Concise Physical Chemistry

c08 JWBS043-Rogers September 13, 2010 11:25 Printer Name: Yet to Come

116 A STATISTICAL APPROACH TO THERMODYNAMICS

Similarly, therotational partition function Qrot, in spectroscopist’s terms is

Qrot=

1

hcBβ

=

0. 695 T

σB

whereBis the rotational resonance frequency, again in units of cm−^1 , andσis

a “symmetry number” included to take into account the fact that a symmetrical

molecule has different configurations that are, from a geometric point of view, the

same. For example, the linear molecule Na–Na rotated byone halfa rotation is

still just Na–Na, indistinguishable from the starting configuration. The difference

between distinguishable and indistinguishable configurations changes the statistics

of the problem. (The statistics of a poker game using a deck in which each ace has

an identical twin is not the same as the statistics using an honest deck.)

The rotational resonance frequency of interest for Na 2 (g) isB= 0 .1547 cm−^1.

Applying the spectroscopist’s formula to the Na–Na molecule with its symmetry

number of 2 gives

Qrot=

0. 695 T

σB

=

0 .695(1000)

2(0.1547)

= 2246

This kind of calculation will be useful to us in Section 8.8. Further discussion ofQ

determination is given in more detailed works (Nash, 2006; Maczek, 1998).

8.7 THE PARTITION FUNCTION FOR DIFFERENT MODES

OF MOTION

A molecule can absorb energy by increasing its translational motion (displacement

of the entire molecule), increasing its rotational motion, and increasing its vibrational

motion, as well as by electronic excitation. The total energy of a molecule is the sum

of energy contributions from all of its modes of motion:

ε=εtr+εrot+εvib+εel

In terms of its different modes of motion, the partition function is

Q=

∑

i

e−(εtr+εrot+εvib+εel)/kBT

Exponential sums can be written as productsea+b=eaeb, and so on, soQcan be

broken up into contributing partition functions, one for each mode of motion:

Q=

∑

i

e−(εtr)/kBT

∑

i

e−(εrot)/kBT

∑

i

e−(εvib)/kBT

∑

i

e−(εel)/kBT