300 Chapter Seven Atomic Structure and Periodicity

EXAMPLE 7.5 Electron Energies

Calculate the energy required to remove the electron from a hydrogen atom in its ground
state.
Solution
Removing the electron from a hydrogen atom in its ground state corresponds to taking
the electron from ninitial ⫽ 1 to nfinal ⫽ ⬁. Thus
1 1
¢E ⫽ ⫺2.178 ⫻ 10⫺18 J a ⫺ b
nfinal2 ninitial2
1 1
⫽ ⫺2.178 ⫻ 10⫺18 J a ⫺ 2b
q 1
⫽ ⫺2.178 ⫻ 10⫺18 J10 ⫺ 12 ⫽ 2.178 ⫻ 10⫺18 J
The energy required to remove the electron from a hydrogen atom in its ground state is
2.178 ⫻ 10⫺18 J.

SEE EXERCISES 7.61 AND 7.62

Although Bohr’s model fits the energy At first Bohr’s model appeared to be very promising. The energy levels calculated by
levels for hydrogen, it is a fundamentally Bohr closely agreed with the values obtained from the hydrogen emission spectrum. How-
incorrect model for the hydrogen atom. ever, when Bohr’s model was applied to atoms other than hydrogen, it did not work at all.
Although some attempts were made to adapt the model using elliptical orbits, it was con-
cluded that Bohr’s model is fundamentally incorrect. The model is, however, very impor-
Unplucked string
tant historically, because it showed that the observed quantization of energy in atoms could
be explained by making rather simple assumptions. Bohr’s model paved the way for later
theories. It is important to realize, however, that the current theory of atomic structure is
in no way derived from the Bohr model. Electrons do not move around the nucleus in cir-
cular orbits, as we shall see later in this chapter.
1 half-wavelength

7.5 䉴 The Quantum Mechanical Model of the Atom

By the mid-1920s it had become apparent that the Bohr model could not be made to work.
A totally new approach was needed. Three physicists were at the forefront of this effort:
Werner Heisenberg (1901–1976), Louis de Broglie (1892–1987), and Erwin Schrödinger
2 half-wavelengths (1887–1961). The approach they developed became known as wave mechanics or, more
commonly, quantum mechanics. As we have already seen, de Broglie originated the idea
that the electron, previously considered to be a particle, also shows wave properties. Pur-
suing this line of reasoning, Schrödinger, an Austrian physicist, decided to attack the prob-
lem of atomic structure by giving emphasis to the wave properties of the electron. To
Schrödinger and de Broglie, the electron bound to the nucleus seemed similar to a stand-
ing wave, and they began research on a wave mechanical description of the atom.
3 half-wavelengths

Figure 7.10
The standing waves caused by the
vibration of a guitar string fastened
at both ends. Each dot represents a
node (a point of zero displacement).

Wave-generating apparatus.

Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
 7.5 The Quantum Mechanical Model of the Atom 301

