[UPSC CSE 1989] Thermodynamics (Q223, 20M)

Use the Maxwell relation \left(\frac{\partial s}{\partial v}\right)_T = \left(\frac{\partial p}{\partial T}\right)_v to show that the internal energy, U, of a thermodynamic system depends on the volume according to the expression \left(\frac{\partial U}{\partial v}\right)_T = T\left(\frac{\partial p}{\partial T}\right)_v - p where the symbols have their usual meanings. Use this result to derive Stefan's law for a black-body radiator.