Physical Chemistry , 1st ed.

the system will be represented by each drawing in Figure 17.6. Because there

are more ways of making a system with the occupation numbers (1, 0, 1, 1),

there will be more microstates having that arrangement. This idea becomes

very important when considering large numbers of gas particles that exist in

any real system.

17.4 The Most Probable Distribution:

Maxwell-Boltzmann Distribution

Let us now consider a canonical ensemble that describes an isolated system

in which Nj,Vj, and Tare the same for all microstates. The energies of each

microstate,Ej, are not equal, but we are still assuming that discrete energy

levels exist, as dictated by quantum mechanics. Previously, we showed that

the number of ways to distribute particles among the energy levels of the

microstates is



j

gjNj

There are some constraints on this equation. The total energy of the system

must be equal to the energy of each quantized state, (^) i, times the number of
particles in that energy level,Ni:
E
i
Ni i (17.10)
Furthermore, the sum of all the Nivalues must equal the total number of par-
ticles in the system:
N
i
Ni (17.11)
(Note the slight difference in the definitions of the number of particles.Nirep-
resents the number of particles in each energy level, and Nj(different sub-
script) represents the number of particles in each microstate. In a canonical
ensemble,Njis the same for each microstate, but there is no requirement that
Niis the same for all microstates.)
There is still the problem that the set of occupation numbers Njcan be any-
thing, according to the principle of equal a priori probabilities. In fact, the to-
tal possible arrangements are truly astronomical, but we will ignore all but one:
the most probable arrangement.
Consider a bar graph that plots the number of ways a combination can be
made versus what we will call the compactness of the arrangement. By “com-
pactness,” we mean a general understanding of how many different microsys-
tems participate in the specific arrangement. For instance, in Example 17.2
there were two particular arrangements in which two energy levels were
populated and one arrangement in which three energy levels were populated.
We can say that the three-level population was less compact than the other
two, which were more compact in either a lower energy level or a higher en-
ergy level. Notice that the less compact arrangement had six possible combi-
nations, but the two more compact arrangements had only three possible
combinations each. If we plot the number of combinations versus the
compactness—with each side of the plot representing either extreme in com-
pactness and the middle representing the least compact arrangement—we
get a graph like Figure 17.7.

N!





j

Nj!

17.4 The Most Probable Distribution: Maxwell-Boltzmann Distribution 593

Figure 17.7 If we plot the number of possi-

bilities versus the set of occupation numbers, we

see the beginnings of a curve. See Figure 17.8 to

see how this curve evolves.

W

(1, 0, 1, 1) (0, 1, 2, 0)

6

3

(0, 2, 0, 1)