kind of
angular momentum is associated with rotation around each equation, so the vector p changes with time as
axis. The magnitude J of the angular momentum is
dp Newton’s second law
Magnitude of angular momentum
=F [vector form]
(5b)
J I (4) dt
[definition]
Equation 5 is the equation of motion of the particle, the equa-
tion that has to be solved to calculate its trajectory.
Brief illustration 2
A CO2 molecule is linear, and the length of each CO bond

is 116 pm. The mass of each 16O atom is 16.00mu, where Brief illustration 3
mu = 1.661 × 10−27 kg. It follows that the moment of inertia of
the molecule around an axis perpendicular to the axis of the According to ‘Hooke’s law’, the force acting on a particle under-
molecule and passing through the C atom is going harmonic motion (like that in Brief illustration 2) is
proportional to the displacement and directed opposite to the
I mO R2 0 mO R2 2mO R2 direction of motion, so in one dimension
2 (16.00 1.661 1027 kg ) (1.16 1010 m)2 Fx kf x
46 2
7.15 10 kg m where x is the displacement from equilibrium and kf is the
‘force constant’, a measure of the stiffness of the spring (or
chemical bond). It then follows that the equation of motion
of a particle undergoing harmonic motion is dpx/dt = −kfx.
(c) Newton’s second law of motion Then, because px = mvx and vx = dx/dt, it follows that dpx/dt =
mdvx/dt = md2x/dt2. With this substitution, the equation of
The central concept of classical mechanics is Newton’s second motion becomes
law of motion, which states that the rate of change of momen-
tum is equal to the force acting on the particle. This law underlies d2 x
m kx
the calculation of the trajectory of a particle, a statement about dt 2
where it is and where it is moving at each moment of time. Like
ENERGY A First Look xxxv
Equations of this kind, which are called ‘differential equations’, Brief illustration 4
are solved by special techniques. In most cases in this text, the
solutions are simply stated without going into the details of Suppose that when a bond is stretched from its equilibrium
how they are found. value Re to some arbitrary value R there is a restoring force
Similar considerations apply to rotation. The change in proportional to the displacement x = R – Re from the equilib-
angular momentum of a particle is expressed in terms of the rium length. Then
torque, T, a twisting force. The analogue of eqn 5b is then
Fx kf (R Re ) kf x
dJ The constant of proportionality, kf, is the force constant intro-
= T (6)
dt duced in Brief illustration 3. The total work needed to move
an atom so that the bond stretches from zero displacement
Quantities that describe translation and rotation are analogous, (xinitial = 0), when the bond has its equilibrium length, to a dis-
as shown below: placement xfinal = Rfinal – Re is
Integral A.1

x final x final
Property Translation Rotation won an atom ( k f x ) dx k f x dx
0 0
Rate linear velocity, v angular velocity, ω 2
12 kf x final 21 kf (Rfinal Re )2
Resistance to change mass, m moment of inertia, I
Momentum linear momentum, p angular momentum, J (All the integrals required in this book are listed in the Resource
Influence on motion force, F torque, T section.) The work required increases as the square of the dis-
placement: it takes four times as much work to stretch a bond
through 20 pm as it does to stretch the same bond through
10 pm.
2 Energy
Energy is a powerful and essential concept in science; never- (b) The definition of energy
theless, its actual nature is obscure and it is difficult to say what Now we get to the core of this discussion. Energy is the capacity
it ‘is’. However, it can be related to processes that can be meas- to do work. An object with a lot of energy can do a lot of work;
ured and can be defined in terms of the measurable process one with little energy can do only little work. Thus, a spring that
called work. is compressed can do a lot of work as it expands, so it is said
to have a lot of energy. Once the spring is expanded it can do
(a) Work only a little work, perhaps none, so it is said to have only a little
energy. The SI unit of energy is the same as that of work, namely
Work, w, is done in order to achieve motion against an op- the joule, with 1 J = 1 kg m2 s−2.
posing force. The work needed to be done to move a particle A particle may possess two kinds of energy, kinetic energy
through the infinitesimal distance dx against an opposing and potential energy. The kinetic energy, Ek, of a particle is the
force Fx is energy it possesses as a result of its motion. For a particle of
mass m travelling at a speed v,
Work
dwon the particle Fx dx [definition]
(7a)
Kinetic energy
Ek = 12 mv 2 [definition]
(8a)
When the force is directed to the left (to negative x), Fx is nega-
A particle with a lot of kinetic energy can do a lot of work, in
tive, so for motion to the right (dx positive), the work that must
the sense that if it collides with another particle it can cause it to
be done to move the particle is positive. With force in newtons
move against an opposing force. Because the magnitude of the
and distance in metres, the units of work are joules (J), with
linear momentum and speed are related by p = mv, so v = p/m,
1 J = 1 N m = 1 kg m2 s–2.
an alternative version of this relation is
The total work that has to be done to move a particle from
xinitial to xfinal is found by integrating eqn 7a, allowing for the p2
possibility that the force may change at each point along Ek = (8b)
2m
the path:
It follows from Newton’s second law that if a particle is initially
x final stationary and is subjected to a constant force then its linear
won the particle Fx dx Work (7b)
xinitial momentum increases from zero. Because the magnitude of the
xxxvi ENERGY A First Look
applied force may be varied at will, the momentum and there- e nergy can be transformed from one form to another, its total
fore the kinetic energy of the particle may be increased to any is constant.
value. An alternative way of thinking about the potential en-
The potential energy, Ep or V, of a particle is the energy it ergy arising from the interaction of charges is in terms of the
possesses as a result of its position. For instance, a stationary potential, which is a measure of the ‘potential’ of one charge to
weight high above the surface of the Earth can do a lot of work affect the potential energy of another charge when the second
as it falls to a lower level, so is said to have more energy, in this charge is brought into its vicinity. A charge Q 1 gives rise to a
case potential energy, than when it is resting on the surface of Coulomb potential ϕ1 (phi) such that the potential energy of
the Earth. the interaction with a second charge Q 2 is Q 2ϕ1(r). Comparison
This definition can be turned around. Suppose the weight is of this expression with eqn 11 shows that
returned from the surface of the Earth to its original height. The
work needed to raise it is equal to the potential energy that it Q1 Coulomb potential
1 (r ) [in a vacuum]
(13)
once again possesses. For an infinitesimal change in height, dx, 4 0 r
that work is −Fxdx. Therefore, the infinitesimal change in po-
tential energy is dEp = −Fxdx. This equation can be rearranged The units of potential are joules per coulomb, J C–1, so when the
into a relation between the force and the potential energy: potential is multiplied by a charge in coulombs, the result is the
potential energy in joules. The combination joules per coulomb
dE p dV Relation of force occurs widely and is called a volt (V): 1 V = 1 J C–1.
Fx or Fx to potential energgy
(9) The language developed here inspires an important alterna-
dx dx
tive energy unit, the electronvolt (eV): 1 eV is defined as the
No universal expression for the dependence of the potential potential energy acquired when an electron is moved through
energy on position can be given because it depends on the type a potential difference of 1 V. The relation between electronvolts
of force the particle experiences. However, there are two very and joules is
important specific cases where an expression can be given. For
a particle of mass m at an altitude h close to the surface of the 1 eV 1.602 1019 J
Earth, the gravitational potential energy is
Many processes in chemistry involve energies of a few electron-
Gravitational potential energy volts. For example, to remove an electron from a sodium atom
Ep (h) Ep (0) mgh [close to surrface of the Earth]
(10)
requires about 5 eV.
where g is the acceleration of free fall (g depends on location,

