### 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)

i.e.,

( 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

and

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