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I am struggling with a thermodynamics question given $$S(U,V,N)=C_VNK_\mathrm b\ln U/U_0+NK_\mathrm b\ln V/V_0$$ (where $U_0$ and $V_0$ are the reference energies and volumes) and the first law of thermodynamics which is $\mathrm dU=T\,\mathrm dS-p\,\mathrm dV+\mu\,\mathrm dN$. The question asks to use the two expressions above to derive the pressure $p$ as a function of $T$, $U$, $V$ and $N$.

Would anyone be able to give me a hint as to where to start with this question?

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    $\begingroup$ What are your thoughts so far? $\endgroup$
    – Chet Miller
    Aug 31, 2022 at 10:53
  • $\begingroup$ Start by writing $dS=\left(\frac{\partial S}{\partial U}\right)_{V,N}dU+\left(\frac{\partial S}{\partial V}\right)_{U,N}dV+\left(\frac{\partial S}{\partial N}\right)_{U,V}dN$ $\endgroup$
    – Chet Miller
    Sep 1, 2022 at 20:12

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