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
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
and the final result for the entropy is
Here I have defined .
For 1,2, and 3 dimensions, we have
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