The most familiar example of standing waves occurs in association with musical in-
struments such as guitars or violins, where a string attached at both ends vibrates to pro-
n=4 duce a musical tone. The waves are described as “standing” because they are stationary; the
waves do not travel along the length of the string. The motions of the string can be explained
as a combination of simple waves of the type shown in Fig. 7.10. The dots in this figure in-
(a)
dicate the nodes, or points of zero lateral (sideways) displacement, for a given wave. Note
that there are limitations on the allowed wavelengths of the standing wave. Each end of the
string is fixed, so there is always a node at each end. This means that there must be a whole
number of half wavelengths in any of the allowed motions of the string (see Fig. 7.10).
Standing waves can be illustrated using the wave generator shown in the photo.
n=5 A similar situation results when the electron in the hydrogen atom is imagined to be
a standing wave. As shown in Fig. 7.11, only certain circular orbits have a circumference
into which a whole number of wavelengths of the standing electron wave will “fit.” All
(b)
other orbits would produce destructive interference of the standing electron wave and are
not allowed. This seemed like a possible explanation for the observed quantization of the
hydrogen atom, so Schrödinger worked out a model for the hydrogen atom in which the
electron was assumed to behave as a standing wave.
Mismatch It is important to recognize that Schrödinger could not be sure that this idea would
n = 4 13 work. The test had to be whether or not the model would correctly fit the experimental
data on hydrogen and other atoms. The physical principles for describing standing waves
(c) were well known in 1925 when Schrödinger decided to treat the electron in this way. His
mathematical treatment is too complicated to be detailed here. However, the form of
Figure 7.11 Schrödinger’s equation is
The hydrogen electron visualized as a
standing wave around the nucleus. The
Ĥc ⫽ Ec
circumference of a particular circular or- where ␺, called the wave function, is a function of the coordinates (x, y, and z) of the elec-
bit would have to correspond to a whole
number of wavelengths, as shown in
tron’s position in three-dimensional space and Ĥ represents a set of mathematical instruc-
(a) and (b), or else destructive interference tions called an operator. In this case, the operator contains mathematical terms that produce
occurs, as shown in (c). This is consis- the total energy of the atom when they are applied to the wave function. E represents the
tent with the fact that only certain elec- total energy of the atom (the sum of the potential energy due to the attraction between the
tron energies are allowed; the atom is proton and electron and the kinetic energy of the moving electron). When this equation is
quantized. (Although this idea encour-
aged scientists to use a wave theory, it
analyzed, many solutions are found. Each solution consists of a wave function ␺ that is char-
does not mean that the electron really acterized by a particular value of E. A specific wave function is often called an orbital.
travels in circular orbits.) To illustrate the most important ideas of the quantum (wave) mechanical model of
the atom, we will first concentrate on the wave function corresponding to the lowest en-
ergy for the hydrogen atom. This wave function is called the 1s orbital. The first point of
interest is to explore the meaning of the word orbital. As we will see, this is not a trivial
matter. One thing is clear: An orbital is not a Bohr orbit. The electron in the hydrogen 1s
orbital is not moving around the nucleus in a circular orbit. How, then, is the electron
moving? The answer is quite surprising: We do not know. The wave function gives us no
information about the detailed pathway of the electron. This is somewhat disturbing. When
we solve problems involving the motions of particles in the macroscopic world, we are
able to predict their pathways. For example, when two billiard balls with known veloci-
ties collide, we can predict their motions after the collision. However, we cannot predict
the electron’s motion from the 1s orbital function. Does this mean that the theory is wrong?
Not necessarily: We have already learned that an electron does not behave much like a
billiard ball, so we must examine the situation closely before we discard the theory.
To help us understand the nature of an orbital, we need to consider a principle dis-
covered by Werner Heisenberg, one of the primary developers of quantum mechanics.
Heisenberg’s mathematical analysis led him to a surprising conclusion: There is a funda-
mental limitation to just how precisely we can know both the position and momentum of
a particle at a given time. This is a statement of the Heisenberg uncertainty principle.
Stated mathematically, the uncertainty principle is
h
¢x ⴢ ¢1my2 ⱖ
4p

Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
302 Chapter Seven Atomic Structure and Periodicity

where ⌬x is the uncertainty in a particle’s position, ⌬(m␷) is the uncertainty in a particle’s

momentum, and h is Planck’s constant. Thus the minimum uncertainty in the product
¢x ⴢ ¢1my2 is h兾4␲. What this equation really says is that the more accurately we know a
particle’s position, the less accurately we can know its momentum, and vice versa. This
limitation is so small for large particles such as baseballs or billiard balls that it is unno-
ticed. However, for a small particle such as the electron, the limitation becomes quite im-
portant. Applied to the electron, the uncertainty principle implies that we cannot know the
exact motion of the electron as it moves around the nucleus. It is therefore not appropri-
ate to assume that the electron is moving around the nucleus in a well-defined orbit, as in
the Bohr model.

The Physical Meaning of a Wave Function

Given the limitations indicated by the uncertainty principle, what then is the physical
meaning of a wave function for an electron? That is, what is an atomic orbital? Although
the wave function itself has no easily visualized meaning, the square of the function does
Probability is the likelihood, or odds, that have a definite physical significance. The square of the function indicates the probability
something will occur. of finding an electron near a particular point in space. For example, suppose we have
two positions in space, one defined by the coordinates x1, y1, and z1 and the other by the
coordinates x2, y2, and z2. The relative probability of finding the electron at positions 1
and 2 is given by substituting the values of x, y, and z for the two positions into the wave
function, squaring the function value, and computing the following ratio:
3c1x1, y1, z1 2 4 2 N1
⫽
3c1x2, y2, z2 2 4 2 N2
The quotient N1兾N2 is the ratio of the probabilities of finding the electron at positions
1 and 2. For example, if the value of the ratio N1兾N2 is 100, the electron is 100 times
more likely to be found at position 1 than at position 2. The model gives no informa-
tion concerning when the electron will be at either position or how it moves between
the positions. This vagueness is consistent with the concept of the Heisenberg uncer-
tainty principle.
The square of the wave function is most conveniently represented as a probability
distribution, in which the intensity of color is used to indicate the probability value near
a given point in space. The probability distribution for the hydrogen 1s wave function (or-
bital) is shown in Fig. 7.12(a). The best way to think about this diagram is as a three-
(a) dimensional time exposure with the electron as a tiny moving light. The more times the
electron visits a particular point, the darker the negative becomes. Thus the darkness of a
point indicates the probability of finding an electron at that position. This diagram is also
known as an electron density map; electron density and electron probability mean the
same thing. When a chemist uses the term atomic orbital, he or she is probably picturing
Probability (R2 )

an electron density map of this type.

