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CHAPTER 7
Quantum Theory of the Atom
CHAPTER TERMS AND DEFINITIONS
Numbers in parentheses after definitions give the text sections in which the terms are explained. Starred
terms are italicized in the text. Where a term does not fall directly under a text section heading,
additional information is given for you to locate it.
wave*
continuously repeating change (oscillation) in matter or in a physical field (7.1)
electromagnetic radiation*
travel through space (7.1)
wavelength (λ)
distance between any two adjacent identical points of a wave (7.1)
frequency (ν)
second) (7.1)
hertz (Hz)*
energy in the form of oscillating electric and magnetic fields that can
number of wavelengths of a wave that pass a fixed point in one unit of time (usually 1
unit of frequency; 1 Hz = /s, or s1 (7.1)
electromagnetic spectrum
range of frequencies or wavelengths of electromagnetic radiation (7.1)
diffraction*
spreading out of waves when they encounter an obstruction or opening the size of the
wavelength (7.2)
Planck’s constant (h) constant of proportionality relating energy and frequency of vibration or
oscillation; h = 6.63  1034 J ∙ s (7.2)
quantum numbers*
quantized*
integers specifying energy states of quantized particles (7.2)
limited to certain values (7.2)
photons particles of electromagnetic energy, with energy E proportional to the observed frequency
of the light (7.2)
photoelectric effect
ejection of electrons from the surface of a metal or from another material when
light of the proper frequency shines on it (7.2)
threshold value*
metal (7.2)
absorbed*
minimum frequency of light that will produce the photoelectric effect for a given
taken in wholly, as a photon and its energy taken up by an electron (7.2)
wave–particle duality* idea that wave and particle models are complementary views of light (and
all electromagnetic radiation) (7.2)
continuous spectrum
spectrum containing light of all wavelengths (7.3)
line spectrum spectrum showing only certain colors or specific wavelengths of light; each element
gives a distinctive spectrum (7.3)
energy levels
specific energy values that an electron can have in an atom (7.3)
RH* constant used to derive the energy levels of the electron in the hydrogen atom;
RH = 2.179  1018 J (7.3)
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150
n*
Chapter 7: Quantum Theory of the Atom
symbol for the principal quantum number (7.3)
transition*
change in energy level of an electron in an atom from Ei to Ef (7.3)
emission*
release of energy from an atom as an electron’s energy level changes from an upper to a
lower one (7.3)
excited*
said of an electron gaining energy and boosted to a higher energy level (7.3)
absorption*
taking in of energy by an atom that can raise an electron to a higher energy level (7.3)
monochromatic* of one color; describing light of very narrow wavelength (A Chemist Looks at:
Lasers and Compact Disc Players)
laser* acronym meaning light amplification by stimulated emission of radiation (A Chemist Looks
at: Lasers and Compact Disc Players)
radiationless transitions* electron transitions in an atom or ion from a higher to lower energy level
that release energy as heat rather than as light (A Chemist Looks at: Lasers and Compact Disc
Players)
spontaneous emission* release of light energy from an electron transition without external cause (A
Chemist Looks at: Lasers and Compact Disc Players)
stimulated emission* release of a photon of light from an excited atom or ion that encounters a
photon of the same wavelength (A Chemist Looks at: Lasers and Compact Disc Players)
coherent* describes light in which all waves forming the beam are in phase; that is, the waves have
their maxima and minima at the same points in space and time (A Chemist Looks at: Lasers and
Compact Disc Players)
the wavelength associated with a particle of matter given by the equation λ =
de Broglie relation
h/mv (7.4)
electron microscope* microscope in which a beam of electrons, rather than light, is focused on the
object to be magnified; electrons passing through or emitted from the object are focused by magnetic
lenses on a fluorescent screen (7.4)
resolving power*
ability of a microscope to distinguish detail (7.4)
quantum (wave) mechanics
submicroscopic particles (7.4)
branch of physics that mathematically describes the wave properties of
uncertainty principle the product of the uncertainty in position and the uncertainty in momentum of
a particle can be no smaller than Planck’s constant (h) divided by 4π:
(Δx) (Δpx) ≥
statistical*
probability*
h
(7.4)
4p
of or pertaining to numerical values (7.4)
likelihood of finding an electron at a certain point in an atom at a specified time (7.4)
wave function (ψ)* mathematical expression providing information about a particle associated with
a given energy level (such as an electron in an atom) (7.4)
ψ 2*
square of the wave function; gives the probability of finding the associated particle within a
region of space (7.4)
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Chapter 7: Quantum Theory of the Atom
151
scanning tunneling microscope* microscope in which a very fine-pointed tungsten needle is
moved close to a sample so that electrons tunnel to the sample and produce an electric current, the
voltage variations of which give rise to an image (Instrumental Methods: Scanning Tunneling
Microscopy)
tunneling* movement of an electron from one atom to another without extra energy being supplied,
owing to quantum mechanical effects (Instrumental Methods: Scanning Tunneling Microscopy)
piezoelectric*
describes the conversion of the variable compression or expansion of a material into a
variable voltage (Instrumental Methods: Scanning Tunneling Microscopy)
atomic orbital
wave function for an electron in an atom, pictured as the region of space about the