but its ‘standard value’ is close to 9.81 m s–2). The zero of poten-
tial energy is arbitrary. For a particle close to the surface of the 3 Temperature
Earth, it is common to set Ep(0) = 0.
The other very important case (which occurs whenever A key idea of quantum mechanics is that the translational en-
the structures of atoms and molecules are discussed), is the ergy of a molecule, atom, or electron that is confined to a re-
electrostatic potential energy between two electric charges Q 1 gion of space, and any rotational or vibrational energy that a
and Q 2 at a separation r in a vacuum. This Coulomb potential molecule possesses, is quantized, meaning that it is restricted
energy is to certain discrete values. These permitted energies are called
Q1Q 2 energy levels. The values of the permitted energies depend on
Coulomb potential energy
Ep (r ) [in a vacuum]
(11) the characteristics of the particle (for instance, its mass) and for
4 0 r
translation the extent of the region to which it is confined. The
Charge is expressed in coulombs (C). The constant ε0 (epsilon allowed energies are widest apart for particles of small mass
zero) is the electric constant (or vacuum permittivity), a fun- confined to small regions of space. Consequently, quantization
damental constant with the value 8.854 × 10–12 C2 J–1 m–1. It is must be taken into account for electrons bound to nuclei in
conventional (as in eqn 11) to set the potential energy equal to atoms and molecules. It can be ignored for macroscopic bodies,
zero at infinite separation of charges. for which the separation of all kinds of energy levels is so small
The total energy of a particle is the sum of its kinetic and that for all practical purposes their energy can be varied virtu-
potential energies: ally continuously.
Figure 3 depicts the typical energy level separations associ-
E Ek Ep , or E Ek V Total energy (12) ated with rotational, vibrational, and electronic motion. The
separation of rotational energy levels (in small molecules,
A fundamental feature of nature is that energy is conserved; about 10−21 J, corresponding to about 0.6 kJ mol−1) is smaller
that is, energy can neither be created nor destroyed. Although than that of vibrational energy levels (about 10−20–10−19 J, or
ENERGY A First Look xxxvii
Translation Rotation Vibration Electronic T=0 T=
1 aJ (1000 zJ)
10–100 zJ
Continuum
1 zJ
Energy
Figure 4 The Boltzmann distribution of populations (represented
Figure 3 The energy level separations typical of four types of by the horizontal bars) for a system of five states with different
system. (1 zJ = 10–21 J; in molar terms, 1 zJ is equivalent to about energies as the temperature is raised from zero to infinity. Interact
0.6 kJ mol–1.) with the dynamic version of this graph in the e-book.
6–60 kJ mol−1), which itself is smaller than that of electronic en- olecule are zero. As the value of T is increased (the ‘tempera-
m
ergy levels (about 10−18 J, corresponding to about 600 kJ mol −1). ture is raised’), the populations of higher energy states increase,
and the distribution becomes more uniform. This behaviour is
illustrated in Fig. 4 for a system with five states of different en-
(a) The Boltzmann distribution ergy. As predicted by eqn 14a, as the temperature approaches
The continuous thermal agitation that molecules experience in infinity (T → ∞), the states become equally populated.
a sample ensures that they are distributed over the available en- In chemical applications it is common to use molar energies,
ergy levels. This distribution is best expressed in terms of the Em,i, with Em,i = NAεi, where NA is Avogadro’s constant. Then
occupation of states. The distinction between a state and a level eqn 14a becomes
is that a given level may be comprised of several states all of
which have the same energy. For instance, a molecule might be Ni ( E /N E /N ) /kT ( E E ) /N kT ( E E )/RT
e m,i A m,j A e m,i m,j A e m,i m,j (14b)
rotating clockwise with a certain energy, or rotating counter- Nj
clockwise with the same energy. One particular molecule may
be in a state belonging to a low energy level at one instant, and where R = NAk. The constant R is known as the ‘gas constant’;
then be excited into a state belonging to a high energy level a it appears in expressions of this kind when molar, rather than
moment later. Although it is not possible to keep track of which molecular, energies are specified. Moreover, because it is simply
state each molecule is in, it is possible to talk about the average the molar version of the more fundamental Boltzmann con-
number of molecules in each state. A remarkable feature of na- stant, it occurs in contexts other than gases.
ture is that, for a given array of energy levels, how the molecules
are distributed over the states depends on a single parameter,
the ‘temperature’, T.
The population of a state is the average number of mol- Brief illustration 5
ecules that occupy it. The populations, whatever the nature of
Methylcyclohexane molecules may exist in one of two confor-
the states (translational, rotational, and so on), are given by
mations, with the methyl group in either an equatorial or axial
a formula derived by Ludwig Boltzmann and known as the
position. The equatorial form lies 6.0 kJ mol–1 lower in energy
Boltzmann distribution. According to Boltzmann, the ratio of
than the axial form. The relative populations of molecules in
the populations of states with energies εi and εj is the axial and equatorial states at 300 K are therefore
Ni ( ) /kT N axial ( E E ) /RT

