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Physical Chemistry

Thomas Engel, Philip Reid

Chapter 22

Quantum States for Many-Electron Atoms and Atomic Spectroscopy - all with Video Answers

Educators

Chapter Questions

01:31

Problem 1

The principal line in the emission spectrum of potassium is violet. On close examination, the line is seen to be a doublet with wavelengths of 393.366 and $396.847 \mathrm{nm}$ Explain the source of this doublet.

Keshav Singh
Keshav Singh
Numerade Educator
02:10

Problem 2

The absorption spectrum of the hydrogen atom shows lines at $5334,7804,9145,9953,$ and $10,478 \mathrm{cm}^{-1} .$ There are no lower frequency lines in the spectrum. Use the graphical methods discussed in Example Problem 22.6 to determine $n_{\text {initial }}$ and the ionization energy of the hydrogen atom in this state. Assume values for $n_{\text {initial}}$ of $1,2,$ and 3.

Crystal Wang
Crystal Wang
Numerade Educator
01:59

Problem 3

Using Table 22.3, which lists the possible terms that arise from a given configuration, and Hund's rules, write the term symbols for the ground state of the atoms $\mathrm{H}$ through $\mathrm{F}$ in the form $^{(2 S+1)} L_{J}$.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
02:28

Problem 4

In this problem, you will supply the missing steps in the derivation of the formula $E_{\text {singlet}}=E_{1 s}+E_{2 s}+J+K$ for the singlet level of the $1 s^{1} 2 s^{1}$ configuration of He.
a. Expand Equation (22.17) to obtain
$$\begin{aligned}
E_{\text {singlet}}=\frac{1}{2} \iint[1 s(1) 2 s(2)+2 s(1) 1 s(2)]\left(\hat{H}_{1}\right) \\
+\frac{1}{2} \iint_{[1 s(1) 2 s(2)+2 s(1) 1 s(2)] d \tau_{1} d \tau_{2}}(10(1) 2 s(2)+2 s(1) 1 s(2)]\left(\hat{H}_{2}\right) \\
+\frac{1}{2} \iint[1 s(1) 2 s(2)+2 s(1) 1 s(2)] \times \\
&\left(\frac{e^{2}}{4 \pi \varepsilon_{0}\left|r_{1}-r_{2}\right|}\right) \times \\
&[1 s(1) 2 s(2)+2 s(1) 1 s(2)] d \tau_{1} d \tau_{2}
\end{aligned}$$
b. Starting from the equations $\hat{H}_{i} 1 s(i)=E_{1 s} 1 s(i)$ and $\hat{H}_{i} 2 s(i)=E_{2 s} 2 s(i),$ show that $E_{\text {singlet}}=E_{1 s}+E_{2 s}$
$$\begin{array}{c}
+\frac{1}{2} \iint[1 s(1) 2 s(2)+2 s(1) 1 s(2)]\left(\frac{e^{2}}{4 \pi \varepsilon_{0}\left|r_{1}-r_{2}\right|}\right) \times \\
{[1 s(1) 2 s(2)+2 s(1) 1 s(2)] d \tau_{1} d \tau_{2}}
\end{array}$$
c. Expand the previous equation using the definitions
$$\begin{array}{c}
J=\frac{e^{2}}{8 \pi \varepsilon_{0}} \iint[1 s(1)]^{2}\left(\frac{1}{\left|r_{1}-r_{2}\right|}\right)[2 s(2)]^{2} d \tau_{1} d \tau_{2} \text { and } \\
K=\frac{e^{2}}{8 \pi \varepsilon_{0}} \iint[1 s(1) 2 s(2)]\left(\frac{1}{\left|r_{1}-r_{2}\right|}\right)[1 s(2) 2 s(1)] \times
\end{array}$$
to obtain the desired result, $E_{\text {singlet}}=E_{1 s}+E_{2 s}+J+K$.

Adriano Chikande
Adriano Chikande
Numerade Educator
01:28

Problem 5

What $J$ values are possible for a $^{6}$ H term? Calculate the number of states associated with each level and show that the total number of states is the same as that calculated from the term symbol.