nucleus in which there is high probability of finding an electron (7.5)
spin*
magnetic property of electrons (7.5)
principal quantum number (n)
any positive integer (1, 2, 3, and so on) specifying the major
energy level of an electron in an atom (7.5)
shell*
energy level of an atom (7.5)
angular momentum (azimuthal) quantum number (l)
of an orbital (7.5)
subshells*
integer from 0 to n – 1 denoting the shape
sublevels of an energy level of an atom (7.5)
magnetic quantum number (ml) integer from l to +l designating the orientation in space of a
specific orbital within a subshell (7.5)
orientation*
position (7.5)
spin quantum number (ms) fraction, either +½ or ½, designating one of the two possible
orientations of the spin axis of an electron (7.5)
99% contour* region of space about an atomic nucleus in which an electron of a given energy is
expected to be found 99% of the time (7.5)
CHAPTER DIAGNOSTIC TEST
Previously, we arranged diagnostic test questions in the same sequence as the presentation of material
in the chapter. Now that you have had some experience with problem solving, the diagnostic test and
post-test questions will be given in no specific order.
1.
Calculate the frequency of electromagnetic radiation having the wavelength of 684.9nm.
2.
When an electron moves from level n = 4 to level n = 5 in an excited hydrogen atom, what
amount of energy is required for this electronic transition?
3.
Indicate whether each of the following statements is true or false. If a statement is false, change it
so that it is true.
a.
The azimuthal quantum number describes the main energy level of an electron in an atom.
True/False: ________________________________________________________________
_______________________________________________________________________
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152
Chapter 7: Quantum Theory of the Atom
b.
In an excited hydrogen atom, an electron described by the quantum numbers n = 4, l = 1,
ml = 1 is at higher energy than an electron described by the quantum numbers n = 4, l = 1,
ml = 1. True/False: _________________________________________________________
__________________________________________________________________________
c.
The square of the wave function for an electron is related to the charge density of that
electron at a point in space. True/False: __________________________________________
__________________________________________________________________________
d.
Bohr’s theory accounted for the line spectra of excited atoms in terms of the electromagnetic
radiation absorbed or released from electrons moving in circular orbits about the nucleus.
True/False:
__________________________________________________________________________
__________________________________________________________________________
e.
Assuming that nature was symmetrical, de Broglie postulated that a wavelength could be
associated with a moving particle and calculated from the relationship λ = h/mv. True/False:
__________________________________________________________________________
__________________________________________________________________________
4.
Which of the following five sets of quantum numbers is (are) not allowed?
a.
b.
c.
d.
e.
n=3
n=1
n=4
n=2
n=3
l=1
l=0
l=3
l=2
l=2
ml = 0
ml = 0
ml = 1
ml = 2
ml = 2
ms =1/2
ms = 3/2
ms = 1/2
ms = 1/2
ms = 1/2
5.
Describe the types of orbitals you are likely to find in the n = 3 shell.
6.
Radiation remains visible to the eye down to a frequency of 4.29  1014/s. What are the
wavelength and the energy of one of these photons?
7.
Which of the following statements concerning the second main energy level is (are) incorrect?
a.
Its principal quantum number is 2.
b.
It contains s, p, and d orbitals.
c.
It can contain an electron having the quantum numbers n = 2, l = 1, ml = 1, ms = 1/2.
d.
It has a bilobed (dumbbell) shape.
e.
It cannot contain any f orbitals.
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Chapter 7: Quantum Theory of the Atom
8.
9.
153
Match each term in the left-hand column with the appropriate expression in the right-hand
column.
_____
(1)
Schrödinger
a.
wave nature of particles
_____
(2)
main energy level
b.
wave equation
_____
(3)
hν
c.
stability of nuclear atom
_____
(4)
de Broglie
d.
principal quantum number
_____
(5)
Bohr
e.
energy of a photon
_____
(6)
orbitals
f.
azimuthal quantum number
A typical wavelength for light in the visible region of the electromagnetic spectrum is 4.8  103
pm. If a particle weighs 1.5  1025 g, at what velocity would it have to be moving to exhibit this
wavelength?
ANSWERS TO CHAPTER DIAGNOSTIC TEST
If you missed an answer, study the text section and problem-solving skill (PS Sk.) given in parentheses
after the answer.
1.
4.38  1014/s (7.1, PS Sk. 1)
2.
4.905  1020 J (7.3, PS Sk. 3)
3.
a.
False. The principal quantum number describes the main energy level of an electron in an
atom. (7.5)
b.
False. In an excited hydrogen atom, an electron described by the quantum numbers n = 4, l =
1, ml = 1 is of the same energy as an electron described by the quantum numbers n = 4, l = 1,
ml = 1. (7.5)
c.
True. (7.4)
d.
False. Bohr’s theory accounted for the line spectra of excited atoms in terms of the
electromagnetic radiation absorbed or released when electrons move from one energy level
to another in an atom. (7.3)
e.
True. (7.4)
4.
b, d (7.5, PS Sk. 5)
5.
At n = 3, the l quantum number has values of 0, 1, and 2. Thus we find s, p, and d orbitals in this
shell. (7.5)
6.
λ = 6.99  107 m (699 nm) (7.1, PS Sk. 1);
E = 2.84  1019 J (7.2, PS Sk. 2)
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154
7.
Chapter 7: Quantum Theory of the Atom
b, d (7.5)
8.
(1) b (7.4)
(2) d (7.3, 7.5)
(3) e (7.2)
(4) a (7.4)
(5) c (7.3)
(6) f (7.5)
9.
9.2 m/s (7.2, 7.4)
SUMMARY OF CHAPTER TOPICS
7.1 The Wave Nature of Light
Learning Objectives

