3.D.2. Coexistence Curves: Clausius-Clapeyron Equation

The molar Gibbs energies (chemical potentials), g= G n =μ , of 2 coexisting phases I and II, must be equal.  Thus,

        d g I =d g II          along the coexistence curve


For a PVT system, this means

        v I dP s I dT= v II dP s II dT                     (3.12)


        ( dP dT ) coex = s I s II v I v II = Δs Δv                 (3.13)

                        = Δh TΔv                                      (3.14)

where Δh=TΔs  is the molar latent heat.  Eq(3.14) is called the Clausius- Clapeyron equation.

Exercise 3.1

Prove that Δh>0  in a transition from a low to a high T phase.


Let II be the low and I be the high T phase.

Since the stable state has the lowest G, we have

        G I > G II           for T< T C

        G I < G II           for T> T C

Coupled with the stability conditions of section 2.H.3, we obtain a situation as depicted in the Figure.  Thus,

        ( G I T ) P{ n j } < ( G II T ) P{ n j }          for all T

so that

        S I > S II                     for all T


        ΔH= T C ΔS= T C ( S I S II )>0


3.D.2.a.      Vaporization Curve

3.D.2.b.      Fusion Curve

3.D.2.c.      Sublimation Curve