• The Clausius–Clapeyron relation is a way of characterizing a discontinuous phase transition between two phases of matter of a single constituent.

• On a pressure–temperature (P–T) diagram, the line separating the two phases is known as the coexistence curve.

• The Clausius–Clapeyron relation gives the slope of the tangents to this curve. Mathematically,

$$\frac{dP}{dT} = \frac{L}{TΔv}$$

Where,

dP/dT is the slope of the tangent to the coexistence curve at any point,

L is the specific latent heat,

T is the temperature,

Δv is the specific volume change of the phase transition.

Proof

β = expansion coefficient = change in volume per unit original volume per unit temperature change

β = dV/dT/V

V = dV/dT/β = V/(Tβ)

V = TβV

K = isothermal compressibility = volumetric strain per unit change in pressure

K = dV/V/dP = βdT/dp

dP = dV/(VK); P = βT/K

$Cp-Cv = R = PV/T = βT/K x TβV /T = β^2TV/K$