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Physical Chemistry

Robert J. Silbey, Robert A. Alberty, Moungi G. Bawendi

Chapter 13

Rotational and Vibrational Spectroscopy - all with Video Answers

Educators

Chapter Questions

03:11

Problem 1

Since the energy of a molecular quantum state is divided by $k T$ in the Boltzmann distribution, it is of interest to calculate the temperature at which $k T$ is equal to the energy of photons of different wavelengths. Calculate the temperature at which $k T$ is equal to the energy of photons of wavelength $10^{3} \mathrm{cm}, 10^{-1} \mathrm{cm}$ $10^{-3} \mathrm{cm},$ and $10^{-5} \mathrm{cm}$.

Ernest Nachaki
Ernest Nachaki
Numerade Educator
07:49

Problem 2

Most chemical reactions require activation energies ranging between 40 and $400 \mathrm{kJ} \mathrm{mol}^{-1} .$ What are the equivalents of 40 and $400 \mathrm{kJ} \mathrm{mol}^{-1}$ in terms of $(a) \mathrm{nm},(b)$ wave numbers, and $(c)$ electron volts?

Ernest Nachaki
Ernest Nachaki
Numerade Educator
02:32

Problem 3

$(a)$ What vibrational frequency in wave numbers corresponds to a thermal energy of $k T$ at $25^{\circ} \mathrm{C} ?(b)$ What is the wavelength of this radiation?

Ernest Nachaki
Ernest Nachaki
Numerade Educator
04:23

Problem 4

Show that equation 13.17 is a solution of equation 13.9 by differentiating equation 13.17 and substituting it into equation 13.9.

Mayank Tripathi
Mayank Tripathi
Numerade Educator
03:14

Problem 5

Calculate the reduced mass and the moment of inertia $\operatorname{of} \mathrm{D}^{35} \mathrm{Cl},$ given that $R_{\mathrm{e}}=127.5 \mathrm{pm}$.

Ernest Nachaki
Ernest Nachaki
Numerade Educator
03:27

Problem 6

The H-O-H bond angle for $^{1} \mathrm{H}_{2} \mathrm{O}$ is $104.5^{\circ},$ and the $\mathrm{H}-\mathrm{O}$ bond length is $95.72 \mathrm{pm} .$ What is the moment of inertia of $\mathrm{H}_{2} \mathrm{O}$ about its $\mathrm{C}_{2}$ axis?

Ernest Nachaki
Ernest Nachaki
Numerade Educator
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Problem 7

Some of the following gas molecules have pure microwave absorption spectra and some do not: $\mathrm{N}_{2}, \mathrm{HBr}, \mathrm{CCl}_{4}$ $\mathrm{CH}_{3} \mathrm{CH}_{3}, \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}, \mathrm{H}_{2} \mathrm{O}, \mathrm{CO}_{2}, \mathrm{O}_{2} .$ What is the gross selection rule for rotational spectra, and which molecules satisfy it?

Andrew Eddins
Andrew Eddins
Emory University
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Problem 8

Calculate the frequency in wave numbers and the wavelength in $\mathrm{cm}$ of the first rotational transition $(J=0 \rightarrow 1)$ for $\mathrm{D}^{35} \mathrm{Cl}$.

Andrew Eddins
Andrew Eddins
Emory University
12:46

Problem 9

The pure rotational spectrum of $^{12} \mathrm{C}^{16} \mathrm{O}$ has transitions at 3.863 and $7.725 \mathrm{cm}^{-1}$. Calculate the internuclear distance in $^{12} \mathrm{C}^{16} \mathrm{O} .$ Predict the positions, in $\mathrm{cm}^{-1},$ of the next two lines.

Tianyu Li
Tianyu Li
Numerade Educator
12:46

Problem 10

Assume the bond distances in $^{13} \mathrm{C}^{16} \mathrm{O},^{13} \mathrm{C}^{17} \mathrm{O},$ and $^{12} \mathrm{C}^{17} \mathrm{O}$ are the same as in $^{12} \mathrm{C}^{16} \mathrm{O}$. Calculate the position, in $\mathrm{cm}^{-1},$ of the first rotational transitions in these four molecules. (Use the information in Problem $13.9 .)$

Tianyu Li
Tianyu Li
Numerade Educator
View

Problem 11

The far-infrared spectrum of HI consists of a series of equally spaced lines with $\Delta \tilde{\nu}=12.8 \mathrm{cm}^{-1} .$ What is $(a)$ the moment of inertia and $(b)$ the internuclear distance?

