[UPSC CSE 2012] Thermodynamics (Q99, 20M)

Establish the relation \left(\frac{\partial T}{\partial V}\right)_P=-\left(\frac{\partial P}{\partial S}\right)_T and then derive \left(\frac{\partial C_P}{\partial P}\right)_T=-T\left(\frac{\partial^2 V}{\partial T^2}\right)_P. Hence show that the heat capacity C_P of an ideal gas is independent of pressure P.