Physical Chemistry Third Edition

22.5 The Equilibrium Populations of Molecular States 945

Solution

From Example 22.5,νe 6. 5049 × 1013 s−^1 :

E 10 −E 00

kBT



hνe

kBT



(

6. 6261 × 10 −^34 Js

)(

6. 5049 × 1013 s−^1

)

(1. 3807 × 10 −^23 JK−^1 )( 298 .15 K)

 10. 47

N(1, 0)

N(0, 0)

e−^10.^47  2. 84 × 10 −^5

The populations of excited vibrational states are very small at room temperature for

typical diatomic molecules. However, the difference is smaller with molecules that

have smaller vibrational frequencies.

Exercise 22.11

Find the ratio of the population of theν1 vibrational level to that of theν0 vibrational

level for the I 2 molecule at 500.0 K.

The rotational and vibrational energies are independent of each other. It is a fact

of probability theory that the probability of the occurrence of two independent events

is the product of the probabilities of the two events. If we denote the probability of a

vibrational levelνbypvib(ν) and the probability of a rotational level byprot(J), the

probability that these two levels are simultaneously occupied is

pvib,rot(ν,J)pvib(ν)prot(J) (22.5-10)

The same result can be obtained by combining the two contributions to the energy.

EXAMPLE22.14

For CO at 298.15 K, find the ratio of the population of the level withν1,J2tothe

ν0,J0 level.

Solution

From the previous two examples,

Probability(4.728)(2. 84 × 10 −^5 ) 1. 34 × 10 −^4

The Rotation and Vibration of Polyatomic Molecules

The states of polyatomic molecules are governed by the same Boltzmann probability

distribution as those of atoms and diatomic molecules. The rotational levels of poly-

atomic molecules are generally large enough that many rotational states are occupied.

The rotation of a linear polyatomic molecule such as acetylene or cyanogen is just