Use the first law of thermodynamics to derive the pressure as a function of T, U, V and N

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?

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