Physical Chemistry Third Edition

25.4 The Calculation of Molecular Partition Functions 1069

The value in Example 25.7 is typical for molecules of moderate size near room

temperature, and is large enough to be a good approximation.

The Vibrational Partition Function

If the zero-point vibrational energy is included as part of the electronic energy, the

vibrational energy of a diatomic molecule is given by

εvibεvhνevhcν ̃ev (25.4-14)

wherevis the vibrational quantum number. The vibrational frequencyνeis given by

Eq. (22.2-30):

νe

1

2 π

√

k

μ

(25.4-15)

wherekis the vibrational force constant andμis the reduced mass of the nuclei. The

parameter ̃νeis equal toνe/cwherecis the speed of light.

Since the vibrational levels are nondegenerate, the vibrational partition function of

a diatomic molecule is

zvib

∑∞

v 0

e−hνev/kBT

∑∞

v 0

av (25.4-16)

whereaexp(−hν/kBT). This sum is ageometric progressionwith infinitely many

terms, given by a well-known formula

∑∞

v 0

av

1

1 −a

(25.4-17)

This formula is valid if|a|<1, which is satisfied byaexp(−hνe/kBT). The vibra-

tional partition function is given by

zvib

1

1 −e−hνe/kBT



1

1 −e−hcv ̃e/kBT

(25.4-18)

EXAMPLE25.8

Calculate the vibrational partition function of^35 Cl 2 at 298.15 K. The vibrational frequency

is 1. 6780 × 1013 s−^1.

Solution

Letxhνe/kBT:

x

(6. 6261 × 10 −^34 J s)(1. 6780 × 1013 s−^1 )

(

1. 3807 × 10 −^23 JK−^1

)

( 298 .15 K)

 2. 701

zvib

1

1 −e−x



1

1 −e−^2.^701

 1. 0720