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I'm struggling with solving this question:

Show that the entropy per particle ($\ S/N$) is only a function of $\mu /KT$ for an ideal gas of Fermions and Bosons.

Assume that the gas is not relativistic, with known fixed values of $\ V \ (volume)\ ,\ T \ (teperature)\ ,\ m \ (mass)\ ,\ N \ (number\ of\ particles) $.

You don't need to solve explicitly.

Can someone help me with this questions?

Thank you!

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The Hamiltonian of an ideal bose/fermion gas: $$\hat{H}=\frac{\hat{P}}{2m}$$ Then you can compute the great partition function: $$Z = Tr \ {e^{-\beta(\hat{H}-\mu \hat{N})}}$$ and after that the grand potential: $$\Omega = -kT\ln Z$$ When you have this you can easly compute the entropy with: $$S=-(\frac{\partial \Omega}{\partial T})_{V,\mu}$$ I hope this helps, there are intermediate steps but this is only an outline. If you want more details I could recommend reading eather David Tong's lecture notes (a bit easier) or more advanced Feynman's Statistical Mechanics.

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