Physical Chemistry , 1st ed.

tronic, translational, rotational, vibrational, and other quantum states. We will

denote the energy level of the gas particle as (^) j: 0 is the ground (that is, min-
imum-energy) state, 1 is the first excited state, 2 is the second excited state,
and so on. If we are going to understand the thermodynamics of the macrostate,
we will need to keep track of how many gas particles have which energy in
which microstate.
If we assume that N 0 gas particles are in the energy state 0 ,N 1 particles are
in energy state 1 ,N 2 particles are in energy state 2 , and so on, then how many
ways can the particles be distributed with this total energy? This is a combina-
tion type of problem, just like putting balls into boxes. If we denote the num-
ber of ways we can make this arrangement as W, then by applying equation
17.1, we have
W (17.7)
This is just the number of ways of arranging the right number of particles in
the specified energy states. If we consider the degeneracy of each energy state,
gj, then the total number of possible ways must include the degeneracies as a
factor. If there are Njparticles with degeneracy gj, then the number of ways the
Njparticles can be arranged among these degenerate states is (gj)Nj.[For ex-
ample, having two particles with doubly degenerate wavefunctions allows for
four possible ways (2^2 ) the particles can have particular wavefunctions.] For all
particles and all degeneracies, the total number of arrangements due to de-
generacies is the product of the individual degeneracies:
Wdeg(g 0 )N^0 (g 1 )N^1 (g 2 )N^2 (g 3 )N^3 
j
(gjNj) (17.8)
in which the degeneracy of the ground state,g 0 , is taken N 0 times (for the N 0
particles that have energy 0 ), and so on. Keep in mind that degeneracies can
be very large; for translational states of a mole of atoms, the degeneracy is on
the order of 10^20. The total number of ways these Nparticles might exist in
that arrangement, denoted , is the product ofWand Wdeg:

WWdeg
j
gjNj (17.9)
N 0 ,N 1 ,N 2 , etc., are the occupation numbers for each particular energy level.
As you might expect, is a hugenumber. Additionally, we have assumed a spe-
cific set of occupation numbers Nj. Can the system have some other set ofNj
values? Of course it can. One of the questions of statistical thermodynamics is
whether it can predict whichset of occupation numbers is most probable.
Example 17.2
Assume that you have a three-particle system that has four possible energy
states, as shown in Figure 17.5. Your system has a total of 5 energy units (5 EU’s)
to distribute among the particles. How many different distinguishable distri-
butions can there be? Can you use the occupation numbers to verify equa-
tion 17.7?
Solution
Refer to Figure 17.6. Each drawing shows a way to distribute the particles in
the possible quantum states so that the complete system has 5 EU’s. Under

N!





j

Nj!

N!

N 1! N 2! N 3! 

17.3 The Ensemble 591

Energy E  0 EU

E  1 EU

E  2 EU

E  3 EU

Figure 17.5 Refer to Example 17.2. How many
ways are there of distributing three distinguish-
able particles so that there are 5 energy units in
the system?