Andrew Eddins
Andrew Eddins
Emory University
01:51

Problem 12

For $\mathrm{H}^{35} \mathrm{Cl}$ calculate the relative populations of rotational levels, $f_{J} / f_{0},$ for the first three levels at $300 \mathrm{K}$ and $1000 \mathrm{K}$

Adriano Chikande
Adriano Chikande
Numerade Educator
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Problem 13

Using equation 13.44, show that $J$ for the maximally populated level is given by
\[
J_{\max }=\sqrt{\frac{k T}{2 h c B}}-\frac{1}{2}
\]

Andrew Eddins
Andrew Eddins
Emory University
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Problem 14

Using the result of Problem 13.13, find the $J$ nearest $J_{\max }$ at room temperature for $\mathrm{H}^{35} \mathrm{Cl}$ and $^{12} \mathrm{C}^{16} \mathrm{O}$. ( $a$ ) What is the ratio of the population at that $J$ to the population at $J=0 ?(b)$ What is the energy of that $J$ relative to $J=0$ in units of $k T ?$

Andrew Eddins
Andrew Eddins
Emory University
View

Problem 15

The moment of inertia of $^{16} \mathrm{O}^{12} \mathrm{C}^{16} \mathrm{O}$ is $7.167 \times$ $10^{-46} \mathrm{kg} \mathrm{m}^{2} .(a)$ Calculate the CO bond length, $R_{\mathrm{CO}}$ in $\mathrm{CO}_{2}$
(b) Assuming that isotopic substitution does not alter $R_{\mathrm{CO}},$ calculate the moments of inertia of $(1)^{18} \mathrm{O}^{12} \mathrm{C}^{18} \mathrm{O}$ and (2) $^{16} \mathrm{O}^{13} \mathrm{C}^{16} \mathrm{O}$

Andrew Eddins
Andrew Eddins
Emory University
05:49

Problem 16

Derive the expression for the moment of inertia of a symmetrical tetrahedral molecule such as $\mathrm{CH}_{4}$ in terms of the bond length $R$ and the masses of the four tetrahedral atoms. The easiest way to derive the expression is to consider an axis along one CH bond. Show that the same result is obtained if the axis is taken perpendicular to the plane defined by one group of three atoms HCH.

Averell Hause
Averell Hause
Carnegie Mellon University
02:53

Problem 17

What are the values of $\tilde{A}$ and $\tilde{B}$ (from equation 13.62 ) for the symmetric top $\mathrm{NH}_{3}$ if $I_{\|}=4.41 \times 10^{-47} \mathrm{kg} \mathrm{m}^{2}$ and $I_{\perp}=$ $2.81 \times 10^{-47} \mathrm{kg} \mathrm{m}^{2} ?$ What is the wavelength of the $J=0$ to $J=$ 1 transition? What are the wavelengths of the $J=1$ to $J=2$ transitions (remember the selection rules, $\Delta J=\pm 1, \Delta K=0$ and find all allowed transitions)?

Satpal Satpal
Satpal Satpal
Numerade Educator
02:14

Problem 18

Consider a linear triatomic molecule, ABC. Find the center of mass (which by symmetry lies on the molecular axis). Show that the moment of inertia is given by
\[
I=\frac{1}{M}\left[R_{\mathrm{AB}}^{2} m_{\mathrm{A}} m_{\mathrm{B}}+R_{\mathrm{BC}}^{2} m_{\mathrm{B}} m_{\mathrm{C}}+\left(R_{\mathrm{AB}}+R_{\mathrm{BC}}\right)^{2} m_{\mathrm{A}} m_{\mathrm{C}}\right]
\]
where $R_{\mathrm{AB}}$ is the $\mathrm{AB}$ bond distance, $R_{\mathrm{BC}}$ is the BC bond distance, $m_{i}$ are the masses of the atoms, and $M=m_{\mathrm{A}}+m_{\mathrm{B}}+m_{\mathrm{C}}$ Show that if $R_{\mathrm{AB}}=R_{\mathrm{BC}}$ and $m_{\mathrm{A}}=m_{\mathrm{C}},$ then $I=2 m_{\mathrm{A}} R_{\mathrm{AB}}^{2}$.

