Concise Physical Chemistry

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

118 A STATISTICAL APPROACH TO THERMODYNAMICS

allowed. Now, however, electronic excitation, while quantum mechanically permit-

ted, does not occur because the thermal energy at this temperature is insufficient to

drive the electrons out of their ground state. The equilibrium expression is somewhat

simpler than one might expect:

Keq=

[Qtr(Na)Qel(Na)]^2

Qtr(Na 2 )Qrot(Na 2 )Qvib(Na 2 )

e−ε^0 /RT

The first partition function we need isQtrfortranslationalmotion. That simply

means the partition function for sodium molecules or atoms (or anything else) fly-

ing around and bouncing off the walls of some container. Imposition of the laws

of quantum mechanics brings about an astonishing phenomenon: Simply by being

confined to a container or “box,” the particles suddenly have allowed and forbidden

energy levels. Their energy isquantized. Energy levels of a particle in a box form

exactly the same kind of energy-level manifold we have been talking about except

that, unlike the harmonic oscillator example, they are not evenly spaced. Instead, they

go up according to the equation (Chapter 19)

E=

n^2 h^2

8 ma^2

The integersn= 1 , 2 , 3 ,...are called thequantum numbers,h= 6. 626 × 10 −^34 Js

is Planck’s constant in joule seconds,mis the mass of the particle, andais the length

of one edge of the box (taken to be cubic for simplicity).

When this energy restriction is placed on translational motion, it influences the

partition function for a cubic box of volumeV=a^3 according to the equation

Qtr=

(

2 πm

h^2 β

)^32

a^3 =

(

2 πm

h^2 β

)^32

V

where we have again used a convenience variable, ubiquitous in this field,β= 1 /kBT.

Inverting terms in the parentheses, we obtain

Qtr=

(

h^2 β

2 πm

)−^32

V=

V

(

h^2 β

2 πm

)^32 ≡

V

3

In the jargon of this field, the cube root of the denominator is called thethermal

wavelengthand given the special symbol :

=

(

h^2 β

2 πm

)^12