Another way of representing the electron probability distribution for the 1s wave func-
tion is to calculate the probability at points along a line drawn outward in any direction
from the nucleus. The result is shown in Fig. 7.12(b). Note that the probability of finding
the electron at a particular position is greatest close to the nucleus and drops off rapidly
as the distance from the nucleus increases. We are also interested in knowing the total
Distance from nucleus (r)
probability of finding the electron in the hydrogen atom at a particular distance from the
(b)
nucleus. Imagine that the space around the hydrogen nucleus is made up of a series of
Figure 7.12 thin spherical shells (rather like layers in an onion), as shown in Fig. 7.13(a). When the
(a) The probability distribution for the total probability of finding the electron in each spherical shell is plotted versus the dis-
hydrogen 1s orbital in three-dimensional
tance from the nucleus, the plot in Fig. 7.13(b) is obtained. This graph is called the radial
space. (b) The probability of finding the
electron at points along a line drawn probability distribution.
from the nucleus outward in any direction The maximum in the curve occurs because of two opposing effects. The probability
for the hydrogen 1s orbital. of finding an electron at a particular position is greatest near the nucleus, but the volume

Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
 7.6 Quantum Numbers 303

Radial probability (4πr 2 R2 )

Figure 7.13
(a) Cross section of the hydrogen 1s
orbital probability distribution divided
into successive thin spherical shells.
(b) The radial probability distribution. A
plot of the total probability of finding
the electron in each thin spherical Distance from nucleus (r)
shell as a function of distance from (a) (b)
the nucleus.

of the spherical shell increases with distance from the nucleus. Therefore, as we move
1 Å ⫽ 10⫺10 m; the angstrom is most away from the nucleus, the probability of finding the electron at a given position decreases,
often used as the unit for atomic radius but we are summing more positions. Thus the total probability increases to a certain ra-
because of its convenient size. Another
convenient unit is the picometer: dius and then decreases as the electron probability at each position becomes very small.
For the hydrogen 1s orbital, the maximum radial probability (the distance at which the
1 pm ⫽ 10⫺12 m electron is most likely to be found) occurs at a distance of 5.29 ⫻ 10⫺2 nm or 0.529 Å
from the nucleus. Interestingly, this is exactly the radius of the innermost orbit in the Bohr
model. Note that in Bohr’s model the electron is assumed to have a circular path and so
is always found at this distance. In the quantum mechanical model, the specific electron
motions are unknown, and this is the most probable distance at which the electron is found.
One more characteristic of the hydrogen 1s orbital that we must consider is its size.
As we can see from Fig. 7.12, the size of this orbital cannot be defined precisely, since the
probability never becomes zero (although it drops to an extremely small value at large val-
ues of r). So, in fact, the hydrogen 1s orbital has no distinct size. However, it is useful to
have a definition of relative orbital size. The definition most often used by chemists to de-
scribe the size of the hydrogen 1s orbital is the radius of the sphere that encloses 90% of
the total electron probability. That is, 90% of the time the electron is inside this sphere.
So far we have described only the lowest-energy wave function in the hydrogen atom,
the 1s orbital. Hydrogen has many other orbitals, which we will describe in the next sec-
tion. However, before we proceed, we should summarize what we have said about the
meaning of an atomic orbital. An orbital is difficult to define precisely at an introductory
level. Technically, an orbital is a wave function. However, it is usually most helpful to
picture an orbital as a three-dimensional electron density map. That is, an electron “in” a
particular atomic orbital is assumed to exhibit the electron probability indicated by the
orbital map.

7.6 䉴 Quantum Numbers

When we solve the Schrödinger equation for the hydrogen atom, we find many wave func-
tions (orbitals) that satisfy it. Each of these orbitals is characterized by a series of num-
bers called quantum numbers, which describe various properties of the orbital:
The principal quantum number (n) has integral values: 1, 2, 3, . . . . The principal
quantum number is related to the size and energy of the orbital. As n increases, the
orbital becomes larger and the electron spends more time farther from the nucleus.
An increase in n also means higher energy, because the electron is less tightly bound
to the nucleus, and the energy is less negative.
The angular momentum quantum number (ᐉ) has integral values from 0 to n ⫺ 1
for each value of n. This quantum number is related to the shape of atomic orbitals.
The value of ᐉ for a particular orbital is commonly assigned a letter: ᐉ ⫽ 0 is called s;

Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.