In equilibrium, we know that the first-derivative of thermodynamic quantities of two systems should be equal, for this specifies a maxima or minima of the original thermodynamic function. In terms of phases, the best variable to examine is μ, specifically for when a gas and liquid phase of a system coexist:
μg(p,T) = μl(p,T)
Also, because we are examining a physical continuous system:
μg(p + dp,T + dT) ~= μl(p + dp,T + dT)
If we take these and play around with them using a Taylor's expanaion, limiting dp and dT small enough that higher order terms fall out, and play with them algebraically:
dp/dT = ((∂μl/∂T)p - (∂μg/∂T)p)/((∂μl/∂p)T - (∂μg/∂p)T)
Using some Maxwell relations and the fact that for a single-specie system G=Nμ(p,t) (in order to satisfy G's differential equation) we can obtain:
dp/dT=(sg - sl)/(vg - vl)
But if we make the following definitions:
L := T(sg - sl)
Δv= = (vg - vl)
which is know as the Clausius-Clapeyron equation, which tells the form of the vapor-pressure line where the gas and liquid phases of a material meet.
Another way of looking at the phase transition point is to examine the Gibbs free energy. There is a free energy curve for the material for each of its phases. What phase it's in is determined by the curve with the lowest free energy (as said in the previous chapter). Therefore, the place of coexistance is where Gg=Gl.
Let's take a look at Gibbs' differential form:
N=1 and is constant, and let's assume T is constant, then:
Integrate over the curve so that:
Gg - Gl = ∫Vdp
But since we are at a phase equilibrium position:
∫Vdp = 0
over such a curve for a material. This relation allows us to determine something called the 'coexistance line' on a P-V diagram, and is known as Maxwell's Construction.