Define the wavelength and frequency of a wave.

Relate the wavelength, frequency, and speed of light. (Examples 7.1 and 7.2)

Describe the different regions of the electromagnetic spectrum.
Problem-Solving Skill
1.
Relating wavelength and frequency of light. Given the frequency of light, calculate the
wavelength, or vice versa (Examples 7.1 and 7.2).
The chart on the following page presents information about the electromagnetic spectrum that you may
find useful in this and later courses. (More information is included than you now need, but we will refer
to it later.) For now, you should become familiar with the magnitude of the wavelengths for the various
“boundaries” in the electromagnetic spectrum.
You should memorize the formula for the speed of light (c = λν) and the value of c (3.00  108 m/s). If
the units of frequency, /s or s1, are confusing, you can think of this as “waves” per second, although we
do not write it that way. The units of wavelength are meters (m) or nanometers (nm), although other
units, such as picometers (pm), are sometimes used.
Exercise 7.1
The frequency of the strong red line in the spectrum of potassium is 3.91  1014/s. What is the
wavelength of this light in nanometers?
Wanted: λ (nm)
Given:
ν = 3.91  1014/s
Known:
c = λν; c = 3.00  108 m/s; 109 nm = 1 m
Solution: Rearrange c = λν to get
λ=
3.00  10 8 m/s
c
10 9 nm
=

= 767 nm

m
3.91  10 14 /s
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Chapter 7: Quantum Theory of the Atom
155
Note that high energy correlates with high frequency but low wavelength. Even though this may seem
easy to grasp, it is a concept that many students stumble over when the equations are not in front of
them. You may see a large value for wavelength and automatically think “high energy,” which is
incorrect. Practice writing wavelength values and putting them in order of increasing energy to
emphasize this for yourself.
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156
Chapter 7: Quantum Theory of the Atom
Exercise 7.2
The element cesium was discovered in 1860 by Robert Bunsen and Gustav Kirchhoff, who found two
bright blue lines in the spectrum of a substance isolated from a mineral water. One of these spectral
lines of cesium has a wavelength of 456 nm. What is its frequency?
Solution: Rearrange c = λν to get
ν=
c
10 9 nm
3.00  10 8 m/s
=

= 6.58  1014 /s

456 nm
m
7.2 Quantum Effects and Photons
Learning Objectives

State Planck’s quantization of vibrational energy.

