Kelter P., Mosher M., Scott A. Chemistry. The Practical Science

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FIGURE 6.46

Periodic table with the electron conﬁgurations of all atoms in the gas phase. The “partial” electron conﬁgurations

are shown when the cores of the electron conﬁguration can be deduced from the earlier periods.

248 Chapter 6 Quantum Chemistry: The Strange World of Atoms

104

105

106

107

108

110

111

112

Uub

109

100

101

102

103

d-Transition elements

Period number, highest occupied electron level

Noble

gases

Lanthanides

Actinides

Representative

elements

Group

numbers

IIA

IIIA

IVA

VIA

VIIA

VIIIA

Representative elements

f-Transition elements

13 14 15 16 17

8 9 10 11 1234567

periodic table have the conﬁguration [core]ns

, where [core] is the electron con-

ﬁguration of the noble gas directly preceding the element in the periodic table.

The n is the row number that the element occupies in the periodic table, and n is

also the principal quantum number of the ns electron. It is this similarity in

ground-state electron conﬁguration that gives all of the elements in this column

similar chemical properties, including metallic behavior, malleability, tendency to

form ions with a charge of +1, and high reactivity with water.

What happened to the 3d electrons? When we were talking about the effect of

adding more than one electron to the orbital energies of the hydrogen atom, we

made the point that interactions among the electrons removed the degeneracy

among the electrons in the n-shell orbitals, so that orbitals with increasing value

of l had increasing energy. The 3s is lower in energy than the 3p, which, in turn,

is lower than the 3d. The 3d orbital energy is so destabilized by the electron–

electron interactions that it actually rises in energy to be just a little higher in

energy than the 4s orbital for the atom in the gas phase. To form the lowest energy

electron conﬁguration, the 4s subshell ﬁlls before the 3d subshell. The same prob-

lem is noted when we begin ﬁlling the f orbital subshell. The ground-state con-

ﬁgurations for the elements as gas-phase atoms are given in Figure 6.46.

6.13 Electron Conﬁgurations and the Aufbau Principle 249

The order of ﬁlling is systematic and is illustrated by the diagram shown in Fig-

ure 6.47. For example, we can follow the arrows in the ﬁgure and add the

electrons to each subshell to obtain the ground-state electron conﬁguration of

sulfur (16 electrons):

S: 1s

= [Ne]3s

Using this information, we can obtain the electron conﬁguration of iodine:

I: 1s

= [Kr]5s

EXERCISE 6.8 Practice with Electron Conﬁgurations

Write the ground-state electron conﬁguration for vanadium and for tellurium.

First Thoughts

The structure of the periodic table gives us guidance for writing the ground-state

electron conﬁguration of an element. The two key questions we ask are “In what

period is our element?” and “Is its highest partially ﬁlled energy orbital an s, p, d,or

f orbital?”

Solution

Vanadium, atomic number 23, is in Period 4 and is the third element in the 3d series

that begins with scandium, as shown in Figure 6.46. It will therefore have an argon

electron core with two 4s and three 3d electrons. Alternatively, you can use the tri-

angle in Figure 6.47 and ﬁll the orbitals to an electron count of 23.

V (23 electrons): [Ar]4s

= 1s

Tellurium, atomic number 52, is in Period 5 and Group VIA, in which p orbitals are

being ﬁlled. It will have a krypton electron core, with (reading across Group V in

Figure 6.46) two 5s electrons, ten 4d electrons, and six 5p electrons. Alternatively,

you can use Figure 6.47, as with vanadium, although counting electrons in this way

can get clerically messy.

Te (52 electrons): [Kr]5s

= 1s

Further Insights

Write the ground-state electron conﬁguration of chromium.You will get [Ar]4s

It is often said that one of the s electrons will move to the d orbital to give an ener-

getically more favorable (half–ﬁlled) 3d

conﬁguration (that is, [Ar]4s

). This re-

organization is noted when we’re talking about chromium atoms in the gas phase.

Molybdenum atoms ([Kr]5s

) and copper atoms ([Ar]4s

) are other exam-

ples where the reorganized electron conﬁguration is observed in the gas phase.

It should be noted that these cases are speciﬁc to the gas-phase electron conﬁg-

uration, where interactions with other atoms cannot stabilize the predicted electron

conﬁgurations. Because these elements are not found in nature or generally used as

individual gas-phase atoms, we can use the predicted electron conﬁgurations in our

normal work.

PRACTICE 6.8

Write the ground-state electron conﬁguration for gallium and strontium.

See Problems 99–108 and 118.