Narayan Hari
Narayan Hari
Numerade Educator
01:59

Problem 6

Using Table $22.3,$ which lists the possible terms that arise from a given configuration, and Hund's rules, write the configurations and term symbols for the ground state of the ions $\mathrm{F}^{-}$ and $\mathrm{Ca}^{2+}$ in the form $^{(2 S+1)} L_{J}$.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
08:08

Problem 7

The Doppler broadening in a gas can be expressed as $\Delta \nu=\left(2 \nu_{0} / c\right) \sqrt{2 \ln 2(R T / M)},$ where $M$ is the molar mass.
For the sodium $3 p^{2} \mathrm{P}_{3 / 2} \longrightarrow 3 s^{2} \mathrm{S}_{1 / 2}$ transition, $\nu_{0}=$
$5.0933 \times 10^{14} \mathrm{s}^{-1} .$ Calculate $\Delta \nu$ and $\Delta \nu / \nu_{0}$ at $500.0 \mathrm{K}$.

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
03:24

Problem 8

Calculate the transition dipole moment, $\mu_{z}^{m n}=\int \psi_{m}^{*}(\tau) \mu_{z} \psi_{n}(\tau) d \tau$ where $\mu_{z}=-e r \cos \theta$ for a transition from the 1 s level to the $2 p_{z}$ level in H. Show that this transition is allowed. The integration is over $r, \theta,$ and $\phi .$ Use
$$\psi_{210}(r, \theta, \phi)=\frac{1}{\sqrt{32 \pi}}\left(\frac{1}{a_{0}}\right)^{3 / 2} \frac{r}{a_{0}} e^{-r / 2 a_{0}} \cos \theta$$
for the $2 p_{z}$ wave function.

Narayan Hari
Narayan Hari
Numerade Educator
03:48

Problem 9

Consider the 1 s $n p^{3} \mathrm{P} \rightarrow 1$ s nd $^{3}$ D transition in He. Draw an energy-level diagram, taking the spin-orbit coupling that splits terms into levels into account. Into how many levels does each term split? The selection rule for transitions in this case is $\Delta J=0, \pm 1 .$ How many transitions will be observed in an absorption spectrum? Show the allowed transitions in your energy diagram.

Suzanne W.
Suzanne W.
Numerade Educator
01:45

Problem 10

Atomic emission experiments of a mixture show a calcium line at $422.673 \mathrm{nm}$ corresponding to a $^{1} \mathrm{P}_{1} \rightarrow^{1} \mathrm{S}_{0}$
transition and a doublet due to potassium $^{2} \mathrm{P}_{3 / 2} \rightarrow^{2} \mathrm{S}_{1 / 2}$ and $^{2} \mathrm{P}_{1 / 2} \rightarrow^{2} \mathrm{S}_{1 / 2}$ transitions at 764.494 and $769.901 \mathrm{nm}$ respectively.
a. Calculate the ratio $g_{\text {upper}} /$g$_{\text {lower}}$ for each of these transitions.
b. Calculate $n_{\text {upper}} / n_{\text {lower}}$ for a temperature of $1600^{\circ} \mathrm{C}$ for each transition.

Ajay Singhal
Ajay Singhal
Numerade Educator
01:02

Problem 11

How many ways are there to place three electrons into an $f$ subshell? What is the ground-state term for the $f^{3}$ configuration, and how many states are associated with this term? See Problem P22.36.

Narayan Hari
Narayan Hari
Numerade Educator
04:00

Problem 12

Calculate the wavelengths of the first three lines of the Lyman, Balmer, and Paschen series, and the series limit (the shortest wavelength) for each series.

Narayan Hari
Narayan Hari
Numerade Educator
04:21

Problem 13

The Lyman series in the hydrogen atom corresponds to transitions that originate from the $n=1$ level in absorption or that terminate in the $n=1$ level for emission. Calculate the energy, frequency (in inverse seconds and inverse centimeters), and wavelength of the least and most energetic transition in this series.