e i j Boltzmann distribution(14a) e m,axial m,equatorial
Nj N equatorial
( 6.0103 J mol 1 ) / ( 8.3145 J K 1 mol 1 )( 300 K )
e
where k is Boltzmann’s constant, a fundamental constant with
0.090
the value k = 1.381 × 10–23 J K–1 and T is the temperature, the
parameter that specifies the relative populations of states, re- The number of molecules in an axial conformation is therefore
gardless of their type. Thus, when T = 0, the populations of just 9 per cent of those in the equatorial conformation.
all states other than the lowest state (the ‘ground state’) of the
xxxviii ENERGY A First Look
The important features of the Boltzmann distribution to bear of motion by using the equipartition theorem. This theorem
in mind are: arises from a consideration of how the energy levels associated
with different kinds of motion are populated according to the
• The distribution of populations is an exponential func-
Boltzmann distribution. The theorem states that
tion of energy and the temperature. As the temperature is
increased, states with higher energy become progressively At thermal equilibrium, the average value of each
more populated. quadratic contribution to the energy is 12 kT .
• States closely spaced in energy compared to kT are more A ‘quadratic contribution’ is one that is proportional to the
populated than states that are widely spaced compared square of the momentum or the square of the displacement
to kT. from an equilibrium position. For example, the kinetic energy
The energy spacings of translational and rotational states are of a particle travelling in the x-direction is Ek = px2 /2m. This
typically much less than kT at room temperature. As a result, motion therefore makes a contribution of 12 kT to the energy.
many translational and rotational states are populated. In con- The energy of vibration of atoms in a chemical bond has two
trast, electronic states are typically separated by much more quadratic contributions. One is the kinetic energy arising from
than kT. As a result, only the ground electronic state of a mol- the back and forth motion of the atoms. Another is the poten-
ecule is occupied at normal temperatures. Vibrational states are tial energy which, for the harmonic oscillator, is Ep = 12 kf x 2 and
widely separated in small, stiff molecules and only the ground is a second quadratic contribution. Therefore, the total average
vibrational state is populated. Large and flexible molecules are energy is 12 kT 12 kT kT .
also found principally in their ground vibrational state, but The equipartition theorem applies only if many of the states
might have a few higher energy vibrational states populated at associated with a type of motion are populated. At tempera-
normal temperatures. tures of interest to chemists this condition is always met for
translational motion, and is usually met for rotational motion.
Typically, the separation between vibrational and electronic
(b) The equipartition theorem states is greater than for rotation or translation, and as only a
For gases consisting of non-interacting particles it is often pos- few states are occupied (often only one, the ground state), the
sible to calculate the average energy associated with each type equipartition theorem is unreliable for these types of motion.
☐ 1. Newton’s second law of motion states that the rate of ☐ 6. The Coulomb potential energy between two charges
change of momentum is equal to the force acting on the separated by a distance r varies as 1/r.
particle. ☐ 7. The energy levels of confined particles are quantized, as
☐ 2. Work is done in order to achieve motion against an are those of rotating or vibrating molecules.
opposing force. Energy is the capacity to do work. ☐ 8. The Boltzmann distribution is a formula for calculating
☐ 3. The kinetic energy of a particle is the energy it possesses the relative populations of states of various energies.
as a result of its motion. ☐ 9. The equipartition theorem states that for a sample at
☐ 4. The potential energy of a particle is the energy it pos- thermal equilibrium the average value of each quadratic
sesses as a result of its position. contribution to the energy is 12 kT .
☐ 5. The total energy of a particle is the sum of its kinetic and
potential energies.
ENERGY A First Look xxxix
Checklist of equations
Property Equation Comment Equation number
Component of velocity in x direction vx = dx /dt Definition; likewise for y and z 1a
Component of linear momentum in x direction px = mvx Definition; likewise for y and z 2
2
Moment of inertia I = mr Point particle 3a
I mi ri2 Molecule 3b
i
Angular momentum J = Iω 4
Equation of motion Fx = dpx /dt Motion along x-direction 5a
F = dp/dt Newton’s second law of motion 5b
T = dJ/dt Rotational motion 6
Work opposing a force in the x direction dw = –Fxdx Definition 7a
Kinetic energy Ek = mv 1
2
2
Definition; v is the speed 8a
Potential energy and force Fx dV /dx One dimension 9

Coulomb potential energy Ep (r ) Q 1Q 2 /4 0r In a vacuum 11
Coulomb potential 1 (r ) Q 1 /4 0r In a vacuum 13
Boltzmann distribution N i /N j e
( i j ) /kT
14a
Atkins’
PHYSICAL CHEMISTRY
FOCUS 1
The properties of gases
A gas is a form of matter that fills whatever container it occu- 1C Real gases
pies. This Focus establishes the properties of gases that are used
throughout the text. The perfect gas is a starting point for the discussion of prop-
erties of all gases, and its properties are invoked throughout
thermodynamics. However, actual gases, ‘real gases’, have prop-
erties that differ from those of perfect gases, and it is necessary
1A The perfect gas to be able to interpret these deviations and build the effects of
molecular attractions and repulsions into the model. The dis-
This Topic is an account of an idealized version of a gas, a ‘per- cussion of real gases is another example of how initially primi-
fect gas’, and shows how its equation of state may be assembled tive models in physical chemistry are elaborated to take into
from the experimental observations summarized by Boyle’s account more detailed observations.
law, Charles’s law, and Avogadro’s principle. 1C.1 Deviations from perfect behaviour; 1C.2 The van der Waals
1A.1 Variables of state; 1A.2 Equations of state equation
1B The kinetic model What is an application of this material?