Define Planck’s constant and photon.

Describe the photoelectric effect.

Calculate the energy of a photon from its frequency or wavelength. (Example 7.3)
Problem-Solving Skill
2.
Calculating the energy of a photon. Given the frequency or wavelength of light, calculate the
energy associated with one photon (Example 7.3).
Diffraction is an interesting phenomenon. To observe it, hold a mesh curtain very close to your eye and
look at a street light through the space between the threads. If you are really observant, you will “see”
black lines. These appear because the light is being spread as it goes through the holes, and some
portions of your retina have no light falling on them.
A good example of a hot solid glowing red is the heating element on your electric stove.
It is interesting that our concept of the particle nature of light and our understanding of the electronic
structure of the atom are interdependent and were worked out together.
You should memorize the formula for the energy of a photon, E = hν.
Exercise 7.3
The following are representative wavelengths in the infrared, ultraviolet, and x-ray regions of the
electromagnetic spectrum, respectively: 1.0  106 m, 1.0  108 m, and 1.0  1010 m. What is the
energy of a photon of each radiation? Which has the greatest amount of energy per photon? Which has
the least?
Wanted: energy of each photon and most-energetic and least-energetic photons
Given:
λ1 = 1.0  106 m, λ2 = 1.0  108 m, λ3 = 1.0  1010 m
Known:
E = hν; c = λν = 3.00  108 m/s; h is given in text.
Solution: Since the energy calculation is the same for each, we will show only the first one.
Rearrange c = λν to solve for ν:
c
=ν

and substitute this value for ν in the expression for energy:
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Chapter 7: Quantum Theory of the Atom
E = hν = h
E1 = h
157
c

6.63  10 -34 J • s  3.00  10 8 m/s
c
=
1
1.0  10 -6 m
= 2.0  1019 J for the infrared photon
E2 = 2.0  1017 J for the ultraviolet photon
E3 = 2.0  1015 J for the x-ray photon
The energy values increase, so the x-ray photon has the greatest amount of energy and the infrared
photon the least.
A Chemist Looks at: Zapping Hamburger with Gamma Rays
Questions for Study
1.
How do gamma rays and x rays compare?
2.
Why are gamma rays effective in killing bacteria and molds in food?
3.
What is the source of gamma rays?
Answers to Questions for Study
1.
Both gamma rays and x rays are forms of electromagnetic radiation, but gamma rays are photons
of higher energy than the photons of x rays.
2.
Gamma rays break up the DNA molecules within the cells of bacteria and molds, which disrupts
the production of vital cell proteins.
3.
Gamma rays used in food irradiation come from the radioactive decay of cobalt-60.
7.3 The Bohr Theory of the Hydrogen Atom
Learning Objectives

State the postulates of Bohr’s theory of the hydrogen atom.

Relate the energy of a photon to the associated energy levels of an atom.

Determine the wavelength or frequency of a hydrogen atom transition. (Example 7.4)

Describe the difference between emission and absorption of light by an atom.
Problem-Solving Skill
3.
Determining the wavelength or frequency of a hydrogen atom transition. Given the initial and
final principal quantum numbers for an electron transition in the hydrogen atom, calculate the
frequency or wavelength of light emitted (Example 7.4). You need the value of RH .
When a hydrogen-gas discharge tube is connected to a source of electric current, several forms of
electromagnetic radiation are emitted. Besides those in the visible portion, emissions are also in the
infrared and ultraviolet portions of the electromagnetic spectrum. Emissions in the visible region are of
greatest interest to us in this course.
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158
Chapter 7: Quantum Theory of the Atom
Today we accept Bohr’s idea about light being emitted by an atom or ion owing to its (electron)
transitions between allowable energy states. According to his theory, the allowable energies of the
hydrogen atom are the allowable energies the electron can have in the atom. A given energy value that
corresponds to an energy state (level) of a hydrogen atom is also the energy an electron has when at the
specific energy level.
Likewise, we speak of energy transitions in reference to the hydrogen atom. But the atom changes in
energy because the electron can absorb energy under certain conditions and thus undergo transitions to
higher-energy states. When the electron changes to a lower-energy state, we say the atom has
undergone a transition to a lower level. Because the energies of atoms other than hydrogen are so
complex, we make reference only to the electron transitions, although, of course, the atom is changing
to different energy levels as its electrons are.
The combinations of visible frequencies emitted by excited atoms are the colors we see in a fireworks
display. The atoms are excited by the energy of the explosion. Their electrons continually undergo
transitions to higher-energy levels and then fall back, emitting specific frequencies that add together to
give a characteristic color. For example, strontium produces a red color and barium a green.
Exercise 7.4
Calculate the wavelength of light emitted from the hydrogen atom when the electron undergoes a
transition from level n = 3 to level n = 1.
Wanted: λ
Given:
Transition is from n = 3 to n = 1.
Known:
The energy of any level is given by
E=
RH
;
n2
R = 2.180  1018 J
The energy of a photon is
Ei – Ef = ΔE = hν
c = λν = 3.00  108 m/s
h = 6.63  10–34 J ∙ s
Solution: First find the energy of the photon:
  RH    RH
 –  2
2 
 3   1
ΔE = En=3  En=1 = 
=
 R H  (9 R H )

