Physical Chemistry , 1st ed.

dynamics must recognize that. It is why we have put off a discussion of statis-

tical thermodynamics until after our treatment of quantum mechanics.

In this chapter, we will develop the tools and apply them to atomic systems.

The monatomic inert gases (He, Ne, Ar, Kr, Xe) will serve as our examples. The

next chapter will deal with diatomic and larger molecules.

17.2 Some Statistics Necessities

In order to understand statistical thermodynamics, it is necessary to review

some statistical ideas. For example, consider a system that is composed of three

boxes that represent smaller subsystems. How many ways are there of putting

a single black ball into the three identical but separate boxes? There are three

ways, as shown in Figure 17.1. How many ways are there of putting two iden-

tical black balls in the three identical boxes, one ball per box? Again, only three.

(Verify this.)

How many ways are there to distribute a black ball and a white ball in the

three identical boxes? There are six ways. The six possibilities are shown in

Figure 17.2. Because the balls are different, the possible arrangements for our

distributions are different than if the balls were identical.

The two different systems of same- and different-colored balls illustrate the

concepts ofdistinguishableversus indistinguishableobjects. When the objects

that are being partitioned into separate subsystems are distinguishable, there

are more possible ways of arranging the objects in the subsystems. However, if

the objects are indistinguishable, there are fewer unique ways.

In considering the arrangements in Figures 17.1 and 17.2, it is common to

express the population of the subsystems (here, the boxes) in terms of proba-

bilities. In Figure 17.1, for example, in one of three total cases the ball is in the

first box. Therefore, if we are considering all possible arrangements, we might

wonder what the probability is that if any specific arrangement were selected

at random a ball would be in the first box. Since one out of three arrangements

satisfies this criterion, we can say that the probability of finding a ball in the

first box is one out of three, which we can express as ^13 or 33%. Probabilities

are often expressed as percentages.

Example 17.1

Referring to Figure 17.2, what is the probability of finding the following?

a.Any ball in the second box

b.A white ball in the third box

Solution

a.Figure 17.2 has four arrangements out of the six that have a ball in the

second box. Therefore, the probability of finding a ball in the second box is

^46 , or 67%.

b.If we specify that the ball in the third box must be a white ball, then only

two arrangements out of the six satisfy this criterion. This probability is

therefore ^26 , or 33%.

The number of possible unique groupings of distinguishable objects into

various subsystems is determined by the combination formula. If there are m

subsystems in the system and Nobjects, and there are n 1 objects in subsystem 1,

17.2 Some Statistics Necessities 587

Figure 17.1 There are only three ways of
putting a single ball in three identical boxes.

Figure 17.2 If two different balls are placed in
three identical boxes, it turns out that there are
now six possibilities. Compare this with the situ-
ation where the balls are identical: in that case
only three distributions are possible among the
three boxes. This illustrates the difference in num-
ber of possible arrangements for distinguishable
versus indistinguishable objects.