Keshav Singh
Keshav Singh
Numerade Educator
02:56

Problem 14

The inelastic mean free path of electrons in a solid, $\lambda,$ governs the surface sensitivity of techniques such as AES and XPS. The electrons generated below the surface must make their way to the surface without losing energy in order to give elemental and chemical shift information. An empirical expression for elements that give $\lambda$ as a function of the kinetic energy of the electron generated in AES or XPS is $\lambda=538 E^{-2}+0.41(l E)^{0.5} .$ The units of $\lambda$ are monolayers, $E$ is the kinetic energy of the electron in eV, and $l$ is the monolayer thickness in nanometers. On the basis of this equation, what kinetic energy maximizes the surface sensitivity for a monolayer thickness of $0.3 \mathrm{nm} ?$ An equation solver would be helpful in obtaining the answer.

Chai Santi
Chai Santi
Numerade Educator
09:14

Problem 15

The effective path length that an electron travels before being ejected into the vacuum is related to the depth below the surface at which it is generated and the exit angle by $d=\lambda \cos \theta,$ where $\lambda$ is the inelastic mean free path and $\theta$ is the angle between the surface normal and the exit direction.
a. Justify this equation based on a sketch of the path that an electron travels before exiting into the vacuum.
b. The XPS signal from a thin layer on a solid surface is given by $I=I_{0}\left(1-e^{-d /(\lambda \cos \theta)}\right),$ where $I_{0}$ is the signal that would be obtained from an infinitely thick layer, and $\lambda$ is defined in Problem $\mathrm{P} 22.14 .$ Calculate the ratio $I / I_{0}$ at $\theta=0$ for $\lambda=2 d .$ Calculate the exit angle required to increase $I / I_{0}$ to 0.50.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
01:09

Problem 16

List the allowed quantum numbers $m_{l}$ and $m_{s}$ for the following subshells and determine the maximum occupancy of the subshells:
a. $2 p$
b. $3 d$
c. $4 f$
d. $5 g$

Crystal Wang
Crystal Wang
Numerade Educator
01:03

Problem 17

What are the levels that arise from the following terms? How many states are there in each level?
a. $^{4} \mathrm{F}$
b. $^{2} \mathrm{D}$
$\mathbf{c} .^{2} \mathrm{S}$
d. $^{4} \mathrm{P}$

David Collins
David Collins
Numerade Educator
07:20

Problem 18

As discussed in Chapter 20, in a more exact solution of the Schrödinger equation for the hydrogen atom, the coordinate system is placed at the center of mass of the atom rather than at the nucleus. In that case, the energy levels for a one-electron atom or ion of nuclear charge $Z$ are given by
$$E_{n}=-\frac{\mathrm{Z}^{2} \mu e^{4}}{32 \pi^{2} \varepsilon_{0}^{2} \hbar^{2} n^{2}}$$
where $\mu$ is the reduced mass of the atom. The masses of an electron, a proton, and a tritium ( $^{3} \mathrm{H}$ or $\mathrm{T}$ ) nucleus are given by $9.1094 \times 10^{-31} \mathrm{kg}, 1.6726 \times 10^{-27} \mathrm{kg},$ and
$5.0074 \times 10^{-27} \mathrm{kg},$ respectively. Calculate the frequency of the $n=1 \rightarrow n=4$ transition in $\mathrm{H}$ and $\mathrm{T}$ to five significant figures. Which of the transitions, $1 s \rightarrow 4 s, 1 s \rightarrow 4 p$, $1 s \rightarrow 4 d,$ could the frequencies correspond to?

Sheh Lit Chang
Sheh Lit Chang
University of Washington
01:14

Problem 19

Derive the ground-state term symbols for the following configurations:
a. $s^{1} d^{5}$
b. $f^{3}$
$\mathbf{c} . g^{2}$

Adriano Chikande
Adriano Chikande
Numerade Educator
09:59

Problem 20

Calculate the terms that can arise from the configuration $n p^{1} n^{\prime} p^{1}, n \neq n^{\prime} .$ Compare your results with those derived in the text for $n p^{2} .$ Which configuration has more terms and why?

Aniket Bajaj
Aniket Bajaj
Numerade Educator
03:53

Problem 21

For a closed-shell atom, an antisymmetric wave function can be represented by a single Slater determinant. For an open-shell atom, more than one determinant is needed. Show that the wave function for the $M_{S}=0$ triplet state of He $1 s^{1} 2 s^{1}$ is a linear combination of two of the Slater
determinants of Example Problem $22.1 .$ Which of the two are needed and what is the linear combination?