7s 7p

6s 6p

5s 5p 5d 5f

4s 4p 4d 4f

3s 3p 3d

2s 2p

FIGURE 6.47

The order in which atomic orbitals are

ﬁlled. The order is determined by writing

the orbitals found in each energy level

in rows and then drawing diagonal lines

up and to the left. The order is 1s,2s,

2p,3s,3p,4s,3d,4p,5s,4d,5p,6s,4f,

5d,6p,....

250 Chapter 6 Quantum Chemistry: The Strange World of Atoms

A Summary of the Ground-State Electron Conﬁguration–Based Structure

of the Periodic Table

The periodic table can be divided into four regions based on the ﬁlling of orbitals

of the elements; see Figure 6.48.

■

Region 1: This region includes the alkali and alkaline earth metals, in which the

s orbitals are being ﬁlled. These Group IA and IIA elements, such as potassium

and calcium, are part of the main-group elements.

■

Region 2: This region includes the remainder of the main-group elements,

Groups IIIA through VIIIA, such as carbon, nitrogen, and chlorine, in which

the p orbitals are being ﬁlled.

■

Region 3: This region includes the transition elements, in which the d orbitals

are being ﬁlled. These include metals such as chromium, silver, and mercury.

■

Region 4: This includes the inner-transition elements, such as uranium and

plutonium, in which the f orbitals are being ﬁlled.

The structure of the periodic table is a very satisfying one for chemists be-

cause it gives rhyme and reason to the physical properties and chemical reactivity

of the elements. Although the periodic table was an outgrowth of Mendeleev’s

periodic law formulated in 1869, this early grouping of elements with similar

properties was done empirically, just by seeing how the material behaved. The

odd shape of the periodic table, however, simply begs for an explanation, and

quantum mechanics has given it! By solving the Schrödinger equation, we have

generated wave functions with unique sets of quantum numbers. By generating

the wave functions and quantum numbers, we have created a complete set of

electron orbitals, and by ﬁlling these orbitals according to the Aufbau principle,

we have generated the complete periodic table—its form, the properties of its

elements, everything!

1s 1s

Main-group

elements

s-block

Main-group

elements

p-block

Transition elements

d-block

Inner transition

elements

f-block

FIGURE 6.48

The periodic table can be divided into

regions (known as blocks) that illustrate

the type of atomic orbital that is being

ﬁlled.

Key Words 251

The Bottom Line

■

We can interconvert among the wavelength, fre-

quency, and energy of electromagnetic radiation.

(Section 6.2)

■

Atoms and molecules absorb and emit electromag-

netic radiation to gain and lose energy, but only cer-

tain wavelengths of radiation can be absorbed and

emitted. (Section 6.3)

■

The wavelengths of electromagnetic radiation ab-

sorbed and emitted by an atom are characteristic of

that atom and can be used to probe the atomic

structure. (Section 6.3)

■

One of the ﬁrst models of atomic structure, the Bohr

model, quantized the energies and spatial locations

of the electrons to explain the hydrogen emission

spectrum. Although the model was not correct in

other details, quantum energy and orbit locations

were breakthrough concepts and are key concepts in

the modern picture of the atom. (Section 6.4)

■

Electronic transitions between the quantized energy

levels are responsible for atomic absorption and emis-

sion spectra. The energies are given by



E = E

− E

where the subscripts i and f stand for initial and ﬁnal

states. (Section 6.4)

■

The energy of electromagnetic radiation absorbed or

emitted in an electronic transition must exactly match

the energy of the transition. (Section 6.4)

■

At the atomic scale, all things show both wave and

particle behavior. This concept is known as wave–

particle duality. (Section 6.5)

■

The wave and particle natures of quantum objects,

such as electrons and photons, are linked in the de

Broglie equation: λ = h/p. (Section 6.6)

■

The Heisenberg uncertainty principle,



p ≥ h/4π,

places limits on how precisely we can simultaneously

measure position and momentum. (Sections 6.7

and 6.8)

■

All quantum systems, like the hydrogen atom, can be

completely described by the Schrödinger equation:

H

= E



. (Section 6.9)

■

The probability distribution function 

∗



tells us

where to ﬁnd electrons in the atom, and the shape

described by this function is called the atomic or-

bital. (Section 6.9)

■

The orbitals are associated with the probability of

ﬁnding the electron within a certain region of space.

(Section 6.9)

■

Each electron wave function has a set of four quan-

tum numbers associated with it. They are n, the

principal quantum number; l, the angular momen-

tum quantum number; m

, the magnetic quantum

number; and m

, the electron spin quantum number.

