3.C. Classification of Phase Transitions

For a system of m kinds of particles,

        G= j=1 m n j μ j                              (3.7)

Since

        dG= ( G T ) Y{ n j } dT+ ( G Y ) T{ n j } dY+ j=1 m ( G n j ) TY{ n i   |  ij } d n j

we have, for constant T and Y,

        ( dG ) YT = j=1 m ( G n j ) TY{ n ij } d n j = j=1 m μ j d n j          (3.8)

 

At a transition point, 2 or more phases can coexist in equilibrium.  The absence of matter exchange means that, for every kind of particles, the values of its chemical potential in the coexisting phases are equal, i.e.,

μ i r = μ i s             for i=1,,m                                  (a)

where r,s  denotes the coexisting phases. 

 

Without loss of generality, we shall restrict our discussion to cases where the entire system is always in a single homogeneous phase.

 

Eqs(a) and (3.7) mean that, at a transition point, G of all coexisting phases are equal.

In other words, the function G( Y,T,x )  is continuous at a transition point.

The manifest differences in the properties of the phases must then appear as discontinuities in some derivatives of G.  If the 1st order derivatives are discontinuous, the transition is called 1st order.  Otherwise, it is called continuous.

 

In an older scheme introduced by Ehrenfest, the order of the transition is taken to be the lowest order of the derivatives that are discontinuous.

 

Typical behavior of G of a PVT system near a 1st order transition is shown in Fig.3.2.

Discontinuity in ( G P ) T{ n j } =V  means a finite difference of volumes of the 2 phases,

        ΔV= V I V II = ( G P ) T{ n j } I ( G P ) T{ n j } II                            (3.9)

Discontinuity in ( G T ) P{ n j } =S  means a finite difference of entropy of the 2 phases,

        ΔS= S I S II = ( G T ) P{ n j } II ( G T ) P{ n j } I                             (3.10)

which means the heat capacities are not defined.

 

Since G is continuous, the difference in enthalpy (latent heat) is given by

        ΔH=Δ( G+TS )=TΔS                           (3.11)

 

The case for a continuous transition is shown in Fig.3.3.

The distinguishing feature is the peak in the heat capacity at the transition point.