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Question

Using the equation of state for an ideal gas, pV = N kbT, and suitable thermodynamic identities, show that the

internal energy of an ideal gas only depends on temperature if the number of particles is fixed (or alternatively the chemical potential is zero). This implies that U = U(T) is a direct consequence of the specific form of the equation of state for an ideal gas.

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Solution:
Entropy and internal energy essentially looks like
the following representation:-
U = U/ T, v, P ) and S = S ( T, V )
Therefore,
as = as aT + as
aT lv
du
du = Ids - Pdv
- Tas dT + Tas av-...

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