 =
9

 RH  9RH
8RH
8  2.180  10 18 J
=
=
= 1.938  1018 J
9
9
9
To find λ, rearrange c = λν to ν = c/λ, and substitute for ν in ΔE = hν to give
ΔE = h
c

Now solve this for λ and put in the known values:
λ=
6.63  10 -34 J • s  3.00  10 8 m/s
hc
=
= 1.03  107 m (103 nm)
E
1.938  10 -18 J
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Chapter 7: Quantum Theory of the Atom
159
Exercise 7.5
What is the difference in energy levels of the sodium atom if emitted light has a wavelength of 589 nm?
Known:
ΔE = hν; c = λν; values of h and c from above; 1 nm = 10–9 m
Solution: Solve c = λν for ν and substitute in ΔE = hν to give
ΔE = h
c
6.63 10 34 J • s  3.00  10 8 m/s
1 nm
=
 -9 = 3.38  10–19 J

589 nm
10 m
7.4 Quantum Mechanics
Learning Objectives

State the de Broglie relation.

Calculate the wavelength of a moving particle. (Example 7.5)

Define quantum mechanics.

State Heisenberg’s uncertainty principle.

Relate the wave function for an electron to the probability of finding it at a location in space.
Problem-Solving Skill
4.
Applying the de Broglie relation. Given the mass and speed of a particle, calculate the
wavelength of the associated wave (Example 7.5).
Quantum mechanics can be a difficult topic to understand. Part of this difficulty is the advanced
mathematics it uses. But much of the difficulty is that what it describes, the submicroscopic world, is
outside our experience. What is common sense in our macroscopic world has no meaning in a world
where 6.02  1023 particles can fit on the point of a pin! As you read this section of your textbook, try to
get an overall idea of the subject. You may have to read the section several times to do this.
Although the term wave function may be new to you, you may recall the term function from your math
courses. When we plot values of x and y on a two-dimensional graph, we write y = f(x), meaning that
the y value is some function of x, say, x + 3.
Quantum mechanics gives us two ways to describe the electron in motion about the nucleus: (1) We
cannot know the path of the electron, but we can calculate its average speed as it moves about the
nucleus; (2) we can know the probability of finding an electron at a certain point at a given distance
from the atomic nucleus.
Exercise 7.6
Calculate the wavelength (in picometers) associated with an electron traveling at a speed of 2.19  106
m/s.
Wanted: wavelength in picometers (pm)
Given:
Speed ν is 2.19  106 m/s.
Known:
λ = h/mν; h = 6.63  10–34 J ∙ s (text); 1 J = kg ∙ m2/s2
(1.8, PS Sk. 6); electron mass = 9.10939  10–31 kg (text Table 2.1);
1 m = 1010 pm (1.6)
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160
Chapter 7: Quantum Theory of the Atom
Solution:
λ=
h
mv
=
6.63 10 34 J • s
9.10939  10 31 kg  2.19  10 6 m/s

10 10 pm
kg • m 2 /s 2

J
m
= 3.32 pm
7.5 Quantum Numbers and Atomic Orbitals
Learning Objectives

Define atomic orbital.

Define each of the quantum numbers for an atomic orbital.

State the rules for the allowed values for each quantum number.