Kai Chen
Kai Chen
Princeton University
03:24

Problem 22

Calculate the transition dipole moment, $\mu_{z}^{m n}=\int \psi_{m}^{*}(\tau) \mu_{z} \psi_{n}(\tau) d \tau$ where $\mu_{z}=-e r \cos \theta$ for a
transition from the 1 s level to the 2 s level in H. Show that this
transition is forbidden. The integration is over $r, \theta,$ and $\phi$.

Narayan Hari
Narayan Hari
Numerade Educator
02:08

Problem 23

Use the transition frequencies shown in Example Problem 22.7 to calculate the energy (in joules and electron-volts) of the six levels relative to the $3 s^{2} \mathrm{S}_{1} / 2$ level. State your answers with the correct number of significant figures.

Suzanne W.
Suzanne W.
Numerade Educator
12:38

Problem 24

Derive the ground-state term symbols for the following atoms or ions:
a. $\mathrm{H}$
b. $\mathrm{F}$
c. $\mathrm{Na}^{+}$
d. $\mathrm{Sc}$

Shalini Tyagi
Shalini Tyagi
Numerade Educator
02:10

Problem 25

The spectrum of the hydrogen atom reflects the splitting of the $1 s^{2} \mathrm{S}$ and $2 p^{2} \mathrm{P}$ terms into levels. The energy difference between the levels in each term is much smaller
than the difference in energy between the terms. Given this information, how many spectral lines are observed in the $1 s^{2} \mathrm{S} \rightarrow 2 p^{2} \mathrm{P}$ transition? Are the frequencies of these transitions very similar or quite different?

Ajay Singhal
Ajay Singhal
Numerade Educator
01:59

Problem 26

Using Table 22.3, which lists the possible terms that arise from a given configuration, and Hund's rules, write the term symbols for the ground state of the atoms $\mathrm{K}$ through $\mathrm{Cu}$, excluding $\mathrm{Cr},$ in the form $^{(2 S+1)} L_{J}$.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
00:50

Problem 27

What atomic terms are possible for the following electron configurations? Which of the possible terms has the lowest energy?
a. $n s^{1} n p^{1}$
b. $n s^{1} n d^{1}$
c. $n s^{2} n p^{1}$
d. $n s^{1} n p^{2}$

Aadit Sharma
Aadit Sharma
Numerade Educator
03:55

Problem 28

Two angular momenta with quantum numbers $j_{1}=3 / 2$ and $j_{2}=5 / 2$ are added. What are the possible values of $J$ for the resultant angular momentum states?

Suzanne W.
Suzanne W.
Numerade Educator
01:14

Problem 29

Derive the ground-state term symbols for the following configurations:
a. $d^{2}$
b. $f^{9}$
c. $f^{d 4}$

Adriano Chikande
Adriano Chikande
Numerade Educator
01:22

Problem 30

The first ionization potential of ground-state He is $24.6 \mathrm{eV} .$ The wavelength of light associated with the $1 s 2 p^{1} \mathrm{P}$ term is $58.44 \mathrm{nm} .$ What is the ionization energy of the He atom in this excited state?

Ajay Singhal
Ajay Singhal
Numerade Educator
04:43

Problem 31

In the Na absorption spectrum, the following transitions are observed:
\[\begin{array}{ll}
4 p^{2} P \rightarrow 3 s^{2} S & \lambda=330.26 \mathrm{nm} \\
3 p^{2} P \rightarrow 3 s^{2} S & \lambda=589.593 \mathrm{nm}, 588.996 \mathrm{nm} \\
5 s^{2} \mathrm{S} \rightarrow 3 p^{2} \mathrm{P} & \lambda=616.073 \mathrm{nm}, 615.421 \mathrm{nm}
\end{array}\]
Calculate the energies of the $4 p^{2} P$ and $5 s^{2}$ S states with respect to the $3 s^{2}$ S ground state.

