[UPSC IFoS 2015] Thermodynamics (Q84, 5+15=20M)

Start from the equation T dS = C_p dT - T \left( \frac{\partial V}{\partial T} \right)_P dP and get the relation \left( \frac{\partial C_p}{\partial P} \right)_T = -T \left( \frac{\partial^2 V}{\partial T^2} \right)_P Hence, show that the third law of thermodynamics requires for the coefficient of thermal expansion of any substance to vanish at T = 0.