Question

Making use of the Carnot theorem, show that in the case of a physically uniform substance whose state is defined by the parameters T and V  

(δU/δV)_T = T (δp/δT)_v - p, 

where U (T, V) is the internal energy of the substance. Instruction. Consider the infinitesimal Carnot cycle in the variables p, V.

Answer 1

We consider an infinitesimal cannot cycle with isothermal  process at temperatures T = dT and T. 

Let δA be the work done in the cycle and δQ, be heat received at the higher temperature.

Then by cannot's theorem