Chai Santi
Chai Santi
Numerade Educator
03:50

Problem 19

The fundamental vibration frequency of $\mathrm{H}^{35} \mathrm{Cl}$ is $8.967 \times$ $10^{13} \mathrm{s}^{-1}$ and that of $\mathrm{D}^{35} \mathrm{Cl}$ is $6.428 \times 10^{13} \mathrm{s}^{-1} .$ What would the
separation be between infrared absorption lines of $\mathrm{H}^{35} \mathrm{Cl}$ and $\mathrm{H}^{37} \mathrm{Cl}$ on one hand and those of $\mathrm{D}^{35} \mathrm{Cl}$ and $\mathrm{D}^{37} \mathrm{Cl}$ on the other, if the force constants of the bonds are assumed to be the same in each pair?

MT
Monica Tran
Numerade Educator
01:06

Problem 20

Find the force constants of the halogens $^{127} \mathrm{I}_{2},^{79} \mathrm{Br}_{2},$ and $^{35} \mathrm{Cl}_{2}$ using the data of Table $13.4 .$ Is the order of these the same as the order of the bond energies?

Lottie Adams
Lottie Adams
Numerade Educator
06:05

Problem 21

Given the following fundamental frequencies of vibration, calculate $\Delta H^{\circ}$ for the reaction
\[
\begin{array}{rl}
\mathrm{H}^{35} \mathrm{Cl}(v=0)+^{2} \mathrm{D}_{2}(v=0)=^{2} \mathrm{D}^{35} \mathrm{Cl}(v=0)+\mathrm{H}^{2} \mathrm{D}(v=0) \\
\mathrm{H}^{35} \mathrm{Cl}: 2989 \mathrm{cm}^{-1} & \mathrm{H}^{2} \mathrm{D}: 3817 \mathrm{cm}^{-1} \\
^{2} \mathrm{D}^{35} \mathrm{Cl}: 2144 \mathrm{cm}^{-1} & ^{2} \mathrm{D}^{2} \mathrm{D}: 3119 \mathrm{cm}^{-1}
\end{array}
\]

Ameer Said
Ameer Said
Numerade Educator
01:18

Problem 22

If the fundamental vibration frequency of $^{1} \mathrm{H}_{2}$ is $4401.21 \mathrm{cm}^{-1},$ compute the fundamental vibration frequency of $^{2} \mathrm{D}_{2}$ and $^{1} \mathrm{H}^{2} \mathrm{D}$ assuming the same force constants. If $D_{0}$ for $^{1} \mathrm{H}_{2}$ is $4.4781 \mathrm{eV}$, what is $D_{0}$ for $^{2} \mathrm{D}_{2}$ and $^{1} \mathrm{H}_{2}$ D? Neglect anharmonicities.

Lottie Adams
Lottie Adams
Numerade Educator
01:39

Problem 23

Using the values for $\tilde{\nu}_{\mathrm{e}}$ and $\tilde{\nu}_{\mathrm{e}} \tilde{x}_{\mathrm{e}}$ in Table 13.4 for $^{1} \mathrm{H}^{35} \mathrm{Cl}$ estimate the dissociation energy assuming the Morse potential is applicable.

Zhuxi Luo
Zhuxi Luo
Numerade Educator
03:46

Problem 24

Apply the Taylor expansion to the potential energy given by the Morse equation $\tilde{V}(R)=D_{\mathrm{e}}\left\{1-\exp \left[-a\left(R-R_{0}\right)\right]\right\}^{2}$ to show that the force constant $k$ is given by $k=2 D_{\mathrm{e}} a^{2}$

Chai Santi
Chai Santi
Numerade Educator
04:49

Problem 25

(a) What fraction of $\mathrm{H}_{2}(\mathrm{g})$ molecules are in the $v=$ 1 state at room temperature?
(b) What fractions of $\operatorname{Br}_{2}(\mathrm{g})$ molecules are in the $v=1,2,$ and 3 states at room temperatures?

