14.20.Lithium hydride,^7 Li^1 H, is a potential fuel for fusion re-
actors because it is one of the few compounds of very small
elements that exists as a solid, and is therefore more dense
than gaseous fuels even under extreme conditions. LiH in the
gas phase has a pure rotational spectrum consisting of lines
spaced by 15.026 cm^1. Calculate the bond distance in LiH.
14.21.Determine the first four absorptions in the pure rota-
tional spectrum of LiH (see exercise 14.20) in units of GHz.
14.22.Determine the most populated rotational level, Jmax, for
a sample of LiH (see problem above) at (a)298 K, (b)1000 K,
(c)5000 K.
14.23.Given a pure rotational spectrum of the HS radical di-
atomic molecule (r 1.40 Å), you notice that the most in-
tense absorption is assigned to the J 8 →J9 transition.
Estimate the sample temperature.
14.24.A gas-phase sample is subjected to an electric field,
and its Stark-effect rotational spectrum is measured. How many
individual lines will be detected for the following transitions?
(a)J 0 →J 1 (b)J 1 →J 2 (c)J 2 →J 3
14.25.From the data in Table 14.2, predict Bfor DCl (D
is^2 H).
14.7 Centrifugal Distortions
14.26.Verify equation 14.26.
14.27.An acquaintance remarks that a rotational spectrum of
I 2 showing the J 200 →J201 transition is predicted very
closely by the rigid rotor equations from quantum mechanics.
Give two reasons why you should question the validity of that
statement.
14.28.Consider the values in Table 14.2 and remark on the
trend of the magnitudes of Band Dversus atomic mass. Does
the trend make sense?
14.29.Using the value of ̃ 4320 cm^1 for diatomic hy-
drogen and the value of Bfor H 2 from Table 14.2, approxi-
mate DJand compare it to the values given in Table 14.2.
14.8 & 14.9 Vibrations and Normal Modes
14.30.Determine the number of total degrees of freedom
and the number of vibrational degrees of freedom for the fol-
lowing molecules. (a)Hydrogen fluoride, HF (b)Hydrogen
telluride, H 2 Te (c)Buckminsterfullerene, C 60 (d)Phenyl-
alanine, C 6 H 5 CH 2 CHNH 2 COOH (e)Naphthalene, C 10 H 8 (f)
The linear isomer of the C 4 radical (g)The bent isomer of the
C 4 radical
14.31.How many total normal modes of vibration do the
molecules in the previous problem have?
14.32.Methane, CH 4 , has only two IR-active vibrational modes.
Comment on the expected number of IR-active vibrational
modes of CH 3 D, where one hydrogen atom is replaced by a
deuterium.
14.10 Quantum Mechanics and Vibrations
14.33.Show that the two expressions in equation 14.30 are
equivalent.
14.34.Verify that the ratio of vibrational frequencies used
originally in Example 14.12 does reduce to ̃*/(2886 cm^1 )
/*.
14.35.Considering reduced mass can sometimes yield useful
approximations even if a replaced atom isn’t an isotope.
Consider CO 2 and OCS. The symmetric CO stretching vi-
bration occurs at 1338 cm^1. Estimate the frequency of the
stretching vibration of the CS bond assumingthat the S
atom is an isotope of oxygen. (It appears at 859 cm^1 .) Is this
a good approximation or not?
14.36.The FeH diatomic molecule absorbs infrared light
having a frequency of 1661.0 cm^1. Assuming that this is
for^56 FeH, calculate the frequency of light that^54 FeH would
absorb.14.11 Vibrational Selection Rules
14.37.Why is nitrogen gas commonly used as a purge gas in
infrared spectrometers?
14.38.From the description in the text of the vibrational mo-
tions of the carbon dioxide molecule, draw arrows on each
atom indicating how the atoms are moving for each normal
mode. Draw a final arrow (if possible) indicating the direction
of any fleeting dipole moment, and state whether each nor-
mal mode will be IR-active. (See Figure 14.30.)
14.39.Differentiate between fundamental vibrations, over-
tone vibrations, and hot bands. What are the selection rules
for the vibrational quantum number for each?
14.40.Are deviations from an ideal harmonic oscillator more
likely to be seen at low energies or high energies? Explain your
answer.14.12 Vibrations of Diatomic
and Linear Molecules
14.41.Prove that ^2 V/r^2 k, the force constant. Show that
the units on V(energy) and k(N/m) are consistent with this
equation.
14.42.Use Figure 14.29 to comment on the variances be-
tween a Morse oscillator and a true molecular potential energy
curve.
14.43.Using the information in Table 14.4, calculate Deand
the Morse potential constant afor HF (k965.1 N/m) and
HBr (411.5 N/m). Express the constant ain units of m^1 and
Å^1. Combining your answer with the value of afor HCl in
Example 14.15, do you see any trends?
14.44.Use the equation for the energy of a Morse oscillator
and calculate the values for the five transitions listed for HCl
in Table 14.3. How close are the predicted vibrational transi-
tions to the experimental values in the table?
14.45.Use the evalues from Table 14.4 to calculate anhar-
monicity constants xeand exefor HBr and CO, and compare
them to the tabulated values. Use D 0 (HBr) 362 kJ/mol and
D 0 (CO) 1071 kJ/mol.516 Exercises for Chapter 14