Physical Chemistry , 1st ed.

If we increase the number of particles as well as the number of possible mi-

crostates, an interesting situation develops. As long as N, the graphs start

looking qualitatively like the curves in Figure 17.8. Of course, as Nincreases,

increases dramatically—the graphs in Figure 17.8 are not drawn to the same

scales! But although the absolute heights of the peaks in Figure 17.8 increase

factorially, the widths of the peaks do not increase at the same rate. Therefore,

as Nincreases, the relative shape of the peaks gets narrower and narrower. This

implies that although the number of possible combinations is growing, the

number of combinations that are populated to any significant extent is getting

progressively smaller. Thus, the combination corresponding to the most prob-

able combination overwhelms any other combination.

What this argument implies is that we don’t have to consider every possible

set of occupation numbers in considering the microstates of an ensemble. For

large N, only the most probable distribution needs to be considered.

If we have an expression for in terms of the Njvalues, we can take the de-

rivative of that expression with respect to the set ofNj’s and set it equal to zero

[recall that at a maximum point in a plot, the derivative (that is, the slope)

equals zero]. We can then derive some expression that might be meaningful.

But we do have an expression for in terms ofNj: equation 17.9. However,

equation 17.9 can’t be maximized by itself. It must be maximized in terms of

the constraints on E(equation 17.10) and N(equation 17.11).

Rather than maximize , we instead maximize ln. We can do this because

ln increases and decreases as increases and decreases, so a maximum value

of corresponds to a maximum value of ln. In addition, by maximizing

ln we can take advantage of Stirling’s approximation. From equation 17.9,

we evaluate ln as

ln 

ln  

j

gjNj

Using the properties of logarithms,* we simplify the right side of the equation

to get

ln 

ln N! 

j

(ln gjNjln Nj!)

By invoking Stirling’s approximation on both factorial terms and applying an-

other property of logarithms,†we get

ln 

Nln NN

j

(Njln gjNjln NjNj)

We can distribute the summation sign through the three terms to get

ln 

Nln NN

j

Njln gj

j

Njln Nj

j

Nj

The summation Njequals N, the total number of particles, so those two

terms cancel on the right side. Again, we can use properties of logarithms and

rearrange the remaining terms inside the summation to get

ln 

Nln N

j

Njln 

N

gj

j

 (17.12)

N!



j

Nj!

594 CHAPTER 17 Statistical Thermodynamics: Introduction

W

(or )

W

(or )

W

(or )

W

(or )

W

(or )

W

(or )

Figure 17.8 When the number of particles
and possible arrangements gets larger and larger,
the plot gets progressively narrower and narrower,
even as the x-axis (representing the possible sets
of occupation numbers) gets larger and larger.

*Specifically, ln (ab/c) ln aln bln c. Another way to express this is ln

jN†jjln Nj.

Specifically, ln abbln a.