(Section 6.10)

■

The principal quantum number n indicates the elec-

tron shell that an electron is in, and n, l, and m

de-

termine the shape and orientation of the orbital

in three-dimensional space. (Section 6.10)

■

Only two possible electron spin states exist:

and m

=−

. (Section 6.10)

■

The Pauli exclusion principle states that in a given

atom, no two electrons can have the same set of

the four quantum numbers − n, l, m

, and m

(Section 6.11)

■

We can envisage multielectron atoms being assem-

bled by placing electrons into the orbitals one by

one, starting with the lowest energy level, according

to the Aufbau principle. The complete listing of oc-

cupied orbitals is called the ground-state electron

conﬁguration. (Section 6.13)

■

Hund’s rule for ground-state electron conﬁgurations

states that when orbitals of equal energy are available,

the lowest energy conﬁguration for an atom is the

one with the maximum number of unpaired elec-

trons with parallel spins. (Section 6.13)

Key Words

absorption spectroscopy The measurement of how atoms

and molecules absorb electromagnetic radiation as a

function of the wavelength or frequency of the radia-

tion. (p. 216)

amplitude The distance between the highest point and

the midpoint (zero) of a wave. (p. 212)

angstrom A unit of distance equal to 10

−10

m. It is sym-

bolized by Å. (p. 211)

angular As a function of the angles that describe the

orientation of the radius in spherical polar coordi-

nates. (p. 234)

angular momentum quantum number The quantum

number, l, that describes the shape of the orbital;

l can be any whole-number value from zero to n −1.

(p. 235)

252 Chapter 6 Quantum Chemistry: The Strange World of Atoms

Aufbau principle As protons are added to the nucleus, in-

creasing the atomic number of the atom, electrons

are added successively to the next highest energy or-

bital. The method of ﬁlling hydrogen atomic orbitals

to get the ground-state electron conﬁguration of a

multielectron atom by adding the electrons one at a

time until all are accommodated; from the German

phase for “building up.” (p. 245)

classical mechanics The physical description of macro-

scopic behavior based on Newton’s equations of

motion. (p. 225)

core electrons Electrons in the conﬁguration of an atom

that are not in the highest principal energy shell.

(p. 247)

d orbital An orbital with quantum number l = 2.

(p. 240)

degenerate orbitals Orbitals that have equal energies.

(p. 236)

electromagnetic spectrum The entire range of radiation

as a function of wavelength, frequency, or energy.

(p. 211)

electron conﬁguration The complete list of ﬁlled orbitals

in an atom. (p. 245)

electron shielding The ability of electrons in lower

energy orbitals to decrease the nuclear charge felt by

electrons in higher energy orbitals. (p. 244)

electron spin quantum number The quantum number, m

that describes the spin of an electron. Also known as

the spin angular momentum quantum number.

(p. 236)

electron subshell The energy level occupied by electrons

that share the same values for both n and l.(p. 235)

electronic transition A change in atomic or molecular

energy level made by an electron bound in an atom

or molecule. (p. 221)

emission spectroscopy The measurement of how atoms

and molecules give off electromagnetic radiation as a

function of the wavelength or frequency of the radia-

tion. (p. 216)

emission spectrum A plot of the intensity of radiation as

a function of wavelength or frequency in an emission

experiment. (p. 217)

empirically derived Derived from experiments and ob-

servations rather than from theory. (p. 219)

energy levels The allowed orbits that electrons may

occupy in an atom. (p. 221)

excited state Any higher energy state than the ground

state characterized by the existence of an electron in

an orbital that violates Hund’s Rule and/or the

Aufbau principle. (p. 221)

f orbital An orbital with quantum number l = 3.

(p. 241)

frequency A descriptor of electromagnetic radiation.

Deﬁned as the number of waves that pass a given

point per second, in units of 1/s (s

−1

). (p. 211)

ground state The lowest energy state of an atom.

(p. 221)

Hamiltonian operator A mathematical function that is

used in the Schrödinger equation. (p. 232)

Heisenberg uncertainty principle There is an ultimate un-

certainty in the position and the momentum of a

particle. Reducing the uncertainty in one increases

the uncertainty in the other. (p. 230)

Hund’s rule When orbitals of equal energy are available,

the lowest energy conﬁguration for an atom has the

maximum number of unpaired electrons with paral-

lel spins. (p. 247)

laser An acronym for “light ampliﬁcation by stimulated

emission of radiation.” (p. 224)

magnetic quantum number The quantum number, m

that describes the orientation of the orbital; m

can be

any whole-number value from −l to +l. Also known

as the orbital angular momentum quantum number.

(p. 235)

molecular orbitals Electron orbitals that are appropriate

for describing bonding between atoms in a molecule.

(p. 244)

node A point in the wave function where the amplitude

is always zero. (p. 233)

orbital The volume of space to which the electron is re-

stricted when it is in a bound atomic or molecular

energy level. (p. 233)

p orbital An orbital with quantum number l = 1.

