Physical Chemistry Third Edition

948 22 Translational, Rotational, and Vibrational States of Atoms and Molecules

ADDITIONAL PROBLEMS

22.51If a molecule is confined in a very small box, its energy
levels can be spaced widely enough that the spacing can
be observed with light absorbed or emitted by transitions
between translational levels. Assume that a CO molecule
is confined in a matrix of solid argon at 75 K, and that the
center of the CO molecule can move in a cubical region
that is 3. 5 × 10 −^10 m on a side.

a.Find the energies and degeneracies of the first three

translational energy levels.

b.Find the wavelength and frequency of the light

absorbed if a molecule makes a transition from the

lowest energy level to the next energy level.

22.52 a.From information in Table A.22, find the value of the
force constant for each of the molecules: N 2 ,O 2 , and
F 2. From the LCAOMO treatment in Chapter 20, find
the bond order for each molecule. Comment on the
relative sizes of these force constants.

b.From information in Table A.22, find the value of the

internuclear distance in each of the molecules: N 2 ,

O 2 , and F 2. Comment on the relative sizes of these

internuclear distances.

22.53 a.From information in Table A.22, find the value of the
force constant for HF and HI. Comment on the
relative sizes of these force constants.

b.From information in Table A.22. find the value of the

internuclear distance for HF and HI. Comment on the

relative sizes of these distances.

22.54For the NO molecule, find the following:

a.The difference in energy between theν1,J 2

level and theν0,J1 level, without corrections

for anharmonicity, centrifugal stretching, and

rotation-vibration interaction.

b.The degeneracies of these two levels.

c.The ratio of the populations of these two levels at

298.15 K and at 1000.0 K.

22.55Identify each of the following statements as either true or

false. If a statement is true only under specific

circumstances, label it as false.

a.The behavior of a molecule confined in a container

will be noticeably different from the behavior of a free

molecule.

b.Although part of the electronic energy in the

Born–Oppenheimer approximation is kinetic energy,

this energy acts as a potential energy for nuclear

motion.

c.Principal axes can be chosen for any object.

d.A linear triatomic molecule will exhibit more distinct

frequencies of vibration than a bent triatomic

molecule.

e.A methane molecule has nine vibrational normal

modes.

f.An SeO 2 molecule has four vibrational normal modes.

g.A normal oxygen molecule^16 O 2 will have rotational

levels that are qualitatively different from those of the

isotopically substituted oxygen molecule^16 O^18 O.

h.The corrections for anharmonicity in the vibrational

energy levels of a diatomic molecule cause the energy

levels to be farther apart than the uncorrected energy

levels.

i.The corrections for centrifugal stretching in the

rotational energy levels of a diatomic molecule cause

the energy levels to be closer together than the

uncorrected energy levels.

j.The corrections for the interaction of rotation and

vibration cause the energy levels to be closer together

than the uncorrected energy levels.

k.Every diatomic molecule can exhibit only even values

ofJor odd values ofJ.

22.56Repeat Problem 22.54 with the corrections for

anharmonicity, centrifugal stretching, and

rotation-vibration interaction.