Apply the rules for quantum numbers. (Example 7.6)

Describe the shapes of s, p, and d orbitals.
Problem-Solving Skill
5.
Using the rules for quantum numbers. Given a set of quantum numbers n, l, ml, and ms, state
whether that set is permissible for an electron (Example 7.6).
If we plot all the points where there is high probability of finding an electron of a given energy, we
have a three-dimensional region of space (a volume) about the atomic nuclei. This volume of space is a
description of an orbital. It is also referred to as an electron cloud. Be sure you do not confuse this with
a Bohr orbit, which is a circular path such as a planetary or satellite orbit.
The first three of the four quantum numbers describing an electron in an atom specify the orbital for the
electron. Thus n specifies the energy level, l the shape of the orbital, that is, the subshell, and ml the
orientation in space, that is, the specific orbital. Do not be concerned about the meaning or derivation of
the names (azimuthal or magnetic). At some later time you can take a course in quantum mechanics and
resolve your unanswered questions. The fourth quantum number, ms, describes the spin orientation of
the electron. You should memorize the rules for the allowable values of these quantum numbers in
order to more easily work the problems associated with them.
Over the years, a number of terms have been used in reference to the energy states of the atom. For
your purposes, energy level, quantum state, and shell mean the same thing.
Exercise 7.7
Explain why each of the following sets of quantum numbers is not permissible for an orbital.
a.
n = 0, l = 1, ml = 0, ms = +½
b.
n = 2, l = 3, ml = 0, ms = ½
c.
n = 3, l = 2, ml = +3, ms = +½
d.
n = 3, l = 2, ml = +2, ms = 0
Known:
The text gives the allowable values for all four quantum numbers.
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Chapter 7: Quantum Theory of the Atom
161
Solution:
a.
The value of n must be a positive whole number.
b.
l cannot be greater than n.
c.
ml cannot be greater than l.
d.
ms can equal only +½ or ½.
ADDITIONAL PROBLEMS
1.
2.
Calculate
a.
the frequency of an x ray with a wavelength of 4.70  10–9 nm.
b.
the wavelength of the transition of cesium-133, frequency 9.193  109/s, the standard for the
SI unit of time.
c.
the energy of a photon of red light, wavelength of 7.00  102 nm, absorbed by chlorophyll a
in the process of photosynthesis.
An energy of 1.09  103 kJ/mol is required to convert gaseous carbon atoms to gaseous C+ ions
and gaseous electrons. Calculate the maximum wavelength (in units of nanometers) of
electromagnetic radiation that can cause the ionization of one carbon atom.
3.
a.
Calculate the wavelength of radiation emitted when the electron in the hydrogen atom
undergoes a transition from level n = 2 to level n = 1. (The value of RH is 2.180  1018 J.)
b.
Refer to Figure 7.5 and determine in which portion of the electromagnetic spectrum this
energy falls.
4.
Calculate the wavelength in nanometers of a vehicle with a mass of 2.200  103 kg that is moving
with a velocity of 1.10  102 km/h.
5.
Explain why each of the following sets of quantum numbers is not permissible for an electron in
an atom.
a.
b.
c.
d.
n = –1,
n = 0,
n = 3,
n = 2,
l = 1,
l = 0,
l = 1,
l = 1,
ml = 0,
ml = 1,
ml = 2,
ml = 1,
ms = +½
ms = ½
ms = +½
ms = 1
ANSWERS TO ADDITIONAL PROBLEMS
If you missed an answer, study the text section and problem-solving skill (PS Sk.) given in parentheses
after the answer.
1.
a.
Because c = λν,
ν=
3.00  10 8 m/s
c
10 9 nm
=

= 6.38  1025 /s (7.1, PS Sk. 1)

m
470  10 9 nm
Copyright © Houghton Mifflin Company. All rights reserved.
162
Chapter 7: Quantum Theory of the Atom
b.
Because c = λν,
λ=
c.
3.00  10 8 m/s
c
=
= 3.26  10–2 m (7.1, PS Sk. 1)
9

9.193  10 /s
Because E = hν and c = λν,
E=
2.
6.63 10 34 J • s  3.00  10 8 m/s
hc
10 9 nm
=

2

m
7.00  10 nm
= 2.84  10–19 J (7.2, PS Sk. 2)
The energy expressed in kJ/mol must be converted to kJ/atom for the ionization of a single carbon
atom.
E=
1.09  10 3 kJ
1 mol

