## Phase Boundaries

The phase boundaries are defined by the fact that they represent the precise conditions of temperature and pressure under which the chemical potentials of the two phases on either side of the boundary are equal. Therefore it is possible to construct equations for the phase boundaries by setting the chemical potentials of the two phases (which are themselves functions of temperature and pressure) equal to each other, and manipulating the resulting expression to give p in terms of T. (In fact, it turns out to be simpler to discuss phase boundaries in terms of their slopes dp/dT.)

We consider a situation where two phases, α and β, are in equilibrium. We can immediately say that they have the same chemical potential. When conditions of pressure and temperature are changed to those of another point on the phase boundary, the chemical potentials of both phases change. However, since the final conditions lie on the phase boundary, the chemical potentials of the two phases must again be equal to each other, and thus the change in chemical potential must be the same for both phases. i.e. dμ_{α} = dμ_{β} . From the definition of chemical potential for a pure species (μ = G_{m}) we know that for each phase dμ = -S_{m} dT + V_{m} dp. Equating the changes in chemical potential for the two phases, we obtain:

– S_{α,m} dT + V_{α,m} dp = – S_{β,m }dT + V_{β,m} dp

which rearranges to

(V_{β,m} – V_{α,m}) dp = (S_{β,m} – S_{α,m}) dT

from which we may simply obtain the** **Clapeyron equation:

(In this expression ΔS_{trs} and ΔV_{trs} are the changes in molar entropy and molar volume respectively that accompany a change of phase.)