(p. 239)

Pauli exclusion principle In a given atom, no two elec-

trons can have the same set of the four quantum

numbers − n, l, m

, and m

.(p. 242)

photon A single unit of electromagnetic radiation.

(p. 210)

Planck’s constant A fundamental constant equal to

6.62608 × 10

−34

s. (p. 215)

principal quantum number The quantum number, n, that

describes the energy level of the orbital; n can be any

whole-number value from 1 to inﬁnity. (p. 234)

principal shell The energy level that is occupied by elec-

trons with the same value for the principal quantum

number (n). (p. 234)

quantum A single unit of matter or energy. (p. 210)

quantum chemistry The study of chemistry on the atomic

and molecular scale. (p. 210)

quantum mechanics The physical laws governing energy

and matter at the atomic scale. (p. 210)

quantum number A number used to arrive at the solu-

tion of an acceptable wave function that describes the

properties of a speciﬁc orbital. (p. 218)

radial Along the radius in spherical polar coordinates.

(p. 234)

rest mass The mass a particle has when it is stationary.

(p. 231)

orbital An orbital with quantum number l =0.

(p. 238)

Schrödinger equation The mathematical expression that

relates the wave function to the energy in a quantum

system. (p. 232)

Focus Your Learning 253

Focus Your Learning

The answers to the odd-numbered problems and some selected

problems appear at the back of the book, as represented by the

blue numbering.

Section 6.1 Introducing Quantum Chemistry

Skill Review

1. At the submicroscopic level we typically discuss the probabil-

ity of events and objects rather than their location and

individual speeds. Describe one event or object that is best

explained using probability. Is this event considered “macro”

or “micro”?

2. Use the analogy of ﬂipping a coin to explain the term

probability.

Section 6.2 Electromagnetic Radiation

Skill Review

3. As you walk through the produce section of a supermarket,

you will see various colors of vegetables. Say the grocer

decided to organize the vegetables in order of increasing fre-

quencies of their reﬂected light—a novel marketing concept.

Indicate in what sequence these vegetables would appear: a

red tomato, a green zucchini squash, a purple eggplant, and

a yellow spaghetti squash.

4. The three colors of a trafﬁc light are red, yellow, and green.

The green light is placed at the bottom, with yellow in the

center and red on top. Are these colors in order, from bottom

to top, by frequency or by wavelength? Justify your answer.

5. a. What is the frequency of an X ray with a wavelength of

1.5 × 10

−2

nm?

b. What is the energy, in joules, associated with a photon of

this frequency?

c. What would be the energy of a mole of such photons?

6. a. What is the frequency of visible light with a 400-nm

wavelength?

b. What is the energy, in joules, associated with a photon of

this frequency?

c. What would be the energy associated with a mole of these

photons?

7. In what region on the electromagnetic radiation spectrum

would a wavelength of 2.5 × 10

nm be placed? What would

be the frequency and energy of this radiation?

8. In what region on the electromagnetic radiation spectrum

would a wavelength of 5.8 × 10

nm be placed? What would

be the frequency and energy of this radiation?

9. Calculate your height in nanometers, meters, and light-years.

Which do you ﬁnd the most convenient unit for this

application?

10. Calculate the length of a 12-in ruler in nanometers, meters,

and lightyears. Which do you ﬁnd the most convenient unit

for this application?

11. Cell phones operate at frequencies of 824 to 894 MHz. What

range of wavelengths is this? To which part of the electro-

magnetic spectrum does this correspond?

12. Helium–neon (HeNe) lasers are both cheap and common.

They are even sold as attachments to novelty key chains. If

the helium–neon mixture lases at 0.632 µm, what is its fre-

quency? To which part of the electromagnetic spectrum does

this correspond?

13. Radio stations broadcast with frequencies given in mega-

hertz, where 1 MHz = 10

−1

. This means that when a sta-

tion advertises itself as Radio WXYZ located at 98.3 on your

dial, it is broadcasting at 98.3 MHz. What wavelength does

Radio WXYZ use in its broadcast?

14. A wireless Internet connection broadcasts its signal in the

1200-MHz range. To what wavelength does this correspond?

15. Chlorophyll, the green pigment found in plants, absorbs visible

radiation best in the red region (at about 675 nm) and in the

blue-violet region (at about 440 nm). What are the energies of

the photons collected by plants at these two wavelengths?

16. What are the energies of photons emitted from a 100-MHz

magnetic resonance imager?

17. How much energy resides in 1 mol of photons whose wave-

length is 440 nm? . . . 675 nm?

