Clausius Clapeyron equation

The temperature at which a phase transition occurs depends on the ambient pressure. For example at atmospheric pressure water would boil at 100 ⁰ C but in a pressure cooker it boils at a much higher temperature because of the high pressures involved. Thus we may plot a curve between the ambient pressure and the corresponding temperature for the phase transition.

The Clausius Clapeyron equation gives the slope of this curve at any point. There are lots of ways to derive it and probably the most physical method is to use the Carnot theorem. However that requires a few diagrams so I'll skip it here. If you are interested you could look at Feynman Vol 1.

Anyway by definition a first order phase transtion is a process where the Gibb's free energy G is continuous but its first derivatives are not. Consider a phase transition at T,P. If g₁ is the specific free energy of one phase and g₂ is the specific free energy of the other then:
g₁ = g₂
For another phase transition at T+dT, P+dP:
g₁ + dg₁ = g₂ + dg₂
or
dg₁ = dg₂
Also
dg = -Sdt + Vdp So
(S₂-S₁)dt = (V₂ - V₁)dp
as delta(S) = L/T where L is the latent heat and T is the temperature so
dp/dt = L/TV_del.
where V_del is the difference in the specific volumes of the two phases.