Stanley Enemuo
Stanley Enemuo
Numerade Educator
06:45

Problem 26

The first three lines in the $R$ branch of the fundamental vibration-rotation band of $\mathrm{H}^{35} \mathrm{Cl}$ have the following frequencies in $\mathrm{cm}^{-1}: 2906.25(0), 2925.78(1), 2944.89(2),$ where the numbers in parentheses are the $J$ values for the initial level. What are the values of $\tilde{\nu}_{0}, B_{v}^{\prime}, B_{v}^{\prime \prime}, B_{\mathrm{e}},$ and $\alpha ?$

Susan Hallstrom
Susan Hallstrom
Numerade Educator
01:08

Problem 27

In Table $13.3, D_{\mathrm{e}}$ for $\mathrm{H}_{2}$ is given as $4.7483 \mathrm{eV}$ or $458.135 \mathrm{kJ} \mathrm{mol}^{-1} .$ Given the vibrational parameters for $\mathrm{H}_{2}$ in Table $13.4,$ calculate the value you would expect for $\Delta_{\mathrm{f}} H^{\circ}$ for $\mathrm{H}(\mathrm{g})$ at $0 \mathrm{K}$.

Ajay Singhal
Ajay Singhal
Numerade Educator
03:32

Problem 28

Calculate the wavelengths in $(a)$ wave numbers and $(b)$ micrometers of the center two lines in the vibration spectrum of HBr for the fundamental vibration. The necessary data are to be found in Table 13.4.

Guilherme Barros
Guilherme Barros
Numerade Educator
01:44

Problem 29

How many normal modes of vibration are there for $(a)$ $\mathrm{SO}_{2}(\text { bent })$ $(b) \mathrm{H}_{2} \mathrm{O}_{2}(\mathrm{bent})$ $(c)$ HC?CH (linear), and $(d)$ $\mathrm{C}_{6} \mathrm{H}_{6} ?$

Lottie Adams
Lottie Adams
Numerade Educator
01:04

Problem 30

List the numbers of translational, rotational, and vibrational degrees of freedom for $(a) \mathrm{Ne},(b) \mathrm{N}_{2},(c) \mathrm{CO}_{2},$ and $(d)$ $\mathrm{CH}_{2} \mathrm{O}$.

Narayan Hari
Narayan Hari
Numerade Educator
01:09

Problem 31

Acetylene is a symmetrical linear molecule. It has seven normal modes of vibration, two of which are doubly degenerate. These normal modes may be represented as follows:
(a) Which are the doubly degenerate vibrations? (b) Which vibrations are infrared active? (c) Which vibrations are Raman active?

Adriano Chikande
Adriano Chikande
Numerade Educator
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Problem 32

Calculate the wave number and wavelength of the pure fundamental $(v=0 \rightarrow 1)$ vibrational transitions for $(a)^{12} \mathrm{C}^{16} \mathrm{O}$ and $(b)^{39} \mathrm{K}^{35} \mathrm{Cl}$ using data in Table 13.4.

Andrew Eddins
Andrew Eddins
Emory University
06:47

Problem 33

$(a)$ Consider the four normal modes of vibration of a linear molecule $\mathrm{AB}_{2}$ from the standpoint of changing dipole moment and changing polarizability. Which vibrational modes are infrared active, and which are Raman active? (Note the exclusion rule.) $(b)$ Consider the three normal modes of a nonlinear molecule $\mathrm{AB}_{2}$. Which vibrational modes are infrared active, and which are Raman active?

Tianyu Li
Tianyu Li
Numerade Educator
05:34

Problem 34

Calculate the fraction of $\mathrm{Cl}_{2}$ molecules $(\tilde{v}=559.7$ $\mathrm{cm}^{-1}$ ) in the $i=0,1,2,3$ vibrational states at $1000 \mathrm{K}$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:13

Problem 35

When CCl $_{4}$ is irradiated with the 435.8 -nm mercury line, Raman lines are obtained at $439.9,441.8,444.6,$ and $450.7 \mathrm{nm}$ Calculate the Raman frequencies of $\mathrm{CCl}_{4}$ (expressed in wave numbers). Also calculate the wavelengths (expressed in $\mu \mathrm{m}$ ) in the infrared at which absorption might be expected.