18. How much energy resides in 2 mol of photons with a fre-

quency of 1.5 × 10

Hz? . . . in 0.75 mol?

spectroscopy

The measurement of how atoms and mol-

ecules interact with electromagnetic radiation as a

function of the wavelength or frequency of the radia-

tion. (p. 216)

standing wave The constructive interference of two or

more waves that results in the presence of nodes in

ﬁxed locations. (p. 226)

valence electrons The electrons in an atom, ion, or mol-

ecule that are in the highest principal energy shell.

(p. 247)

wave function The mathematical equation describing a

system, such as an electron, atom, or molecule, that

contains all physical information that can be ob-

tained for the system by quantum mechanics.

(p. 232)

wave–particle duality The quantum mechanical theorem

that states that all things have both wave and particle

natures simultaneously and that both of these natures

can be observed on the atomic scale. (p. 227)

wavelength A descriptor of electromagnetic radiation.

Deﬁned as the distance from the top of one crest to

the top of the next crest of the electromagnetic wave.

(p. 211)

254 Chapter 6 Quantum Chemistry: The Strange World of Atoms

Chemical Applications and Practices

19. a. The process of photosynthesis is quite complex. However,

one of the main considerations is that plant pigments

absorb visible light to power reactions that convert carbon

dioxide and water into food and produce oxygen. If a plant

absorbs blue light that has a wavelength of 565 nm, what is

the energy per photon that is being absorbed?

b. A typical ratio between photons absorbed and oxygen

molecules produced is 8:1. How much energy is required

to produce one molecule of oxygen in this manner?

20. Ozone (O

) is important in our upper atmosphere because it

aids in ﬁltering out harmful ultraviolet rays. Ultraviolet rays

may be classiﬁed as either UV-A, UV-B, or UV-C. The UV-B

rays cause the most problems for earth-based organisms. For

example, higher incidences of “jumping genes” that cause

mutations may be related to exposure to UV-B.

a. What is the energy in 1 mol of UV-B photons that have a

wavelength of 312 nm?

b. What is the energy of 1 mol of photons with a wavelength

of 600 nm? To what type of radiation does this correspond?

21. UV-B radiation is responsible for “sunburn” in humans. A

helpful advancement in technology is the personal UV detec-

tor. This small device uses a photoelectric response. An ex-

ample is a gallium-based device to convert absorbance into

an electrical signal. If the device gave a maximum reading at

290 nm, what energy is being absorbed? To what frequency

does this correspond?

22. a. Some snakes have the ability to detect infrared radiation

(IR). Are they detecting energy that is higher or lower in

energy than human eyes can see?

b. What is the source of typical IR wavelengths?

c. A television remote control may use IR with a frequency

of 1 × 10

cycles per second. To what energy does this

correspond?

23. One way to gain information about the origin and functions

of stars within the universe is to study the origin and distrib-

ution of the hundreds of gamma ray sources in the sky. If one

such source were producing high-energy gamma rays of

1.6 × 10

−8

J, what wavelength would astronomers have

detected? What is the frequency of this radiation?

24. If an astronomer recorded energy from a distant star at 3.6 ×

−10

J, what wavelength would he have detected? What is the

frequency of this radiation?

Section 6.3 Atomic Emission and Absorption

Spectroscopy, Chemical Analysis and the Quantum

Number

Skill Review

25. When bombarded with high-energy electrons, copper metal

gives off radiation with a wavelength of 1.54 Å. What is the

frequency of this radiation? To which range of the electro-

magnetic spectrum does this correspond?

26. Sodium arc lamps, which are used as automobile headlights

and street lights, are colored by the sodium doublet: electro-

magnetic radiation produced by excited sodium atoms found

at 5895 Å and 5904 Å. What color are these lights?

27. Much of the radiation striking the earth from the sun has a

wavelength of approximately 500 nm. Express this wave-

length in meters, angstroms, centimeters, and inches.

28. Express 280-nm ultraviolet radiation in meters, angstroms,

centimeters, and inches.

29. According to the Balmer equation, the wavelength of emitted

light from hydrogen can be calculated from the whole-

number values of n. The difference between n = 5 and n = 4

is only 1. The difference between n = 2 and n = 1 is also

only 1. Why don’t the two conditions produce the same

wavelength of emitted light in hydrogen?

30. The Balmer equation can be used to calculate the wavelength

of light emitted from excited hydrogen. To what initial value

of n in hydrogen would an emitted wavelength of 5547 Å cor-

respond? Explain why this value of n is never noted for

hydrogen.

31. The ﬁrst two wavelengths of the Balmer series of the hydro-

gen emission spectrum are 6562.1 Å, 4860.8 Å. What are the

next three values in this series? What are the frequencies and

energies of the emission lines?

32. What is the highest frequency of the Balmer series of the

hydrogen emission spectrum? What is the lowest frequency

of this series?