A central feature of physical chemistry is its role in building The perfect gas law and the kinetic theory can be applied to the
models of molecular behaviour that seek to explain observed study of phenomena confined to a reaction vessel or encom-
phenomena. A prime example of this procedure is the develop- passing an entire planet or star. In Impact 1, accessed via the
ment of a molecular model of a perfect gas in terms of a col- e-book, the gas laws are used in the discussion of meteoro-
lection of molecules (or atoms) in ceaseless, essentially random logical phenomena—the weather. Impact 2, accessed via the
motion. As well as accounting for the gas laws, this model can e-book, examines how the kinetic model of gases has a surpris-
be used to predict the average speed at which molecules move in ing application: to the discussion of dense stellar media, such as
a gas, and its dependence on temperature. In combination with the interior of the Sun.
the Boltzmann distribution (see Energy: A first look), the model
can also be used to predict the spread of molecular speeds and
its dependence on molecular mass and temperature.
1B.1 The model; 1B.2 Collisions
➤ Go to the e-book for videos that feature the derivation and interpretation of equations, and applications of this material.
TOPIC 1A The perfect gas
these collisions are so numerous that the force, and hence the
➤ Why do you need to know this material? pressure, is steady.
The SI unit of pressure is the pascal, Pa, defined as 1 Pa =
The relation between the pressure, volume, and tempera-
1 N m−2 = 1 kg m−1 s−2. Several other units are still widely used,
ture of a perfect gas is used extensively in the develop-
and the relations between them are given in Table 1A.1. Because
ment of quantitative theories about the physical and
many physical properties depend on the pressure acting on a
chemical behaviour of real gases. It is also used extensively
sample, it is appropriate to select a certain value of the pressure
throughout thermodynamics.
to report their values. The standard pressure, p , for reporting
⦵
physical quantities is currently defined as p = 1 bar (that is,

⦵
➤ What is the key idea?

105 Pa) exactly. This pressure is close to, but not the same as,
The perfect gas law, which describes the relation between
1 atm, which is typical for everyday conditions.
the pressure, volume, temperature, and amount of sub-
Consider the arrangement shown in Fig. 1A.1 where two
stance, is a limiting law that is obeyed increasingly well as
gases in separate containers share a common movable wall.
the pressure of a gas tends to zero.
In Fig. 1A.1a the gas on the left is at higher pressure than that
➤ What do you need to know already? on the right, and so the force exerted on the wall by the gas on
the left is greater than that exerted by the gas on the right. As
You need to know how to handle quantities and units in
a result, the wall moves to the right, the pressure on the left
calculations, as reviewed in the Resource section.
Table 1A.1 Pressure units*
Name Symbol Value

The properties of gases were among the first to be established pascal Pa 1 Pa = 1 N m−2, 1 kg m−1 s−2
quantitatively (largely during the seventeenth and eighteenth bar bar 1 bar = 105 Pa
centuries) when the technological requirements of travel in atmosphere atm 1 atm = 101.325 kPa
balloons stimulated their investigation. This Topic reviews how torr Torr 1 Torr = (101 325/760) Pa = 133.32 … Pa
the physical state of a gas is described using variables such as
millimetres of mercury mmHg 1 mmHg = 133.322 … Pa
pressure and temperature, and then discusses how these vari-
pounds per square inch psi 1 psi = 6.894 757 … kPa
ables are related.
* Values in bold are exact.
1A.1 Variables of state Movable

wall
The physical state of a sample of a substance, its physical condi-
High Low
tion, is defined by its physical properties. Two samples of the
pressure pressure
same substance that have the same physical properties are said
to be ‘in the same state’. The variables of state, the variables (a)
needed to specify the state of a system, are the amount of sub-
stance it contains, n; the volume it occupies, V; the pressure, p;
Equal Equal
and the temperature, T. pressures pressures
(b)
1A.1(a) Pressure and volume
Figure 1A.1 (a) When a region of high pressure is separated from
The pressure, p, that an object experiences is defined as the a region of low pressure by a movable wall, the wall will be pushed
force, F, applied divided by the area, A, to which that force is into the low pressure region until the pressures are equal. (b) When
applied. A gas exerts a pressure on the walls of its container as the two pressures are identical, the wall will stop moving. At this
a result of the collisions between the molecules and the walls: point there is mechanical equilibrium between the two regions.
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10 For if they fall, the Because if one falls,
one will lift up his fellow: then his fellow sets him
but woe to him that is up again: but sad is it to
alone when he falleth; the solitary when he
for he hath not another slips, for there is no
to help him up. second to set him up.
(10.) For if they fall, the one (singular, following plural, either
one or other fall, that is) then is caused to stand his fellow (Judges
xx. 11, Psalms xlv. 7), and woe to him (in this form at this place
only, perhaps because of the play upon the word ‫‘ ֵא י לֹו‬where is he?’
an equivoke which helps the sense) the single one which falls
(contracted relative, ‘when or as he falls,’) and there is no second
to make him stand.
11 Again, if two lie Moreover, if two lie

together, then they have together they keep each
heat: but how can one other warm: but how can
be warm alone? one be warm alone?
(11.) Moreover (an additional instance of the advantage of

companionship, taken from the passive side, as the other was from
the active side of this matter), if they lie down, two of them, and
heat to them (i.e. ‘there is certainly warmth for them’), but to the
single one, how can he be warmed?
12 And if one prevail And where one would
against him, two shall fail, two will prevail; and
withstand him; and a a threefold cord is not
threefold cord is not quickly broken.
quickly broken.
(12.) And if they prevail over (impersonal, any prevail over)

the single, the doubles will stand before him (plural, the idea is
that there are two to one), and the cord which is the triplex is not
in haste broken (Jeremiah viii. 16; Judges xvi. 9).
13 Better is a poor A Poor and Prudent

and a wise child than an young man is better than
old and foolish king, a Perverse old king, who
¹who will no more be cannot be prevailed on
admonished. to listen to a warning.
¹ Hebrew who
knoweth not
to be
admonished.
(13.) Good is a child, poor (‫מסכן‬, occurs chapter iv. 13, ix.
15, 16 only; the root occurs in the sense ‘profitable,’ see Job xxii. 2;
the idea seems to be, that kind of poverty which is economical and
sparing) and wise from (‘above,’ that is; the ordinary ‫ מ־‬of
comparison;) a king old (‫זקן‬, the alliteration between miscan and
zakan gives pungency. We have rendered this in the paraphrase by
a corresponding alliteration) and befooled, who does not know
how to be warned as yet. (The allusion here to Solomon is
palpable, and this may account for the apparently redundant ‫עוד‬, ‘as
yet,’ at the end of the sentence.)
14 For out of prison For from a prison-house

he cometh to reign; of plotters he comes
whereas also he that is forth to reign, and beside
born in his kingdom in his kingdom is the
becometh poor. birthplace of meanness.
(14.) For from the house of rebels (‫ הסורים‬is considered a

