Physical Chemistry Third Edition

22.5 The Equilibrium Populations of Molecular States 947

Ratioexp

(

− 7. 052 × 10 −^20 J

(

1. 3807 × 10 −^23 JK−^1

)

(298 K)

)

 3. 60 × 10 −^8

This value is typical of the fact that excited vibrational levels are not significantly populated

at room temperature.

PROBLEMS

Section 22.5: The Equilibrium Populations of Molecular
States

22.45Find the ratio of the populations of theν1,J2 level
and theν0,J0 level at 298.15 K in the
rigid-rotor-harmonic oscillator approximation for:
a.H 2
b.HD, where D is deuterium,^2 H
c.D 2

22.46Find the ratio of the population of theν1,J0 level
of CO to the population of theν0,J0 level at
298.15 K.

22.47Find the ratio of the populations of theν1,J1 level
and theν0,J0 state at 298 K for:
a.^1 H^35 Cl
b.^1 H^37 Cl
c.^2 H^35 Cl
d.^2 H^37 Cl

The parameters for^1 H^35 Cl are in Table A.22 of

Appendix A. Assume that isotopic substitution does not

change the bond length or the force constant.

22.48a.For a temperature of 298 K, find the ratio of the

population of theν1 vibrational state to the

population of theν0 vibrational state of H 2.

b.For a temperature of 298 K, find the ratio of the

population of theν1 vibrational state to the

population of theν0 vibrational state of I 2.

c.Explain why the values in parts a and b are so

different.

22.49Find the rotational level of maximum population

for H 2 at 298 K. Do it separately forortho- and

para-hydrogen.

22.50Find the rotational level of maximum population for I 2

at 298 K.

Summary of the Chapter

In addition to electronic motion, atoms can have translational motion, and molecules

can have translational, rotational, and vibrational motions. To a good approximation

the translational energy of a molecule confined in a box is the same as that of a point-

mass particle in the same box. To a first approximation, the vibrational energy of a

diatomic molecule is that of a harmonic oscillator, and the rotation is that of a rigid

rotor. Correction terms can be included if high accuracy is necessary. The rotational

energy of a polyatomic molecule is taken to be that of a rigid rotating body. The

vibrational energy of a polyatomic molecule is taken to be that of normal modes, each

one of which oscillates like a harmonic oscillator. Only a fraction of the conceivable

set of rotational quantum numbers is possible. This fraction is equal to 1/σ, whereσis

the symmetry number of the molecule, equal to the number of equivalent orientations

of the molecule.