Chemical Applications and Practices

33. The presence of cadmium in drinking water is undesirable

because exposure to large amounts has been associated with

weakening of bones and joints. The wavelength of electromag-

netic radiation strongly absorbed by cadmium is 214.439 nm.

a. What is the frequency of that light?

b. In what range of the electromagnetic spectrum would you

classify this frequency?

c. Which other element has an absorbance wavelength

closest to (and therefore possibly difﬁcult to distinguish

from) cadmium? Consult Table 6.3 for additional

information.

34. An adaptation of the Balmer equation makes it possible to

calculate other emitted wavelengths from excited hydrogen.

For example, if the ﬁnal value for n is 3, then emitted light is

in the infrared area of the spectrum. These line spectra are

known as the Paschen series. Calculate the wavelength emit-

ted when an electron in hydrogen drops from the fourth

Bohr level to the third.

Section 6.4 The Bohr Model of Atomic Structure

Skill Review

35. Calculate the energy of an electron in each of these Bohr

energy levels:

a. n = 1b.n = 3c.n = 5d.n = 7

36. Calculate the energy of an electron in each of these Bohr

energy levels:

a. n = 2b.n = 4c.n = 6d.n = 8

37. Calculate the energy a photon released from a hydrogen atom

in each of these transitions:

a. n = 4 to n = 1c.n = 5 to n = 4

b. n = 3 to n = 1d.n = 7 to n = 2

38. Calculate the energy a photon released from a hydrogen atom

in each of these transitions:

a. n = 2 to n = 1c.n = 4 to n = 3

b. n = 4 to n = 2d.n = 8 to n = 2

n = 6

n = 5

n = ∞

n = 4

n = 3

n = 2

n = 1

n = 6

n = 5

n = ∞

n = 4

n = 3

n = 2

n = 1

n = 6

n = 5

n = ∞

n = 4

n = 3

n = 2

n = 1

n = 6

n = 5

n = ∞

n = 4

n = 3

n = 2

n = 1

Focus Your Learning

255

39. Calculate the wavelength of a photon that would cause these

transitions:

a. n = 1 to n = 2c.n = 2 to n = 4

b. n = 3 to n = 5d.n = 1 to n = 6

e. f.

40. Calculate the frequency of a photon that would cause these

transitions:

a. n = 8 to n = 10 c. n = 4 to n = 8

b. n = 3 to n = 6d.n = 4 to n = 5

e. f.

41. What is the shortest wavelength that can be emitted by the

hydrogen atom in the Brackett spectral series? What is the

longest wavelength in this series? Consult Table 6.2 for assis-

tance with this problem.

42. According to the Bohr model of the hydrogen atom, what is

the closest an electron can get to the nucleus? What is the far-

thest it can get from the nucleus? (If the latter answer seems

absurd, it might amuse you to know that it is the same value

predicted by present-day quantum chemistry.)

Chemical Applications and Practices

43. The emission spectrum being given out by a mixture of gases,

possibly including hydrogen, includes an emission line with

an energy of 4.84 × 10

−19

J per photon. Is this emission pos-

sible for hydrogen? (To make this problem more manageable,

consider transitions within the ﬁrst eight energy levels only.)

44. The emission spectrum for a speciﬁc sample of gas includes

an emission line with an energy of 7.38 ×10

−19

J per photon.

Is this emission possible for hydrogen? (To make this prob-

lem more manageable, consider transitions within the ﬁrst

eight energy levels only.)

45. To ionize hydrogen is to remove its single electron. We con-

sider a free electron to have, with respect to an electron in

hydrogen, zero energy. In other words, the initial state is n = 1

and the ﬁnal state is n = ∞. Calculate the energy needed to

ionize an electron from the ground state.

46. Calculate the energy released when an electron is added to a

hydrogen nucleus. Assume the transition is n = ∞ to n = 1.

Section 6.5 Wave–Particle Duality

Skill Review

47. In a classical sense, compare matter and waves in terms of

permissible energy values, spatial positions, and momentum.

48. How did the quantum view change the classical distinction

between matter and waves? What is the quantum view called?

Chemical Applications and Practices

49. What is the momentum of an electron that is moving at 68%

of the speed of light? (Use 9.1 × 10

−31

kg as the mass of the

electron.)

50. What is the speed of an electron that has a momentum of

9.7 × 10

−23

kg ·m · s

−1

51. A freight train locomotive weighs about 415,000 lb and has a

top speed of about 100.0 mi/h. What is its momentum?

52. The average speed of an electron in the ground state of

the hydrogen atom is 2.19 × 10

m/s. What is the average

momentum of an electron in this state?

53. What is the momentum of a photon with a wavelength of

540 nm?