contraction for ‫האסורים‬, ‘prison;’ but, to say the least, this is a violation
of the critical canon, which bids us prefer the harder reading. That
the derivation from ‫סור‬, ‘to turn aside,’ hence ‘revolters,’ is contrary to
the pointing, is not a valid objection, because the Masorets pointed
as they did to explain a difficulty; so also the LXX., who read δεσμῶν
and δεσμίων. If possible, we ought to preserve intact the unpointed
text. The exact and literal meaning is, as the text stands, ‘from the
house of the turners-aside,’ i.e. those conspirators and wicked men,
sycophants, who will be flattering him to promote their own interest,
and this was exactly Solomon’s case) he goes out to reign, for (the
second ‫כי‬, with the meaning ‘so’) also (‘moreover;’ this particle, as
we have seen, usually introduces an additional reason, confirming
the one which went before) in his kingdom is begotten ( ♦ ‫נולד‬,
noled, a play upon ‫ילד‬, jeled, above) want (‫ רש‬is poverty in the sense
of indigence and meanness; compare 1 Samuel xviii. 23, and
Proverbs xix. 1, 7, 22; as, however, ‫ רש‬has the form of a concrete,
we must remember that it involves the idea of a poor man, hence
there is a sarcastic ambiguity, heightened by alliteration. Take the
LXX. and Masorets’ sense, which is merely to allow the obvious play
between ‫ הסורים‬and ‫האסורים‬, and the sentiment is true, and, curiously
enough, equally corresponds with the history).
♦ “‫ ”נלד‬replaced with “‫”נולד‬
The following passage is one of great difficulty, but a very careful

attention to its precise wording and the equivoke it contains, may
perhaps afford a solution.
15 I considered all I have observed of all

the living which walk lives whatsoever, as
under the sun, with the they are progressing in
second child that shall this work-day world, in
stand up in his stead. regard to any
successors which may
arise in their places,
(15.) I have seen (‘observed as matter of fact’), with respect to

all the lives (which the LXX. render σύμπαντας τοὺς ζῶντας), the
proceeding ones (participle, piel plural with the article――LXX. τοὺς
περιπατοῦντας――occurs here and Psalms civ. 3, Proverbs vi. 11; ‘as
they are advancing’ must be the meaning, and hence the
observation was made with regard to the progress of these lives),
under the sun (that is, in this stage of their existence; the limitation
here is excessively important,) together with the child (with the
article, generic, and giving the meaning of that which is ‘begotten of
them,’ of course children primarily, but not exclusively; the ‘heir’ or
‘successors’ would represent the idea), the second (i.e. the
immediate successor) who stands in their stead (plural, which
nevertheless the LXX. render ἀντ’ αὐτοῦ, and rightly, because it is an
instance of a distributive plural, with regard to ‫)הילד‬.
16 There is no end of that no result was ever

all the people, even of all reached by the [moiling]
that have been before multitude in the past:
them: they also that and as to what succeeds
come after shall not them, they will have no
rejoice in him. Surely [earthly] pleasure in that.
this also is vanity and Another instance of
vexation of spirit. evanescence and vexing
of spirit.
(16.) There is nothing of an end (i.e. ‘result,’ occurs chapter iv.

8, 16) to all the people (with the article, τῷ παντὶ λαῷ, LXX.――and
in this book it appears as a collective for the human race――see
chapter xii. 9), to all (repeated, hence with the meaning, ‘that is to all
those’) that (full relative) were before them (but ‘before’ in the
sense of in their ‘presence,’ not in the sense of ‘before their time’),
moreover (introducing an additional reason), the succeeding
ones――(see 2 Chronicles ix. 29, xii. 15, which will give the exact
meaning) not (rather emphatic from its position, ‘not at all’) will they
(i.e. the people before them) rejoice in it (‘it’ is a singular following a
plural, and hence a distributive, ‘any successor’) for also this is a
vanity (an instance of evanescence) and vexing (not ‘vexation,’
because this comes from within) of spirit. Thus the sense is clear; it
is the conclusion of the argument. Koheleth’s observation has regard
to the progress of lives in relation to anything that may or is to be
produced by them in the way of heritage――or, in other words, he
asks how far the present state of things can be explained on the
theory that it is a working for posterity, and he shows that this is not
an explanation, for there is no result obtained by the collective
people in the present, because each age is the same morally as that
which went before it; while, of course, with regard to what is to
succeed, the present generation cannot rejoice in that, because they
will be all dead, and as the argument is limited to what takes place
under the sun, so all so-called progress is but an instance of
evanescence. The idea, if not that contained in the observation of
one who selfishly observed, when requested to care for posterity,
‘that as posterity had done nothing for him, he did not see why he
should do anything for posterity,’ rests on the same facts.
The sentence also, it appears, contains a remarkable equivoke.

‫ לכל העם לכל‬sounds very like ‫לכל העמל כל‬, and this division of the words
will make such good and pungent sense that we can hardly imagine
that the equivoke was unintentional. The equivoke is sought to be
rendered in the paraphrase by the addition of the words enclosed in
the brackets.
At this point we come to another division in the book. Certain

practical exhortations follow, deduced from the previous arguments,
concerning human conduct, under the circumstances above set
forth.
CHAPTER V.
K EEP thy foot when

thou goest to the
Section
IV.――Practical
house of God, and be aphorisms grounded on
more ready to hear, than the foregoing.
to give the sacrifice of
fools: for they consider
not that they do evil. G UARD thou thy
steps as one who
art walking to the House
of the Divinity, and
approach rather to
hearken than to give, as
the fools do, a sacrifice;
who do not know when
evil is being done.
V. (1.) Keep thy feet (the Masorets have altered this to the
singular, but without sufficient reason; yet the LXX. support the Kri)
as when (occurs chapter v. 3 (4), viii. 7; ‘as though’ is the meaning
here) thou walkest (taking up the word from the last clause above)
towards the house of the Deity (the LXX. render, of course
correctly as to sense by the double article, τὸν οἶκον τοῦ
Θεοῦ――‘Thou art walking to the temple of a Divine Providence’ is
the idea), and drawing near to hear (evidently ‘in order to hear’;
hence the LXX. render ἐγγὺς τοῦ ἀκούειν; some, however, with the
Authorized Version, take this as an imperative, but the sense is
better preserved by rendering as the LXX. do), more than giving of
the befooled ones (for we must not lose sight of the hiphil form:
they are deceived either by themselves or others) a sacrifice. (The
curious rendering of the LXX. by no means shows that they did not
understand the meaning, or even would have altered the present
pointing; ὑπὲρ δόμα τῶν ἀφρόνων θυσία σου fulfils their conditions of
rendering, which is, if possible, to preserve both the sense and the
order, ‘above the gift of fools is thy sacrifice’). For they are not those
instructed to the doing of (so the LXX., τοῦ ποιῆσαι) evil. The
sentence is purposely ambiguous and equivocal; it is not clear at first
sight whether the fools are those who do evil, or whether it be the
doing of evil generally which is the point, but the following will seem
to give a fair explanation of this ♦difficult passage. The advice given
after the considerations above, is to walk reverently, and to listen to
what God’s oracle will say, rather than do as fools do,――offer a
sacrifice to avert evil, which they do not after all know to be such,
and which, if it implies dissatisfaction with these divine providential
arrangements, is a foolish, if not sinful, sacrifice. This is further set
forth in the following verses.
♦ “difcult” replaced with “difficult”
2 Be not rash with Do not be hasty with thy

