Concise Physical Chemistry

c01 JWBS043-Rogers September 13, 2010 11:20 Printer Name: Yet to Come

THE SUM-OVER-STATES OR PARTITION FUNCTION 11

or

∑

Ni=

∑

N 0 gie−εi/kBT =N

SinceN 0 is a number appearing in each term of the sum, it can be factored out:

N 0

∑

gie−εi/kBT=N

DividingNi=N 0 gie−εi/kBTbyN 0

∑

gie−εi/kBT=N,gives

Ni

N

=

N 0 gie−εi/kBT

N 0

∑

gie−εi/kBT

=

gie−εi/kBT

Q

where we have given the symbolQto the summation

∑

gie−εi/kBT. This important

summation appears frequently and is given the namesum-over-statesorpartition

function. Rewriting the ratioNi/N,wehave

NiQ=Ngie−εi/kBT

︸ ︷︷ ︸

fixed atT=const

We see that, for a given number of moleculesNat temperatureT, the right hand

side of the equation is fixed for any specific energy state, that is,NiQ=const.

The sum-over-states is then a scaling factor, determining the relative population of a

state,Ni=const/Q.IfQis large, the state is sparsely populated. IfQis small, it is

densely populated. Another way of looking atQis that it is an indicator of the number

of quantum states available to a system. For many available states (largeQ)agiven

state is less densely populated than it would be if only a few states were available

(smallQ).

Theoccupation numberof a quantum state relative to the total number of particles

Ni/Nis, strictly speaking, a probability; however, given the immense number of

particles in a mole of gas, we may treat it as a certainty. The summation of all

possible fractionsNi/Nmust be 1:

∑

i

Ni

N

=

∑

i

Ni

N

=

N

N

= 1

The probability distribution of molecular velocities in 3-space is a collection of

randomly oriented vectors away from an origin of the zero (least squares) estimate.

Plotted in velocity-space, the probability density approaches zero near the origin.

This comes about when we use spherical polar coordinates to express our probability

densities because, near the origin, the volume of the spherical space becomes very

small. As the radial distance is increased, the space becomes larger but the probability