54. What is the momentum of a photon with a frequency of

1.0 × 10

Hz?

Section 6.6 Why Treating Things as “Waves”

Allows Us to Quantize Their Behavior

Skill Review

55. List two particle-like properties of electrons.

56. List two wave-like properties of electrons.

Chemical Applications and Practices

57. The average radius in the second Bohr orbit (n =2) is 2.116 ×

−10

m. Using the equation 2πr = nλ, calculate the wave-

length for the standing electron wave.

58. The approximate wavelength for an electron in hydrogen’s

ﬁrst energy level has been calculated to be 3.3 × 10

−10

What is the ratio of this wavelength to the diameter of ten

hydrogen atoms (7.4 × 10

−11

m)? What might be some rea-

sons why the diameter isn’t closer in size to the wavelength?

59. In badminton, the object being struck by the racket is called

a shuttlecock. Although the shut-

tlecock may be made of various

materials, it mass must be close

to 5.00 g. What is the wavelength

of a served shuttlecock that is

moving at 78 mi/h? Without

doing exact calculations, indicate

whether the wavelength of a soft-

ball moving at the same speed

would be larger or smaller.

A shuttlecock.

256 Chapter 6 Quantum Chemistry: The Strange World of Atoms

60. If a proton, mass = 1.67 × 10

−27

kg, were moving as fast as

the electron in the ground state of hydrogen (2.1 × 10

m/s),

what would be the wavelength?

Section 6.7 The Heisenberg Uncertainty Principle

Skill Review

61. In the uncertainty relationship, what do the symbols p and x

represent? Explain the importance of noting that ∆p and ∆x

multiplied must be equal to or greater than a constant.

62. Taking a photograph of a moving object—for example, a

person sprinting—will cause some blurring of the actual

person’s position when the photograph is developed. There-

fore, the more blur, the better you represent movement. If the

entire scene were re-shot using a faster shutter speed, what

information would you gain and what would you lose in the

photo?

Chemical Applications and Practices

63. Using the mass of an electron as 9.11 ×10

−31

kg and velocity

as 2.1 ×10

m/s, what would you calculate as the uncertainty

in the position,

x

, for the electron if the uncertainty in the

velocity were 5.0%? What would be the answer to this if

the uncertainty in the velocity were 10.0%? How does this

uncertainty compare to the radius of the hydrogen atom

(3.7 × 10

−11

m)?

64. Which of these diagrams of an atom most agrees with the

Heisenberg uncertainty principle? Explain your answer, and

indicate why the other two choices do not conform to the

Heisenberg uncertainty principle.

a. b. c.

Section 6.8 More About the Photon—the de Broglie

and Heisenberg Discussions

Skill Review

65. What is the momentum of an X-ray that has a wavelength of

6.5 × 10

−10

66. How does the momentum of the X-ray in Problem 65 com-

pare to the momentum of a photon of green light that has a

wavelength of 560 nm?

67. What is the momentum of a mole of X-rays with wavelength

6.5 × 10

−10

68. What is the momentum of a mole of photons with wave-

length 560 nm?

Chemical Applications and Practices

69. What is the momentum of a 190-lb chemistry professor

walking through a chemistry lab at 3.0 mi/h?

70. In order for you to see the strolling professor of Problem 69,

light photons would have to bounce off the professor and

enter your eyes. If some of that light had a wavelength of

565 nm, what would be the momentum of the photon?

Section 6.9 The Mathematical Language of Quantum

Chemistry

Skill Review

71. Acceptance of the Schrödinger equation literally knocked the

electrons out of Bohr’s orbit concept. However, the term

orbital was maintained as a connection to the quantization of

energy states proposed in Bohr’s model. Explain the meaning

of an electron orbital.

72. a. What is meant by the term node when it is applied to

waves?

b. What is the relationship among nodes, wavelength, and

energy in the context of standing waves?

Chemical Applications and Practices

73. In both the Bohr model of electronic structure and the

Schrödinger model, electrons appear to change energy levels

by a method that does not show the electron making a grad-

ual transition. Using the wave model, explain why a stable

standing wave is not possible between two adjacent energy

levels. Does this explain why there is no gradual transition

between states?

74. a. What is a photon?

b. In what way does a photon resemble particles, and in what

way does it resemble light?

Section 6.10 Atomic Orbitals

Skill Review

75. The sequence in each line that follows represents values for

the quantum numbers for an electron in a hydrogen atom.

Select any sequence(s) that are not possible and explain the

problem(s).

nl m

a. 3 −10

b. 2 +2 +1

c. 3 +2 +3

d. 1 +1 +1

e. 4 +3 −2

76. The sequence in each line that follows represents values for

the quantum numbers for an electron in a hydrogen atom.

