Question

Prove that raising the pressure will increase melting point (T_m) of a metal. Use the Clausius-Clapeyron eqn., and remember that the enthalpy of a liquid is larger that the enthalpy of solid. Then consider the difference in molar volumes of the liquid and the solid

Answer 1

The derivation will be given for a liquid-vapor equilibrium interface but it equally well applies to the interface between any two phases.

Let s_v and s_l be the specific entropy of the vapor and liquid phases, respectively. The pressure and temperature of the two phases are equal. The chemical potential ? is equal on either side of the phase boundary curve. Therefore the changes d? in the chemical potential for movements along the phase boundary curve are also equal. This means that

d?_v = V_vdp - s_vdT = V_ldp - s_ldT = d?_l

Solving for dp gives  

dp = (s_v - s_l)dT/(V_v - V_l)

or, equivalently
dp/dT = (s_v - s_l)/(V_v - V_l)

But the change in entropy ?s for the phase change is just L/T so

dp/dT = L/(T(V_v - V_l))

This is the Clausius-Clapeyron equation.