Physical Chemistry Third Edition

944 22 Translational, Rotational, and Vibrational States of Atoms and Molecules

The level of maximum population is the level with the maximum value of

(2J+1)exp(−EJ/kBT). The quickest way to find this is to treatJas a continuously

variable quantity, differentiating the function to be maximized with respect toJ, setting

this derivative equal to zero, and solving forJ. A nonintegral value ofJcan result, but

rounding to the nearest integer gives the desired value.

d

dJ

(2J+1)exp

(

−hc ̃BeJ(J+ 1 )

kBT

)

exp

(

−hc ̃BeJ(J+ 1 )

kBT

)(

2 −

hc ̃Be

kBT

( 2 J+ 1 )^2

)

The exponential factor does not vanish except forJ→∞, so we seek the value ofJ

for which the second factor vanishes, which occurs when

(2Jmp+1)^2 

2 kBT

hc ̃Be

(22.5-8)

EXAMPLE22.12

Find the rotational level of the CO molecule with the largest population at 298 K.

Solution

(

2 Jmp+ 1

) 2



2

(

1. 3807 × 10 −^23 JK−^1

)

(298 K)

(

6. 6261 × 10 −^34 Js

)(

2. 9979 × 1010 cm s−^1

)(

1 .931 cm−^1

)

 214. 5

2 Jmp+ 1 

√

214. 5  14. 65

Jmp 6. 82 ≈ 7

For a typical molecule at room temperature, several of the rotational energy levels are

significantly populated and the levels of maximum population correspond to a value ofJ

larger than 1. Molecules with smaller reduced masses have fewer occupied rotational

levels, and molecules with larger reduced masses have more occupied rotational levels.

Exercise 22.10

a.Find the rotational level with the largest population for HF molecules at 500.0 K.

b.Find the rotational level with the largest population for Br 2 molecules at 500.0 K.

The Population of Vibrational States of Diatomic Molecules

Since the vibrational levels of a diatomic molecule are nondegenerate, we have for the

population of a vibrational energy level with quantum numberν

(Population of vibrational levelν)∝e−Ev/kBT (22.5-9)

EXAMPLE22.13

Find the ratio of the population of theν1,J0 state to theν0,J0 state for the

CO molecule at 298 K. Neglect the anharmonicity correctionνexe.