Configurations, Entropy, and Volume

The reasons for the behavior of the dice are not energetic but lie in statistics. Let us use the term state, but not exactly in the quantum mechanical sense of the last section, to refer to any specific arrangement of the dice. A state of the dice may be represented by a list of six integers, n -n, which tells how many dice have 1-6, respectively, showing. Thus, the initial state in our example is n, H2, n, n, nf) = (0, 0, 0, 0, 0, 6). It is 

There is a simple formula that gives the number of configurations for all possible states. If N is the number of dice and C is the number of configurations, then 

C has been written as a function of N and of the integers that specify the state. We expect that each individual configuration is equally likely in any throw of the dice. But this means that the state (1,1,1,1,1,1) is 720 times as likely to turn up as the state (0,0,0,0,0,6). 

for the entropy to be a function of the configurations, that is, for S = f(C), the following relations must hold. 

These relationships are satisfied if and only if S is proportional to the logarithm of C. Notice the effect of taking the logarithm of the expression for Cab- 

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We also know from the analysis of configurations, entropy, and volume in Chapter 1 that the change in entropy (for an isothermal or constant-temperature expansion of an ideal gas) is AS = nRA(ln V) as in Equation 1.7. The corresponding differential expression is dSj = nRd In Vj. Since dlnx = dx/x, this yields... 

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