Prachita Kush
Prachita Kush
Numerade Educator
03:18

Problem 32

The Grotrian diagram in Figure 22.7 shows a number of allowed electronic transitions for He. Which of the
following transitions shows multiple spectral peaks due to a splitting of terms into levels? How many peaks are observed in each case? Are any of the following transitions between energy levels forbidden by the selection rules?
a. $1 s^{2}$$^{1} \mathrm{S} \rightarrow 1 s 2 p^{1} \mathrm{P}$
b. $1 s 2 p^{1} \mathrm{P} \rightarrow 1 s 3 s^{1} \mathrm{S}$
c. $1 s 2 s^{3} \mathrm{S} \rightarrow 1 s 2 p^{3} \mathrm{P}$
d. $1 s 2 p^{3} P \rightarrow 1 s 3 d^{3} D$

Alick Cushing
Alick Cushing
Numerade Educator
01:54

Problem 33

List the quantum numbers $L$ and $S$ that are consistent with the following terms:
a. $^{4} \mathrm{S}$
b. $^{4} \mathrm{G}$
c. $^{3} \mathrm{P}$
d. $^{2} \mathrm{D}$

Morgan Sizemore
Morgan Sizemore
Numerade Educator
03:48

Problem 34

The transition Al[Ne] $(3 s)^{2}(3 p)^{1} \rightarrow$ Al $[\mathrm{Ne}]$ $(3 s)^{2}(4 s)^{1}$ has two lines given by $\tilde{\nu}=25354.8 \mathrm{cm}^{-1}$ and $\widetilde{\nu}=25242.7 \mathrm{cm}^{-1} .$ The transition $\mathrm{Al}[\mathrm{Ne}](3 s)^{2}(3 p)^{1} \rightarrow$
$\mathrm{Al}[\mathrm{Ne}](3 s)^{2}(3 d)^{1}$ has three lines given by $\widetilde{\nu}=32444.8 \mathrm{cm}^{-1}$, $\tilde{\nu}=32334.0 \mathrm{cm}^{-1},$ and $\tilde{\nu}=32332.7 \mathrm{cm}^{-1} .$ Sketch an energy-level diagram of the states involved and explain the source of all lines.]

Suzanne W.
Suzanne W.
Numerade Educator
07:32

Problem 35

Given that the levels in the $^{3} \mathrm{P}$ term for carbon have the relative energies (expressed in wave numbers) of $^{3} \mathrm{P}_{1}-^{3} \mathrm{P}_{0}=16.4 \mathrm{cm}^{-1} \text {and }^{3} \mathrm{P}_{2}-^{3} \mathrm{P}_{1}=27.1 \mathrm{cm}^{-1}, \text {calculate }$
the ratio of the number of $C$ atoms in the $^{3} P_{2}$ and $^{3} P_{0}$ levels at 200.0 and $1000 . \mathrm{K}$.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
05:27

Problem 36

A general way to calculate the number of states that arise from a given configuration is as follows. Calculate the combinations of $m_{l}$ and $m_{s}$ for the first electron, and call that number $n .$ The number of combinations used is the number of electrons, which we call $m$. The number of unused combinations is $n-m .$ According to probability theory, the number of distinct permutations that arise from distributing the $m$ electrons among the $n$ combinations is $n ! /[m !(n-m) !]$.
For example, the number of states arising from a $p^{2}$ configuration is $6 ! /[2 ! 4 !]=15,$ which is the result obtained in Section 22.2. Using this formula, calculate the number of possible ways to place five electrons in a $d$ subshell. What is the ground-state term for the $d^{5}$ configuration and how many states does the term include?

João Gabriel Alencar Caribé
João Gabriel Alencar Caribé
Numerade Educator
03:42

Problem 37

The ground-state level for the phosphorus atom is $^{4} \mathrm{S}_{3 / 2}$. List the possible values of $L, M_{b} S, M_{S}, J,$ and $M_{J}$ consistent with this level.

Tim Blackstad
Tim Blackstad
Numerade Educator
12:38

Problem 38

Derive the ground-state term symbols for the following atoms:
a. $\mathrm{F}$
b. Na
c. $P$

Shalini Tyagi
Shalini Tyagi
Numerade Educator