Physical Chemistry Third Edition

1092 26 Equilibrium Statistical Mechanics. II. Statistical Thermodynamics

Exercise 26.6

a.Carry out the steps of algebra to obtain Eq. (26.2-8).

b.Find the expression for the vibrational energy if the zero-point energy is included in the

vibrational energy, giving the vibrational partition function of Eq. (25.4-20).

For a polyatomic substance, the vibrational energy contains one term for each normal

mode:

UvibNkBT^2

(

∂ln(zvib)

∂T

)

V



3 n∑−5(6)

i 1

NkBT^2

d

dT

[

ln

(

1

1 −e−hνi/kBT

)]

Uvib

3 n∑−5(6)

i 1

Nhνi

ehνi/kBT− 1



3 n∑−5(6)

i 1

Nhcν ̃i

ehcν ̃i/kBT− 1

(

polyatomic

substances

)

(26.2-9)

The upper limit on the sum equals 3n−5 for linear polyatomic molecules and equals

3 n−6 for nonlinear polyatomic molecules.

EXAMPLE26.4

The vibrational frequency of CO is equal to 6.5048× 1013 cm−^1.

a.Find the value of the vibrational partition function at 298.15 K.

b.Find the vibrational contribution to the molar energy at 298.15 K.

Solution

We set the zero of vibrational energy at the lowest vibrational state. We let

x

hν

kBT



(6. 6261 × 10 −^34 J s)(6. 5048 × 1013 s−^1 )

(1. 3807 × 10 −^23 JK−^1 )(298.15 K)

 10. 47

a. zvib 1 −^1 e−x 1. 0000284

b. UvibeNhνx− 1



(6. 022 × 1023 mol−^1 )(6. 6261 × 10 −^34 J s)(6. 5048 × 1013 s−^1 )

e^10.^47 − 1

 0 .737 J mol−^1

The closeness of the partition function to unity and the small value of the vibrational energy

show that the ground vibrational state is the only significantly occupied vibrational state.

Exercise 26.7

a.Repeat the calculation of Example 26.4 with the zero-point energy included in the vibrational

energy, so that the vibrational partition function is given by Eq. (25.4-20).

b.Find the value of the vibrational partition function and the vibrational energy for 1.000 mol

of gaseous iodine at 500 K with the zero of energy taken at the zero-point energy.

c.Find the value of the vibrational partition function and the vibrational energy for 1.000 mol

of gaseous iodine at 500 K with the zero of energy taken at the minimum of the vibrational

potential energy.