I have to prove the identity (∂U∂V)T=T(∂p∂T)V−p
I think that I have to start from the equation for internal energy and then differentiate it using dV - this produces almost the end result, but when I work with the Jacobian I'm always lost.
Any explanations on how to use the Jacobian would be great. I know the basic rules, but cannot solve this.
Could you help me please?
Answer
In a reversible cyclic process the heat absorbed is the work done, hence ∫CTdS=∫CpdV where C is a closed contour. But it is also true that for the enclosed area A within C we have ∫CTdS=∬AdTdS and ∫CpdV=∬AdpdV, so ∬AdTdS=∬AdpdV, and if we go from the variables T,S to p,V one has ∬AdTdS=∬AdpdV∂(T,S)∂(p,V). This implies that the Jacobian must be 1: ∂(T,S)∂(p,V)=1
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