[UPSC CSE 1992] Thermodynamics (Q203, 20M)

Define the thermodynamic energy functions. Using these functions establish the following relations:

(i) \left(\frac{\partial T}{\partial V}\right)_{\phi} = -\left(\frac{\partial P}{\partial V}\right)_V

(ii) \left(\frac{\partial P}{\partial T}\right)_V = \left(\frac{\partial \phi}{\partial V}\right)_T

(iii) \left(\frac{\partial T}{\partial P}\right)_{\phi} = \left(\frac{\partial V}{\partial \phi}\right)_P

(iv) \left(\frac{\partial \phi}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_V