Making use of the Carnot theorem, show that in the same of physically uniform substance whose state is defined by the parameters `T` and `V`
`(del U//delV)_T = T(del p//del T)_v - p`
where `U(T, V)` is the internal energy of the substance.
Consider the infinitesmal Carnot cycle in the variables `p, V`.

Explanation

We consider an infinitesimal carnot cycle with isothermal process at temperatures `T + dT` and `T`.
Let `delta A` be the work done in the cycle and `delta Q`, be the heat received at the higher temperature. Then by Carnot's theorem
`(delta A)/(delta Q_1) = (dT)/(T)`
On the other hand `delta A = dp dV = ((del p)/(del T))_v dT dV`
while `delta Q_1 = dU_12 + pdV = [((del U)/(del V))_T + p]dV`
Hence `((del U)/(del V))_T + p = T ((del p)/(del T))_v`.
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