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Partition function, chemical potential and free energy

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Derive equations: S = -(&#8706;F/&#8706;T)_V, N = Nk [ln (V/N_vQ) + 5/2] - &#8706;F_int/&#8706;T and &#956; = (&#8706;F/&#8706;N)_T, V = -kT ln (VZ_int/N_vQ) for the entropy and chemical potential of an ideal gas.

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We can derive the expressions from the fundamental law of thermodynamics:

dU = T dS -PdV +&#956; dN

If you write.

T dS = d(TS) - S dT

and bring the d(TS) to the left hand side, you get:

dF = -S dT - PdV + &#956; dN

From this equation you read off that:

S = -(&#8706;F/&#8706;T)_V, N

and

&#956; = (&#8706;F/&#8706;N)_T, ...

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