Physical Chemistry Third Edition

25.2 The Probability Distribution for a Dilute Gas 1049

and the most probable distribution resembled each other. Although we do not prove

this fact, the average distribution and the most probable distribution become identical

in the limit of an infinitely large system, which is called thethermodynamic limit.

LetW({N}) be the number of system microstates that correspond to a given

distribution{N}. The sum of theW’s for all distributions that correspond to the correct

values ofE,V, andNobeys

∑

{N}

W({N})Ω (25.2-5)

where the sum includes one term for each distinct distribution. By the second postulate

of statistical mechanics, all system microstates with the correct values ofE,V, andN

are equally probable. The distribution with the largest value ofWis therefore the most

probable distribution.

We now obtain an expression forWfor a given distribution. Consider energy level

numberjwith degeneracygj. LetNjbe the number of molecules occupying states in

levelj. We will assume that our system contains very many molecules. Since we have

already assumed that it is a dilute gas, we must assume that it is confined in an extremely

large volume. Equation (22.1-4) shows that when the dimensions of a rectangular

container become very large, the translational energy levels of a particle confined

in the container become very close together. The degeneracies of the translational

energy levels also become very large, and if necessary, we can group energy levels

of nearly equal energies together to get very large degeneracies. These assumptions

correspond to

Nj 1 (25.2-6)

and

gjNj (25.2-7)

whereNjis the number of molecules occupying molecule energy levelj, andgjis the

degeneracy of the level. The condition of Eq. (25.2-7) is calleddilute occupation.

We first assume that the molecules are fermions, so that no more than one molecule

can occupy each state. We need the number of ways to divide thegjstates of levelj

intoNjoccupied states andgj−Njunoccupied states. We denote this number bytj.

Counting up this number of ways is an elementary statistics problem.^3 Consider first

a set ofNballs andNboxes, with each box able to hold no more than one ball. The

first ball can go into any ofNboxes, the second ball can go into any ofN−1 boxes,

and so forth, until the final ball has only one empty box to go into. Since the choices

are independent, the number of ways of making all of the choices is the product of the

number of ways of making the individual choices:

Number of ways of choosingN(N−1)(N−2)···(3)(2)(1)N! (25.2-8)

whereN! stands forN factorial, the product of all of the integers starting withNand

ranging down to 1. We say thatN! is the number ofpermutationsofNobjects.

Now consider the number of ways to pickNoccupied states fromgpossible states,

whereg>N. This is analogous to having more boxes than balls. There aregchoices

(^3) J. E. Freund,Modern Elementary Statistics, 7th ed., Prentice-Hall, Englewood Cliffs, NJ, 1988,
p. 97ff.