Mehrnaz Siavoshi
Mehrnaz Siavoshi
Numerade Educator
01:42

Problem 36

The first several Raman frequencies of $^{14} \mathrm{N}_{2}$ are 19.908 $27.857,35.812,43.762,51.721,$ and $59.662 \mathrm{cm}^{-1} .$ These lines are due to pure rotational transitions with $J=1,2,3,4,5,$ and 6 The spacing between the lines is $4 B_{\mathrm{e}} .$ What is the inter nuclear distance?

Crystal Wang
Crystal Wang
Numerade Educator
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Problem 37

What Raman shifts are expected for the first four Stokes lines for $\mathrm{CO}_{2} ?$

Andrew Eddins
Andrew Eddins
Emory University
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Problem 38

Some of the following gas molecules have a pure rotational Raman spectrum and some do not: $\mathrm{N}_{2}, \mathrm{HBr}, \mathrm{CCl}_{4}$ $\mathrm{CH}_{3} \mathrm{CH}_{3}, \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}, \mathrm{H}_{2} \mathrm{O}, \mathrm{CO}_{2}, \mathrm{O}_{2} .$ What is the gross selection rule for pure rotational Raman spectra, and which molecules satisfy it?

Andrew Eddins
Andrew Eddins
Emory University
View

Problem 39

Calculate the factors for converting between eV and $\mathrm{cm}^{-1}$ and between $\mathrm{eV}$ and $\mathrm{kJ} \mathrm{mol}^{-1}$.

Andrew Eddins
Andrew Eddins
Emory University
View

Problem 40

Energies in electron volts (eV) may be expressed in terms of temperature by use of the relation $\mathrm{e} \phi=k T,$ where $\phi$ is the difference in potential in $V .$ What temperature corresponds to $1 \mathrm{V} ? 100 \mathrm{V} ? 1000 \mathrm{V} ?$ What is the electron volt equivalent of room temperature?

Andrew Eddins
Andrew Eddins
Emory University
01:52

Problem 41

The internuclear distance in CO is 112.82 pm. Calculate $(a)$ the reduced mass and $(b)$ the moment of inertia.

Lottie Adams
Lottie Adams
Numerade Educator
06:45

Problem 42

Calculate the frequencies in $\mathrm{cm}^{-1}$ and the wavelengths in $\mu \mathrm{m}$ for the pure rotational lines in the spectrum of $\mathrm{H}^{35} \mathrm{Cl}$ corresponding to the following changes in rotational quantum number: $0 \rightarrow 1,1 \rightarrow 2,2 \rightarrow 3,$ and $8 \rightarrow 9$.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
02:01

Problem 43

Assuming that the internuclear distance is $74.2 \mathrm{pm}$ for $(a) \mathrm{H}_{2},(b) \mathrm{HD},(c) \mathrm{HT},$ and $(d) \mathrm{D}_{2},$ calculate the moments of inertia of these molecules.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
08:33

Problem 44

Calculate the energy difference in $\mathrm{cm}^{-1}$ and $\mathrm{kJ} \mathrm{mol}^{-1}$ between the $J=0$ and $J=1$ rotational levels of $\mathrm{OH}$, using the data of Table $13.4 .$ Assuming that OD has the same internuclear distance as OH, calculate the energy difference between $J=0$ and $J=1$ in $\mathrm{OD}$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:33

Problem 45

In the pure rotational spectrum of $^{12} \mathrm{C}^{16} \mathrm{O},$ the lines are separated by $3.8626 \mathrm{cm}^{-1} .$ What is the internuclear distance in the molecule?

Lottie Adams
Lottie Adams
Numerade Educator
01:38

Problem 46

Consider the molecular radicals $^{12} \mathrm{CH}$ and $^{13} \mathrm{CH}$. Calculate their moments of inertia using $R_{\mathrm{e}}$ from Table 13.4 and assuming $R_{\mathrm{e}}$ is the same in both. Using the results of Problem $13.13,$ find the value of $J$ closest to $J_{\max }$ at room temperature and compute the difference in energy between this state and the next higher energy state.

