Physical Chemistry Third Edition

1044 25 Equilibrium Statistical Mechanics. I. The Probability Distribution for Molecular States

over system microstates. Themean valueis the most common type of average. The

mean of a set of valuesw 1 ,w 2 ,...,wNis given by

〈w〉

1

N

(w 1 +w 2 + ··· +wN)

1

N

∑N

i 1

wi (25.1-7)

If some of values are equal to each other, we can write a sum with fewer terms. We

arrange the values so that the values fromw 1 towMare all different from each other

and all of the other values are equal to one or another of the firstMvalues. LetNibe

the number of values equal towi. We can write the mean value with a shorter sum:

〈w〉

1

N

∑M

i 1

Niwi

∑M

i 1

piwi (25.1-8)

wherepiNi/N. The quantitypiis the fraction of the values that equalwiand is also

equal to the probability that a randomly chosen value will equalwi.

According to our first postulate, macroscopic quantities are equal to averages of

appropriate mechanical quantities. The macroscopic energy is defined thermodynam-

ically in terms of work and heat. It is denoted byUand is called the internal energy

or thermodynamic energy. We now assert that the internal energyUis equal to〈E〉,

the average mechanical energy of the system. In our example, we consider only states

withE 4 hν,soU 4 hν. The average energy of an oscillator is given by

〈ε〉

∑^4

v 0

pvεvp 00 +p 1 hν+p 22 hν 3 +p 33 hν+p 44 hν (25.1-9)

This must equalU/4, which is equal tohν.

Exercise 25.4

Calculate〈ε〉using the probabilities in Eq. (25.1-6) and show that to five significant digits

〈ε〉(1.0000)hν.

When we discuss a real system, the average distribution will be impossible to calcu-

late because of the large number of microstates. We will seek another approach, which

we now illustrate with our model system. Return to Table 25.1 and look at theWvalues

of the different distributions. Since all of the system microstates are equally probable,

the distribution with the largestWis themost probable distribution. In our example,

there are two distributions that correspond toW12, so we call the average of these

two distributions the most probable distribution.

Table 25.2 and Figure 25.1 exhibit the average distribution, the most probable dis-

tribution, and the Boltzmann distribution of Eq. (22.5-2) that corresponds to the same

average energy per oscillator. All three distributions have the same general trend, that

molecular states of higher energy are less probable. The Boltzmann distribution does

not stop atv4, because it applies to a system of many oscillators with an average

energy equal tohν.

Exercise 25.5

Show that the most probable distribution gives the correct value of〈ε〉.