Physical Chemistry Third Edition

25.4 The Calculation of Molecular Partition Functions 1065

EXAMPLE25.6

For the case thatg 0 1,ε 0 0 andε 1  1 .00 eV, show thatzel≈1 is an adequate approx-

imation atT300 K.

Solution

ε 1

kBT



(1.00 eV)(1. 602 × 10 −^19 JeV−^1 )

(1. 3807 × 10 −^23 JK−^1 )(300 K)

 38. 7

e−^38.^7  1. 6 × 10 −^17

All terms are negligible except for the first term andzel1 to an excellent approximation.

The result of this example justifies our earlier assumption that there was no significant

electronic excitation in our dilute monatomic gas.

Diatomic Gases

In addition to translational and electronic motion, a diatomic gas has rotational and

vibrational motion. To a good approximation the energy is a sum of four separate

terms, as in Eq. (22.2-37):

εjεjtr+εjrot+εjvib+εjel (25.4-5)

Equation (25.4-5) leads to a factoring of the molecular partition function:

z

∑

jtr

gjtre−βεjtr

∑

jrot

gjrote−βεjrot

∑

jvib

gjvibe−βεjvib

∑

jel

gjele−βεjel

ztrzrotzvibzel

(

diatomic or polyatomic

substances

)

(25.4-6)

The Translational and Electronic Partition Functions

The translational energy is given by the same formula as for atoms, and the translational

partition function is given by Eq. (25.3-21). As with atoms, there is no general formula

for the electronic energy. Part of the Born–Oppenheimer energy acts as a potential

energy of vibration, and the rest is a constant that is assigned to be the electronic energy.

There are two ways to assign this energy. The first is to assign the minimum value of

the Born–Oppenheimer energy asεeland to assign the remainder to be the vibrational

potential energy, as was done in Eq. (22.2-41). The second choice is to include the

zero-point vibrational energy as part of the electronic energy as in Eq. (22.2-43), setting

the vibrational ground-state energy equal to zero. Both these choices are depicted in

Figure 25.3. With either choice, the electronic energy is taken as a different constant

for each electronic state, and the formula in Eq. (25.4-2) is summed explicitly. Except

for a few unusual cases, such as NO, only the first term is significantly different from

zero, as is the case with atoms. For example, the ground-level of O 2 is a^3 Σ−gstate with

a degeneracy of 3, and all excited states are high enough in energy to be ignored at