Chai Santi
Chai Santi
Numerade Educator
03:33

Problem 47

The separation of the pure rotation lines in the spectrum of $\mathrm{CO}$ is $3.86 \mathrm{cm}^{-1}$. Calculate the equilibrium internuclear separation.

Lottie Adams
Lottie Adams
Numerade Educator
02:33

Problem 48

Show that for large $J$ the frequency of radiation absorbed in exciting a rotational transition is approximately equal to the classical frequency of rotation of the molecule in its initial or final state.

Lottie Adams
Lottie Adams
Numerade Educator
00:31

Problem 49

For the rotational Raman effect, what are the displacements of the successive Stokes lines in terms of the rotational constant $B ?$ Is the answer the same for the anti-Stokes lines?

Hunza Gilgit
Hunza Gilgit
Numerade Educator
01:16

Problem 50

Show that the moments of inertia of a regular hexagonal molecule made up of six identical atoms of mass $m$ are given by
\[
I_{\|}=6 m r^{2} \quad \text { and } \quad I_{\perp}=3 m r^{2}
\]
where $r$ is the bond distance.

Adriano Chikande
Adriano Chikande
Numerade Educator
12:46

Problem 51

What are the frequencies of the first three lines in the rotational spectrum of $^{16} \mathrm{O}^{12} \mathrm{C}^{32} \mathrm{S}$ given that the $\mathrm{O}-\mathrm{C}$ distance is $116.47 \mathrm{pm}$, the $\mathrm{C}-\mathrm{S}$ distance is $155.76 \mathrm{pm}$, and the molecule is linear. Atomic masses of isotopes are given inside the back cover. The moment of inertia of a linear molecule ABC is given in Problem 13.18.

Tianyu Li
Tianyu Li
Numerade Educator
12:46

Problem 52

What are the rotational frequencies for the first three rotational lines in $16 \mathrm{O}^{12} \mathrm{C}^{34} \mathrm{S}$, assuming the same bond lengths as in Problem $13.51 ?$

Tianyu Li
Tianyu Li
Numerade Educator
04:33

Problem 53

Ammonia is a symmetric top with $$\begin{array}{l}
I_{x x}=I_{y y}=I_{\perp}=2.8003 \times 10^{-47} \mathrm{kg} \mathrm{m}^{2} \\
I_{z z}=I_{\|}=4.4300 \times 10^{-47} \mathrm{kg} \mathrm{m}^{2}
\end{array}$$ Calculate the characteristic rotational temperatures $\Theta_{\mathrm{r}}$ where
\[
\Theta_{\mathrm{r}}=\frac{h^{2}}{8 \pi^{2} I k}
\]

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:39

Problem 54

Using the Morse potential expression, equation 13.82 estimate $D_{\mathrm{e}}$ for $\mathrm{HBr}, \mathrm{HCl}$, and HI from the data in Table 13.4.

Zhuxi Luo
Zhuxi Luo
Numerade Educator
02:25

Problem 55

Calculate the values of $D_{\mathrm{e}}$ for $\mathrm{HCl}$, HBr, and HI using the data of Table $\left.13.4 \text { and equation } 13.80 \text { (neglect } y_{\mathrm{e}}\right)$.

Adriano Chikande
Adriano Chikande
Numerade Educator
01:45

Problem 56

From the data of Table 13.4 , calculate the vibrational force constants of $\mathrm{HCl}$, HBr, and HI. Are these in the same order as the dissociation energies?

Chai Santi
Chai Santi
Numerade Educator
04:01

Problem 57

Using the Boltzmann distribution (equation 16.17 ), calculate the ratio of the population of the first vibrational excited state to the population of the ground state for $\mathrm{H}^{35} \mathrm{Cl}\left(\tilde{v}_{0}=\right.$ $\left.2990 \mathrm{cm}^{-1}\right)$ and $^{127} \mathrm{I}_{2}\left(\tilde{\nu}_{0}=213 \mathrm{cm}^{-1}\right)$ at $300 \mathrm{K}$.

Nicolas Barroga
Nicolas Barroga
Numerade Educator
01:39

Problem 58

Use the Morse potential to estimate the equilibrium dissociation energy for $79 \mathrm{Br}_{2}$ using $\tilde{\nu}_{\mathrm{e}}$ and $\tilde{\nu}_{\mathrm{e}} x_{\mathrm{e}}$ from Table 13.4.