= 1.811  10–21 kJ/atom
mol
6.023  10 23 atom
Because ΔE = hν = hc/λ, solving for λ gives
λ=
hc
1
1 nm
1 kJ
= 6.63  1034 J ∙ s  3.00  108 m/s  9

 3
 21
E
1.8
1
1

10
kJ
10 J
10 m
= 1.10  102 nm (7.2, PS Sk. 2)
3.
a.
This problem is easier if done algebraically first. The energy of the emitted photon is
E = Ei – Ef =
RH
R
hc
– 2H = hν =
2

ni
nf
Putting in the principal quantum numbers gives
E=
 RH
 RH
3R H
hc
–
=
=

4
4
1
Solving the last equality for λ and putting in values, we get
λ=
4  6.63  10 34 J • s  3.00  10 8 m/s
4hc
=
3R H
3  2.180  10 18 J
= 1.22  10–7 m (7.3, PS Sk. 3)
b.
4.
This is ultraviolet radiation. (7.3)
Using the de Broglie relation, we get
h
60 min
1
hr
λ = mv = 6.63  1034 J ∙ s 


3
2
1 hr
2.200  10 kg
1.10  10 km

kg • m 2
60 sec

1 min
s2 • J

nm
10
9
m

1 km
10 3 m
= 9.86  1030 nm (This value is so small that the wave properties cannot be
observed.) (7.4, PS Sk. 4)
Copyright © Houghton Mifflin Company. All rights reserved.
Chapter 7: Quantum Theory of the Atom
163
5.
a.
The value of n must be a positive whole number.
b.
The value of n cannot be 0, and that of the magnitude of ml must not be greater than that of l.
c.
The value of l must be 0 or a positive whole number.
d.
The value of ms can be only +½ or ½. (7.5, PS Sk. 5)
CHAPTER POST-TEST
1.
The numerical value of the____________________ determines the orbital shape.
2.
The______________ quantum number is most important in determining the energy of the orbital.
3.
The s orbital may be described as having the shape of a
a.
dumbbell.
b.
four-leafed clover.
c.
sphere.
d.
pyramid.
e.
circle.
4.
Briefly discuss the interdependence of the numerical values of the quantum numbers n, l, and ml.
5.
Which of the following sets of quantum numbers is (are) not permitted?
a.
b.
c.
d.
e.
n=1
n=4
n=3
n=2
n=2
l=0
l=3
l=2
l=0
l=1
ml = 0
ml = 3
ml = 3
ml = 0
ml = 0
ms = 1/2
ms = 1/2
ms = +1/2
ms = 1/2
ms = +1/2
6.
Find the wavelength associated with an electron moving at 95.0% the speed of light. (Electron
mass = 9.11  10–31 kg; h = 6.63  10–34 J ∙ s.)
7.
A 68.0-kg runner runs a marathon at an average speed of 4.00 m/s. What is the deBroglie
wavelength of the runner? (See Question 6 for h.)
8.
Green light has a wavelength of 5.3  103 pm. What is the frequency of green light?
9.
When an electron moves from level n = 1 to level n = 3 in a hydrogen atom, what amount of
energy is required for this electronic transition?
10. Molecules undergo electronic transitions similar to those of atoms. If a molecule absorbs radiation
of wavelength 380.0 nm, what is the energy of this transition?
ANSWERS TO CHAPTER POST-TEST
If you missed an answer, study the text section and problem-solving skill (PS Sk.) given in parentheses
after the answer.
1.
azimuthal quantum number, l (7.5)
2.
principal (7.3)
3.
c (7.5)
Copyright © Houghton Mifflin Company. All rights reserved.
164
4.
Chapter 7: Quantum Theory of the Atom
See Section 7.5 in your text for a complete discussion. The numerical values of these quantum
numbers are related as follows:
(1)
n can take on any integral value from 1 to .
(2)
Once the value of n is determined, l can take on any integral value from 0 up to (n – l).
(3)
Once the values of n and l are determined, for any given value of l, ml can take on any
integral value from l to + l. (7.5)
5.
c (7.5, PS Sk. 5)
6.
2.55  10–12 m (2.55  10–3 nm) (7.4, PS Sk. 4)
7.
2.44  1036 m (2.44  1027 nm) (7.4, PS Sk. 4)
8.
5.7  1014/s (7.1, PS Sk. 1)
9.
1.938  10–18 J (7.3, PS Sk. 3)
10. 5.23  10–19 J (7.1, 7.2, PS Sk. 2)
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