Select any sequence(s) that are not possible and explain the

problem(s).

nl m

a. 3 +10

b. 2 +2 +1

c. 3 0 −3

d. 5 +40

e. 2 +3 −2

Electron

Nucleus

Diffuse cloud of electrons

Focus Your Learning 257

77. When an electron is in the ﬁfth energy level, how many sub-

levels are possible? How many orbitals are possible?

78. When an electron is in the fourth energy level, how many

sublevels are possible? How many orbitals are possible?

79. When l = 3, how many degenerate orbitals are possible?

80. When l = 6, how many degenerate orbitals are possible?

81. How many sets of quantum numbers are possible when

n = 5?

82. How many sets of quantum numbers are possible when

n = 3?

83. We often use the phrase the shape of an orbital. Indicate what

restrictions apply when the phrase is used, and explain why

the phrase is a bit of an abstraction.

84. Name two distinguishing features that the three p orbitals in

the same level have in common. What property allows them

to be identiﬁed separately?

85. Deﬁne and give an example of both a radial node and a

planar node.

86. In which energy level would the f orbitals make their ﬁrst

appearance? (Use a quantum mechanical proof for your

answer.)

Chemical Applications and Practices

87. a. The explanation for scanning tunneling microscopic pic-

tures relies on the concept presented in the quantum me-

chanical atomic model. When the tip of the STM probe

nears an atom, what actually touches?

b. What is meant by quantum tunneling?

88. The “photograph” that is produced by a scanning tunneling

microscope is not a photograph in the traditional sense. In

what way does the image differ from a traditional photo-

graph?

Section 6.11 Electron Spin and the Pauli Exclusion

Principle

Skill Review

89. The four quantum numbers representing an electron may be

shown as: (3, 1, 0, ). What is the value of the fourth quan-

tum number? What is the name given to the fourth quantum

number?

90. If the four quantum numbers for an electron were (3, 2, 1,

−

) what would be the four quantum numbers for an elec-

tron in the same orbital as the ﬁrst electron?

Chemical Applications and Practices

91. The magnetic properties of elements are related to the

number of unpaired electrons in the atoms. Of the following,

which would have unpaired electrons?

CCaONeZn

92. The magnetic properties of elements are related to the

number of unpaired electrons in the atoms. Of the following,

which would have unpaired electrons?

NSNaHeSc

93. If the Pauli exclusion principle were not used, show how the

conﬁguration of the electrons in oxygen would appear to

allow no unpaired electrons.

→

94. Among the elements from 21 to 30, which would have the

highest number of unpaired electrons?

Section 6.12 Orbitals and Energy Levels in Multielectron

Atoms

Skill Review

95. Name three things that will be the same in multielec-

tron atoms’ orbital descriptions, and name one critical

difference.

96. How many radial and how many nonradial nodes, respec-

tively, will be possible for each of these electron orbitals?

What will be the letter designation for each of the repre-

sented orbitals?

a. n = 3; l = 2b.n = 3; l = 0c.n = 4; l = 3

Chemical Applications and Practices

97. Would you expect it to be easier to remove the outermost

electron from He or He

? Explain the basis of your answer.

98. Would an electron in the 3p sublevel experience more nu-

clear pull than an electron in the 3d sublevel? (Assume that

the electron is in the same atom and that the atom’s sub-

levels up to 3s are ﬁlled.)

Section 6.13

Electron Conﬁgurations and the Aufbau Principle

Skill Review

99. a. What is the written notation for the ground state of the

nitrogen atom?

b. Nitrogen commonly forms a −3 ion. What is the electron

conﬁguration of the ion?

100. In the ground state of manganese, there are ﬁve electrons

that would occupy the 3d sublevel. Show the way these elec-

trons could be conﬁgured to follow Hund’s rule and a way

that would violate Hund’s rule.

101. Report, for each of the following, which element is being

represented.

a. [Ne]3s

b. [Ne]3s

c. [Ar]4s

d. [Kr]5s

102. Report, for each of the following, which element is being

represented.

a. [Ne]3s

b. [He]2s

c. [He]2s

d. [Ar]4s

103. Write the electron conﬁguration for element 21. Explain

why the correct conﬁguration shows the 4s sublevel ﬁlling

with electrons before the 3d sublevel.

104. Write out the ground-state electron conﬁguration for ele-

ment 19.

Chemical Applications and Practices

105. Which, if any, of the following contain unpaired electrons

in their ground state?

KCaFeZnNe

106. Which, if any, of the following contain unpaired electrons?

107. Evidence indicates that copper has no unpaired electrons

in the 3d sublevel. What would have to be the ground-state

electron conﬁguration of copper to make this possible?