### Entropy of a d-dimensional ideal gas

I’m slowly reading through Kittel and Kroemer to refresh my knowledge of basic statistical mechanics and thermodynamics. They have a nice little problem at the end of Chapter 3 (Problem 11), which is to calculate the entropy of a one-dimensional ideal gas (using the methods outlined in that chapter). It is only a brief step further to calculate the general expression for the entropy of a -dimensional ideal gas, which I do here.

The single-particle partition function is calculated using the “particle-in-a-box” solution from quantum mechanics, and the end result is that

where

Define the -dimensional quantum concentration as , then by plugging in we get

The free energy is (using the Stirling approximation)

and thus the entropy is

Now, we can use the properties of the logarithm to make taking the derivative really easy, since

where I have dropped all of the terms that don’t depend on the temperature. So

and the final result for the entropy is

Here I have defined .

For 1,2, and 3 dimensions, we have

and

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