Zhuxi Luo
Zhuxi Luo
Numerade Educator
01:07

Problem 59

The wave numbers of the first several lines in the $R$ branch of the fundamental $(v=0 \rightarrow 1)$ vibrational band for $^{2} \mathrm{H}^{35} \mathrm{Cl}$ have the following frequencies in $\mathrm{cm}^{-1}: 2101.60(0)$ $2111.94(1), 2122.05(2),$ where the numbers in parentheses are the $J$ values for the initial level. What are the values of $\tilde{B}_{v}^{\prime}, \tilde{B}_{v}^{\prime \prime}, \tilde{B}_{\mathrm{e}},$ and $\alpha ?$ How does the internuclear distance compare with that for $^{1} \mathrm{H}^{35} \mathrm{Cl}$ ?

Ajay Singhal
Ajay Singhal
Numerade Educator
01:50

Problem 60

Gaseous HBr has an absorption band centered at about $2645 \mathrm{cm}^{-1}$ consisting of a series of lines approximately equally spaced with an interval of $16.9 \mathrm{cm}^{-1} .$ For gaseous DBr estimate the frequency in wave numbers of the band center and the interval between lines.

Narayan Hari
Narayan Hari
Numerade Educator
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Problem 61

Some of the following gas molecules have infrared absorption spectra and some do not: $\mathrm{N}_{2}, \mathrm{HBr}, \mathrm{CCl}_{4}, \mathrm{CH}_{3} \mathrm{CH}_{3}$ $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}, \mathrm{H}_{2} \mathrm{O}, \mathrm{CO}_{2}, \mathrm{O}_{2} .$ What is the gross selection rule for vibrational spectra, and which molecules satisfy it?

Andrew Eddins
Andrew Eddins
Emory University
02:38

Problem 62

List the numbers of translational, rotational, and vibrational degrees of freedom of $\mathrm{Cl}_{2}, \mathrm{H}_{2} \mathrm{O},$ and $\mathrm{C}_{2} \mathrm{H}_{2}$

Prachita Kush
Prachita Kush
Numerade Educator
02:25

Problem 63

List the numbers of translational, rotational, and vibrational degrees of freedom of $\mathrm{NNO}$ (a linear molecule) and $\mathrm{NH}_{3}$.

Narayan Hari
Narayan Hari
Numerade Educator
02:12

Problem 64

The rotational Raman spectrum of hydrogen gas is measured using a 488 -nm laser. Stokes lines are observed at 355 $588,815,$ and $1033 \mathrm{cm}^{-1}$. since these transitions are of the type $J \rightarrow J+2,$ it may be shown that the wave numbers of these lines are given by $$\Delta \tilde{\nu}_{\mathrm{R}}=4 \tilde{B}_{\mathrm{e}}\left(J+\frac{3}{2}\right)$$ where $J$ is the rotational quantum number of the initial state $(0,1,2, \text { and } 3,$ respectively, for the above lines) and $\tilde{B}_{\mathrm{e}}$ is given by equation $13.34 .$ What is $R_{\mathrm{e}} ?[\mathrm{L} . \text { C. Hoskins, } J .$ Chem. Educ. $54: 642(1977) .]$

Mayukh Banik
Mayukh Banik
Numerade Educator
03:45

Problem 65

The rotational Raman spectrum of nitrogen gas shows Raman shifts of $19,27,34,53, \ldots \mathrm{cm}^{-1},$ corresponding to rota tional quantum numbers of the initial state of $J=1,2,3,4, \ldots$ since the spacing is $4 B_{\mathrm{e}}$ ignoring centrifugal distortion, what is $R_{\mathrm{e}} ?[\mathrm{L} . \mathrm{C} . \text { Hoskins, } J . \text { Chem. Educ. } 52: 568(1975) .]$
13.66 $\quad$ Calculate $\Delta H^{\circ}(298 \mathrm{K})$ for the reaction
\[
\mathrm{H}_{2}+\mathrm{D}_{2}=2 \mathrm{HD}
\]
assuming that the force constant is the same for all three molecules.

Lottie Adams
Lottie Adams
Numerade Educator