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Consider a system that consists of two phases, which we shall denote by 1 and 2. For example, these might be
solid and liquid, or liquid and gas. Suppose that the system is in equilibrium with a reservoir at the constant temperature
, and the
constant pressure
, so that the system always has the temperature
and the mean pressure
. However, the
system can exist in either one of its two phases, or some mixture of the two. Let us begin by finding the conditions that allow
the two phases to coexist in equilibrium with one another.
In accordance with the discussion in Section 9.4, the equilibrium condition is that the
Gibbs free energy,
, of the system is a minimum:

(9.59) 
Let
be the number of moles of phase
present in the system, and let
be the
Gibbs free energy per mole of phase
at the temperature
and the pressure
. It follows that

(9.60) 
Furthermore, the conservation of matter implies that the total number of moles,
, of the substance
remains constant:

(9.61) 
Thus, we can take
as the one independent parameter that is free to vary. In equilibrium, Equation (9.59) requires that
be stationary for small variations in
. In other words,

(9.62) 
because
, as a consequence of Equation (9.61). Hence, a necessary condition for equilibrium between the two
phases is

(9.63) 
Clearly, when this condition is satisfied then the transfer of a mole of substance from one phase to another does not
change the overall Gibbs free energy,
, of the system. Hence,
is stationary, as required. Incidentally, the
condition that
is a minimum (rather than a maximum) is easily shown to reduce to the requirement that the
heat capacities and isothermal compressibilities of both phases be positive, so that each phase is stable to temperature and volume fluctuations.
Figure 9.1:
Pressuretemperature plot showing the domains of relative stability of two phases, and the
phaseequilibrium line.

Now, for a given temperature and pressure,
is a welldefined function characteristic of phase 1. Likewise,
is a
welldefined function characteristic of phase 2. If
and
are such that
then the minimum value of
in Equation (9.59)
is achieved when all
moles of the substance transform into phase 1, so that
. In this case, phase 1 is the stable
one. On the other hand, if
then the minimum value of
is achieved when all
moles of the substance transform into phase 2, so that
. In this case, phase 2 is the stable
one. Finally, if
then the condition (9.59) is automatically satisfied, and any amount,
, of phase 1 can coexist with the
remaining amount,
, of phase 2. The locus of the points in the

plane where
then
represents a phaseequilibrium line along which the two phases can coexist in equilibrium. This line
divides the

plane into two regions. The first corresponds to
, so that phase 1 is stable. The
second corresponds to
, so that phase 2 is stable. See Figure 9.1.
Next: ClausiusClapeyron Equation
Up: MultiPhase Systems
Previous: Stability of SinglePhase Substance
Richard Fitzpatrick
20160125