Physical Chemistry Third Edition

1072 25 Equilibrium Statistical Mechanics. I. The Probability Distribution for Molecular States

zrot

8 π^2 kBT

h^2



kBT

hBe



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

(

6. 6261 × 10 −^34 Js

)(

5. 0123 × 1010 s−^1

) 123. 95

hve

kBT



(6. 6261 × 10 −^34 J s)(5. 7086 × 1013 s−^1 )

(

1. 3807 × 10 −^23 JK−^1

)

( 298 .15 K)

 9. 1889

zvib

1

1 −e−hve/kBT



1

1 −e−^9.^1889

 1. 0001

zelec≈ 2 + 2 e−ε^1 /kBT 2 +2 exp

(

− 2. 380 × 10 −^21 J

(

1. 3807 × 10 −^23 JK−^1

)

( 298 .15 K)

)

≈ 2 + 2 e−^0.^57815  3. 1218

zztrzrotzvibzelec(3. 8893 × 1030 )(123.95)(1.00010)(3.1218)

 1. 5052 × 1033

b.ztris unchanged. A calculation as in part a shows thatzvibfor the excited state is

equal to that of the ground-state to the number of significant digits given. Denote par-

tition functions for the ground-state by the subscript (gs) and those for the excited state

by (ex):

zrot(ex)

8 π^2 kBT

h^2



kBT

hBe



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

(

6. 6261 × 10 −^34 Js

)(

5. 1569 × 1010 s−^1

)

 120. 47

zztr

[

(zrot(gs)zvib(gs))(2)+(zrot(ex)zvib(ex))(2)e−^0.^57815

]

 3. 8893 × 1030

[

(123.95)(1.00010)(2)+(120.47)(1.0001)(2)(.5609325)

]

 1. 4900 × 1033

The partition function of part a is in error by about 1.0%.

The molecular partition function for a diatomic gas can also be corrected by going

beyond the harmonic oscillator–rigid rotor approximation, including the correction

terms in Eq. (22.2-45) for the energy levels.^6 We do not discuss these corrections.

Polyatomic Gases

In the harmonic oscillator–rigid rotor approximation polyatomic molecules obey the

same separation of their energy into four independent terms as in Eq. (25.4-5). In this

approximation the molecular partition function of a polyatomic substance factors into

the same four factors as in Eq. (25.4-6). The translational partition function is given by

(^6) N. Davidson,op. cit., p. 116ff (note 2).