Entropy of gases

To relate entropy to statistically based quantities we follow the lead of Fast (ENTROPY, Philips Technical Library, 1962) who noted that from thermodynamics the change in entropy upon increase in volume of a gas of N molecules from V ₁ to V ₂ is

and upon mixing of n ₁ molecules of gas 1 of volume V ₁ and n ₂ molecules of gas 2 of volume V ₂ by removing a partition separating the two volumes is

Now consider that the volume V ₁, containing gas is partitioned to produce many equal volumes, the number of which is much larger than the number of molecules in the volume, and the size of which is such that no cell contains more than one molecule. As a result of the constant thermal motion of the molecules, different cells become occupied with time. Suppose there are N ₁ cells and n molecules. Thus, n cells are occupied and N ₁ ?n are unoccupied. The total number of different configurations is then

because the first molecule can fill N ₁ boxes, the second N ₁ ? 1, and so on, but not all these configurations are different because there are n indistinguishable molecules and N ₁ ? n indistinguishable cells.

Using Stirling s approximation for large numbers, ln N! = N ln N ? N then

Suppose the volume containing the n molecules were increased to...