Physical Chemistry Third Edition

1042 25 Equilibrium Statistical Mechanics. I. The Probability Distribution for Molecular States

Table 25.1 Vibrational States for a System of Four Harmonic Oscillators

System State No. (k) v 1 v 2 v 3 v 4 N 0 N 1 N 2 N 3 N 4 W({N})

1 111104000 1

2 0112 12100 12

3 012112100

4 021112100

5 101212100

6 102112100

7 201112100

8 210112100

9 120112100

10 110212100

11 211012100

12 121012100

13 112012100

14 002220200 6

15 020220200

16 200220200

17 022020200

18 202020200

19 220020200

20 0013 21010 12

21 003121010

22 010321010

23 030121010

24 100321010

25 300121010

26 013021010

27 031021010

28 103021010

29 301021010

30 130021010

31 310021010

32 000430001 4

33 004030001

34 040030001

35 400030001

The Postulates of Statistical Mechanics

Although we know the macrostate of our model system, we have no information about

which of the 35 microstates the system occupies. We make an assumption that is called

thefirst postulate of statistical mechanics:

Postulate 1. A macroscopic property of a system can be equated to the average (mean

value) of the corresponding property over all of the system microstates that are compatible

with the macroscopic state of the system.

To define the type of average to be taken, we make another fundamental assumption,

called thesecond postulate of statistical mechanics: