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between T, P, and the mole fractions, which we can solve for c(p 1) of these vari- Section 7.1
ables, thereby eliminating c(p 1) intensive variables. The Phase Rule
We started out with pc 2 intensive variables in (7.1). We eliminated p of them
using (7.2) and c(p 1) of them using (7.3) to (7.6). Therefore the number of indepen-
dent intensive variables (which, by definition, is the number of degrees of freedom f ) is
f pc 2 p c1p 12

f c p 2 no reactions (7.7)
Equation (7.7) is the phase rule, first derived by Gibbs.
Now let us drop assumption 2 and allow for the possibility that one or more chem-
ical species might be absent from one or more phases. An example is a saturated aque-
ous salt solution in contact with pure solid salt. If species i is absent from phase d, the
d
number of intensive variables is reduced by 1, since x is identically zero and is not a
i
variable. However, the number of relations between the intensive variables is also re-
d
duced by 1, since we drop m from the set of equations (7.3) to (7.6). Recall that when
i
d
substance i is absent from phase d, m need not equal the chemical potential of i in the
i
other phases [Eq. (4.91)]. Therefore, the phase rule (7.7) still holds when some species
do not appear in every phase.
EXAMPLE 7.1 The phase rule

Find f for a system consisting of solid sucrose in equilibrium with an aqueous
solution of sucrose.
The system has two chemical species (water and sucrose), so c 2. The sys-
tem has two phases (the saturated solution and the solid sucrose), so p 2. Hence

f c p 2 2 2 2 2
Two degrees of freedom make sense, since once T and P are specified, the equilib-
rium mole fraction (or concentration) of sucrose in the saturated solution is fixed.

Exercise
Find f for a system consisting of a liquid solution of methanol and ethanol in
equilibrium with a vapor mixture of methanol and ethanol. Give a reasonable
choice for the independent intensive variables. (Answer: 2; T and the liquid-
phase ethanol mole fraction.)

Once the f degrees of freedom have been specified, then any scientist can prepare
the system and get the same value for a measured intensive property of each phase of
the system as any other scientist would get. Thus, once the temperature and pressure
of an aqueous saturated sucrose solution have been specified, then the solution’s den-
sity, refractive index, thermal expansivity, molarity, and specific heat capacity are all
fixed, but the volume of the solution is not fixed.
An error students sometimes make is to consider a chemical species present in two
phases as contributing 2 to c. For example, they will consider sucrose(s) and
sucrose(aq) as two chemical species. From the derivation of the phase rule, it is clear
that a chemical species present in several phases contributes only 1 to c, the number
of chemical species present.

The Phase Rule in Systems with Reactions
We now drop assumption 1 and suppose that chemical reactions can occur. For each inde-
pendent chemical reaction, there is an equilibrium condition n m 0 [Eq. (4.98)],
i i i
where the m ’s and n ’s are the chemical potentials and stoichiometric coefficients of
i i