7.2 The Gibbs Phase Rule
In the previous chapter, we introduced the Gibbs phase rule for a single com-
ponent. Recall that the phase rule gives us the number of independent vari-
ables that must be specified in order to know the condition of an isolated sys-
tem at equilibrium. For a single-component system, only the number of stable
phases in equilibrium is necessary to determine how many other variables, or
degrees of freedom,are required to specify the state of the system.
If the number of components is greater than one, then more information is
necessary to understand the state of the system at equilibrium. Before we con-
sider how much more information is necessary, let us review what information
we do have. First, since we are assuming that the system is at equilibrium, then
the system’s temperature,Tsys, and the system’s pressure,psys, are the same for
all components. That is,
Tcomp.1Tcomp.2Tcomp.3 Tsys (7.1)
pcomp.1pcomp.2pcomp.3 psys (7.2)We also have the requirement, from the previous chapter, that the temperatures
and pressures experienced by all phases are the same:Tphase1Tphase2
and pphase1pphase2 . Equation 7.2 is not meant to imply that the par-
tialpressures of individual gas components are the same. It means that every
component of the system, even gaseous components, are subject to the same
overall system pressure. We will also assume that our system remains at con-
stant volume (is isochoric) and that we know the total amount of material,
usually in units of moles, in our system. After all, the experimenter controls the
initial conditions of the system, so we will always begin by knowing the initial
amount of material.
With this understanding, how many degrees of freedom must be specified
in order to know the state of a system at equilibrium? Consider a system that
has a number of components Cand a number of phases P. To describe the rel-
ative amounts (like mole fractions) of the components,C1 values must be
specified. (The amount of the final component can be determined by subtrac-
tion.) Since the phase of each component must be specified, we need to know
(C1) Pvalues. Finally, if temperature and pressure need to be specified,
we have a total of (C1) P2 values that we need to know in order to de-
scribe our system.
But if our system is at equilibrium, the chemical potentials of the different
phases of each component must be equal. That is,
1,sol1,liq1,gas 1,other phaseand this must hold for every component, not just component 1. This means
we can remove P1 values for every component C, for a total of (P1) C
values. The number of values remaining represents the degrees of freedom,F:
(C1) P 2 (P1) C,or
FCP 2 (7.3)Equation 7.3 is the more complete Gibbs phase rule.For a single component,
it becomes equation 6.17. Note that it is applicable only to systems at equilib-
rium. Also note that although there can be only one gas phase, due to the
mutual solubility of gases in each other, there can be multiple liquid phases
(that is, immiscible liquids) and multiple solid phases (that is, independent,
nonalloyed solids in the same system).
7.2 The Gibbs Phase Rule 167