Physical Chemistry Third Edition

946 22 Translational, Rotational, and Vibrational States of Atoms and Molecules

like that of a diatomic molecule. The rotational energies of the rotational states of a

spherical top molecule are given in Eq. (22.4-10) and the degeneracies are given in

Eq. (22.4-12). The degeneracies of the rotational levels of symmetric top and asym-

metric top molecules are more complicated, and we do not discuss them.

EXAMPLE22.15

If we assume that the bond lengths in SF 6 are equal to 1. 80 × 10 −^10 m, the three principal

moments of inertia are equal to 4. 18 × 10 −^45 kg m^2.

a.Find the ratio of the population of one of the state of theJ2 level to the population of

theJ0 level at 298 K.

b.Find the ratio of the population of theJ2 level to the population of theJ0 level at

298 K.

Solution

a. Ratioe−E^2 /kBT

e−E^0 /kBT

e−(E^2 −E^0 )/kBT

E 2 −E 0  ̄

h^2

2 IA

2(2+1)

3 h ̄^2

IA



3 h^2

4 π^2 IA



3

(

6. 6261 × 10 −^34 Js

) 2

4 π^2

(

4. 18 × 10 −^45 kg m^2

)

 7. 98 × 10 −^24 J

Ratioexp

(

− 7. 98 × 10 −^24 J

(

1. 3807 × 10 −^23 JK−^1

)

(298 K)

)

e−^0.^00194

 0. 9981

b. Ratio

g 2 e−E^2 /kBT

g 0 e−E^0 /kBT

(25)e−^0.^00194  24. 95

The vibrational energy levels of polyatomic molecules are the sum of harmonic

oscillator energy level expressions, as in Eq. (22.4-19).

EXAMPLE22.16

Find the ratio of the population of the vibrational state of SO 2 withν 1 1,ν 2 2, andν 3  1

to the population of the ground vibrational state. The frequencies divided by the speed of

light are given in Figure 22.6.

Solution

Evib(6. 6261 × 10 −^34 J s)(2. 9979 × 1010 cm s−^1 )

×[1151.2cm−^1 +2(519 cm−^1 )+1361 cm−^1 ]

 7. 052 × 10 −^20 J