thy mouth, and let not lips, nor in thought hurry
thine heart be hasty to forth a word against the
utter any ¹thing before Almighty, for that
God: for God is in Almighty is in the
heaven, and thou upon heavens, and thou art
earth: therefore let thy upon the earth: on this
words be few. account let thy words be
sparing. Because just as
3 For a dream there comes dreaming
cometh through the through a multitude of
multitude of business; anxieties, so there
and a fool’s voice is comes the voice of a
known by multitude of befooled through a
words. multitude of reasonings.
¹ Or, word.
(2, 3.) Do not hasten (the hastiness of vexation, see Job iv. 5,
xxiii. 15, Psalms vi. 10) upon thy mouth (the preposition is by no
means redundant), and thy heart do not hurry (the usual word
denoting the hurry of want of time. The meaning then is, do not
speak, no, do not even think, hastily) to cause to send out a word
(with the usual meaning of ‘a reason to be acted on’) before the
Deity, because the Deity (as this is a repetition, the word becomes
emphatic, ‘that Deity’) in the heavens and thou (emphatic) on the
earth, therefore be thy words a few (i.e. diminished rather than
increased, hence the following). For comes the dream in the
multitude of anxiety, and a voice of a befooled one (for it is
without the article) in the multitude of words (or ‘reasons,’ as
above). The argument now passes over from rash speeches to rash
vows. A vow is a favourite resource with the foolish for obtaining the
accomplishment of their wishes: they think to bribe Providence with
gifts and offerings.
4 When thou vowest (2.) Shouldst thou
a vow unto God, defer have vowed specially to
not to pay it; for he hath God, do not be slow to
no pleasure in fools: pay pay it; because there is
that which thou hast no providence with the
vowed. befooled ones: just what
thou hast vowed pay.
(4.) When thou hast vowed a vow (‘If by any means thou hast
done this,’ for considerable emphasis is given by the repetition of
‘vow,’ according to the well-known Hebrew idiom) to God, do not
defer to pay it (the alacrity with which men vow is commonly in
strong contrast with the tardiness with which they pay), because
there is nothing of providence (‫חפץ‬, with its usual technical
meaning, and also equivocal, in the sense of ‘pleasure’) in befooled
ones: with respect to what thou hast vowed, pay (the LXX. render
σὺ οὖν ‘thou then,’ but the emphasis given by ‫ את אשר‬may easily
account for this).
5 Better is it that thou For it is better that thou

shouldest not vow, than shouldst not vow, than
that thou shouldest vow that thou shouldst be
and not pay. vowing and not pay.
(5.) A good is it that thou shouldst not vow (the sentence is

ambiguous, but the equivoke is ‘thou hadst better not vow’), than
that thou shouldst vow and not pay.
6 Suffer not thy Do not allow thy mouth

mouth to cause thy flesh to cause thy body to sin;
to sin; neither say thou and say not in the
before the angel, that it presence of God’s
was an error: wherefore messenger, ‘It was but
should God be angry at an inadvertence:’ why
thy voice, and destroy should the Almighty be
the work of thine hands? angry with your prattle,
and put an arrest on the
work of your hands?
(6.) Do not give with respect to thy mouth (the ‫ את‬is not
redundant, ‘do not appoint,’ which is the meaning of ‫)תתן‬, to cause
to make to sin with respect to thy flesh (the meaning then must
be, ‘do not so arrange matters as to cause thy mouth to make thy
flesh sin,’ by, that is, preferring the ease, pleasure, of the flesh or the
like, to the sacrifice caused by a redemption of the vow), and do not
say in the presence of the angel (with the article; had this been
noticed as it ought, less difficulty would have been felt in the
interpretation of this passage; the angel is the messenger of
Providence who comes to require the vow, and whom, of course,
with or without sufficient reason, the person bound by the vow
expects) that (‫ )כי‬an error it is: (see Leviticus iv. 2, 22, 27, and
Numbers xv. 24, 25, 29; when too this passage is compared with
Leviticus iv. 2, we can have no doubt that ‫ לפ׳ מא׳‬here is the
equivalent of ‫ לפ׳ יי׳‬there) why (LXX. ἵνα μὴ, ‘so that not’), should be
angry (Genesis xl. 2, Deuteronomy i. 24) the Deity over thy voice
(Ginsburg, excellently, ‘with thy prattle’), and destroy (as this word is
used to signify the ‘giving a pledge,’ this peculiar signification
conveys the idea, ‘destroy by exacting a pledge,’ ‘make thee
bankrupt by insisting upon payment’) with respect to the work of
your hands?
7 For in the multitude For in the multitude of

of dreams and many dreams and vanities
words there are also even so reasons are
divers vanities: but fear multiplied that God is to
thou God. be feared.
(7.) For in a multitude of dreams (‘conjectures’ probably) and

vanities and reasonings, the much (i.e. these reasonings are
increased); for (‫ כי‬is repeated, and this repetition makes it
emphatic――‘so indeed’) with respect to the Deity fear. The
probable meaning is, ‘fear God under all circumstances: vanity and
conjectures only increase the reasons for so doing,’――thus is
revealed the real conclusion of the whole treatise.
Koheleth now takes up a subject ineffectually discussed before,

and solves it with this principle just enunciated: Fear God.
8 ¶ If thou seest the (3.) If violent

oppression of the poor, oppression of the poor,
and violent perverting of and wresting of justice
judgment and justice in a and right, should be
province, marvel not ¹at observed by you in a
the matter: for he that is jurisdiction, do not be
higher than the highest surprised at the
regardeth; and there be providence; for the lofty
higher than they. are watched by one
loftier still, and these
¹ Hebrew at
lofty ones
the will, or are――subjects.
purpose.
(8.) If oppression of the poor (see chapter iv. 1, 3, etc.), and

wresting of judgment and right, thou seest in a province (‫במדינה‬,
this has been considered a late word, and a sign, moreover, that the
writer lived in the country and not in the city, as he says, chapter
i. 12; but though it occurs in the later Hebrew [1 Kings xx. 14 is the
first instance] it is quite regularly formed, and is clearly in place
here), do not marvel (Psalms xlviii. 6, Jeremiah iv. 9, to ‘be
astonished,’ ‘struck with astonishment’) over the providence (‫החפץ‬
with the article; the LXX. render τῷ πράγματι in this instance, the
word, however, occurs in the technical meaning it has all through the
book, see chapter iii. 1, v. 4 (3)); for high from above the high
(which the LXX. render word for word, ὑψηλὸς ἐπάνω ὑψηλοῦ) keeps
and high ones above them (the sentence is enigmatic, perhaps
proverbial, though the meaning is clear. Is it possible that a play was
intended between ‫ ֵמ ַעל‬and ‫ַמ ַעל‬, Leviticus v. 15, a ‘transgression,’ ‫גבה‬
being taken in the meaning of swelling up, thus――‘Increasing
transgression is increasing regard?’ In the same way the ‫ מ‬at the end
of ‫ גבהים‬would unite with the word following in utterance, and so help
the equivoke).
9 ¶ Moreover the And besides,
profit of the earth is for
all: the king himself is (i.) The produce of the
served by the field. earth is all in all: a
king is a subject to
the field.
(9.) And the profit (as this is joined by a conjunction with the
former, we must look upon it as a further argument in the same chain
of reasoning; the meaning will then be ‘and besides the produce’) of
earth (not the earth, the article is wanting) in all (the LXX. render
this by ἐπὶ with a dative, hence they understood the preposition here
to mean ‘for all,’ which our version follows) it is (feminine, in close
apposition therefore with the noun, but this noun must be ‫יתרון‬, which
is feminine, and the meaning is that it exists subjectively, or is always
there playing its part) a king (again, not the king: any king, therefore,
however great,――Solomon himself, or any other) to a field (again,
not the field, equivalent to some field; the LXX. render by the simple
genitive) is served (niphal; this occurs only twice in the past tense,
here and at Ezekiel xxxvi. 9, both in the sense of tilling; and the
niphal future twice, at Deuteronomy xxi. 4 and Ezekiel xxxvi. 34,
again with the same meaning――no doubt ‫ עבד‬is used with the
signification ‘to serve generally’ in a vast number of places. It must
be observed, however, that a niphal is not exactly the same as a
passive, but has an objective signification, so that it is often nearer in
meaning to the Greek middle voice than our passive. Bearing this in
mind, we can have no further doubt over this passage as to its
principal scope,――‘the king is served of,’ or ‘a subject to the field.’
The idea is that the very highest are really in a state of abject
dependence――a single day’s starvation would have been sufficient
to have brought to the dust Solomon or Nebuchadnezzar. The other
possible rendering, that ‘the king is served by the field,’ is only the
other side of the same truth, and the sentence is equivocal, being
ingeniously constructed so as to read either way).
10 He that loveth (ii.) A lover of money no

silver shall not be money ever
satisfied with silver; nor satisfied; and who
he that loveth that loved profusion
abundance with ever had sufficient
increase: this is also income? Another
vanity. instance of the
evanescent.
(10.) Loving silver (the Masorets point as a participle, but

however correct this may be, the participial notion is in Ecclesiastes
apparently not so prominent, as it is when the poel is used written
full) not satisfies (i.e. as the nominative follows, ‘shall not be
satisfied with’) silver (silver is doubled here, and used of course in
the sense of money――the meaning being that ‘a lover of money no
money ever satisfies’), and who loving in a multitude (i.e. setting
his desires in a multitude of goods, or anything else) not (but the
LXX. in place of ‫ לא‬possibly read ‫לֹו‬, ‘to him,’ and this makes far
better and more pungent sense――‘to him’ emphatic will then be the
meaning) a revenue (Numbers xviii. 30, Deuteronomy xxxiii. 14,
Proverbs iii. 14, xviii. 20; or, still better, for the word is derived from
the root ‫בוא‬, ‘to come,’ ‘an income.’ Thus it is seen that the two
clauses are aimed respectively against niggardliness and
extravagance. The miser and the spendthrift both never have
enough); also this is vanity (another instance of the transitory and
evanescent, as indeed it is, because these riches look satisfactory
and are not).
11 When goods (iii.) As property
increase, they are increases, so
increased that eat them: increases
and what good is there consumption too;
to the owners thereof, and what success
saving the beholding of then has ownership,
them with their eyes? but just the right of
beholding it?
(11.) In the multitudes of the good (an abstract, with the

article, and hence the meaning is ‘In the very increase of the
property itself, and as it increases,’ this being the meaning of the
plural, which is distributive) multiply the eatings of it (or, for the ‫ה‬
may be considered paragogic, and so making, as it were, an abstract
of the poel participle, ‘consumers’), and what is the success (‫כשרון‬,
see ii. 21, references) to the owners of it (i.e. to ownership), except
seeing (‫ראית‬, this the Masorets alter to ‫ראות‬, but unnecessarily, for
there is a slight difference in the sense here, which will account for
the unusual grammatical form; a causative or hiphil notion is implied
by it; hence the LXX. ἀρχὴ τοῦ ὁρᾶν, ‘the priority to see,’) his eyes?
(i.e. each one with his eyes, singular following plural).
12 The sleep of a (iv.) How sweet is the

labouring man is sweet, sleep of the slave, if
whether he eat little or a little, or if much he
much: but the eats: but a
abundance of the rich sufficiency to one
will not suffer him to who is
sleep. enriched――does
not cause rest to
him so that he
sleeps.
(12.) Sweet (but the participial form of the noun must not be
overlooked, nor the feminine termination, equivalent to a
‘sweetness,’) is the sleep of the slave (‘of the toiler,’ with the
article), if a little, or if the much he eats (there is a peculiar force in
contrasting ‘the much,’ ‫הרבה‬, with the article, with ‫ מעט‬without it; even
if he should eat to the much [i.e. as large a quantity as he can] it will
do him no harm: no nightmare will trouble him who has earned his
hearty meal by his hard work), but the satisfaction (as contrasted
with ‫ )הרבה‬to the enriched it is not that which is causing rest
(hiphil participle) to him (emphatic) to sleep (an equivoke here is to
be found in ‫ השבע‬and ‫לעשיר‬, remembering that ‫שבע‬, ‘seven,’ is used
so commonly for ‘completeness,’ and ‫עשר‬, ‘ten,’ as ‘rich’ and
‘overflowing;’ seven with ten has a peculiar meaning in the
symbolism of numbers).
13 There is a sore (v.) There is this evil

evil which I have seen infirmity which I
under the sun, namely, have observed in
riches kept for the this work-day world:
owners thereof to their Riches kept by an
hurt.
owner to his own
injury;
(13.) There is an evil (abstract, a particular kind of evil), a

sickness (another abstract) I have seen under the sun――wealth
keeping to (i.e. being kept by) its possessors to their hurt.
14 But those riches for the wealth itself

perish by evil travail: and perishes in an
he begetteth a son, and uncertainty which is
there is nothing in his distressing: so that
hand. when he begets an
heir, he has in his
hand just nothing at
all.
(14.) And perishes, that riches, that same (as we should say,
‘those very same riches’) in an uncertainty (‫בענין‬, another instance
of this word; we see that in this case also [see chapter i. 13,
references], the meaning ‘anxious uncertainty’ exactly suits the
context), which is an evil (this anxious care, instead of doing any
good, is but a simple mischief), and he is caused to beget a son
(to whom, of course, he would have wished to bequeath his wealth),
and there is nothing in his hand at all (which the LXX. render by a
double negative, and hence we must render ‘and has in his hand
even nothing at all’).
15 As he came forth For naked as when
of his mother’s womb, he came forth from
naked shall he return to the womb of his
go as he came, and mother does he go
shall take nothing of his out of the world
labour, which he may again; and nothing
carry away in his hand. whatever does he
take from his care,
which he can hold in
his hand.
(15.) And as he came out from the womb of his mother

naked (which is reserved to the end of the clause, making it
emphatic; it is moreover written full, so that a slight additional
emphasis is given by this to the ‘state of nakedness’ existing), he
returns to go back (somewhat stronger than goes back――he
comes to this state through intermediate stages) just as he came
(‘as he was at the first, so now is he at the last’), and nothing at all
does he not lift up (Genesis vii. 17, ‘bear’ as a burden) in his toil
(as we say, ‘have for his pains,’ observing the meaning of ‫עמל‬, not
the labour but the anxiety which causes, or results from, the labour)
which he takes in his hand.
16 And this also is a Moreover, in this is

sore evil, that in all discovered that evil
points as he came, so infirmity, that

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Constant Symbol Value

USING THE BOOK

The twelfth edition of Atkins’ Physical Chemistry has been

Innovative structure TOPIC 2A Internal energy

➤ What is the key idea?

text: differentiation and integration, including a table of the Contents

PART 4 Character tables 910

ready reference. Heat transfer at constant pressure dH = dqp, ΔH = qp No additional work

C(s, graphite) 16.86 4.77 −8.54

questions you will be well prepared to tackle more challenging

Exercises and problems

Exercises are provided throughout the main text and, along −2

volume is constant, on a subsequent summer’s day when the temperature is

typically involve constructing a more detailed answer. For this

the same location as they appear in print.

tailed solutions to the odd-numbered questions are provided Problems

available only to lecturers. Exercises and problems 139

knowledge creatively in a variety of ways.

TAKING YOUR LEARNING FURTHER

TO THE INSTRUC TOR

Photograph by Nathan Pitt,

ENERGY A First Look xxxiii FOCUS 12 Magnetic resonance 499

FOCUS 1 The properties of gases 3 FOCUS 13 Statistical thermodynamics 543

FOCUS 2 The First Law 33 FOCUS 14 Molecular interactions 597

FOCUS 3 The Second and Third Laws 75 FOCUS 15 Solids 655

FOCUS 4  hysical transformations of pure

Conventions xxvii FOCUS 2 The First Law 33

List of Impacts xxxii 2A.2 The definition of internal energy 36

FOCUS 1 The properties of gases 3 (b) Expansion against constant pressure 38

(e) Osmosis 162 (b) Exergonic and endergonic reactions 207

Checklist of concepts 165 6A.2 The description of equilibrium 207

5C.2 Temperature–composition diagrams 168 Checklist of equations 213

Checklist of concepts 181 6C.4 The determination of thermodynamic functions 224

(a) Ideal–dilute solutions 187 Checklist of concepts 230

FOCUS 7 Quantum theory 237 Checklist of concepts 282

Checklist of concepts 338 (c) Density functional theory 381

diatomic molecules 359 10A.3 Some immediate consequences of symmetry 404

9D.1 Polar bonds and electronegativity 366 (d) Characters 411

9D.2 The variation principle 367 10B.3 Character tables 412

9E.2 Applications 376 10C.2 Applications to molecular orbital theory 420

(b) Aromatic stability 378 (b) Symmetry-adapted linear combinations 421

9E.3 Computational chemistry 379 10C.3 Selection rules 422

FOCUS 13 Statistical thermodynamics 543 TOPIC 13F Derived functions 582

FOCUS 14 Molecular interactions 597

(a) The translational contribution 562 (b) Dipole–dipole interactions 609

(d) The electronic contribution 564 14B.2 Hydrogen bonding 613

TOPIC 14C Liquids 618 (c) Scattering factors 666

(a) Surface pressure 624 (b) Electronic structure of metals 675

Checklist of concepts 661 16A.2 The transport parameters 710

FOCUS 4 hysical transformations of pure

Translation Rotation Vibration Electronic T=0 T=

Ni ( ) /kT N axial ( E E ) /RT

Potential energy and force Fx dV /dx One dimension 9