Page 225 - Physical Chemistry

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Chapter 7 Before specializing to one-component systems, we want to answer the general ques-
One-Component Phase Equilibrium tion of how many independent variables are needed to define the equilibrium state of
and Surfaces
a multiphase, multicomponent system.
To describe the equilibrium state of a system with several phases and several
chemical species, we can specify the mole numbers of each species in each phase and
the temperature and pressure, T and P. Provided no rigid or adiabatic walls separate
phases, T and P are the same in all phases at equilibrium. Specifying mole numbers is
not what we shall do, however, since the mass of each phase of the system is of no real
interest. The mass or size of each phase does not affect the phase-equilibrium position,
since the equilibrium position is determined by equality of chemical potentials, which
are intensive variables. (For example, in a two-phase system consisting of an aqueous
solution of NaCl and solid NaCl at fixed T and P, the equilibrium concentration of dis-
solved NaCl in the saturated solution is independent of the mass of each phase.) We
shall therefore deal with the mole fractions of each species in each phase, rather than
a
a
a
with the mole numbers. The mole fraction of species j in phase a is x n /n , where
j
tot
j
a
a
n is the number of moles of substance j in phase a and n tot is the total number of
j
moles of all substances (including j) in phase a.
The number of degrees of freedom (or the variance) f of an equilibrium system
is defined as the number of independent intensive variables needed to specify its in-
tensive state. Specification of the intensive state of a system means specification of
its thermodynamic state except for the sizes of the phases. The equilibrium intensive
state is described by specifying the intensive variables P, T, and the mole fractions in
each of the phases. As we shall see, these variables are not all independent.
We initially make two assumptions, which will later be eliminated: (1) No chem-
ical reactions occur. (2) Every chemical species is present in every phase.
Let the number of different chemical species in the system be denoted by c, and
let p be the number of phases present. From assumption 2, there are c chemical species
in each phase and hence a total of pc mole fractions. Adding in T and P, we have
pc 2 (7.1)
intensive variables to describe the intensive state of the equilibrium system. However,
these pc 2 variables are not all independent; there are relations between them. First
of all, the sum of the mole fractions in each phase must be 1:
a
a
x x p x 1 (7.2)
a
2
1
c
a
where x is the mole fraction of species 1 in phase a, etc. There is an equation like
1
(7.2) for each phase, and hence there are p such equations. We can solve these equa-
a
b
tions for x , x ,..., thereby eliminating p of the intensive variables.
1
1
In addition to the relations (7.2), there are the conditions for equilibrium. We have
already used the conditions for thermal and mechanical equilibrium by taking the
same T and the same P for each phase. For material equilibrium, the following phase-
equilibrium conditions [Eq. (4.88)] hold for the chemical potentials:
b
g
a
m m m p (7.3)
1
1
1
g
a
b
m m m p (7.4)
2
2
2
........................ (7.5)
g
a
b
m m m p (7.6)
c
c
c
Since there are p phases, (7.3) contains p 1 equality signs and p 1 independent
equations. Since there are c different chemical species, there are a total of c(p 1)
equality signs in the set of equations (7.3) to (7.6). We thus have c(p 1) independent
relations between chemical potentials. Each chemical potential is a function of T, P,
a
a
a
a
and the composition of the phase (Sec. 4.6); for example, m m (T, P, x , ..., x ).
1 1 1 c
Hence the c(p 1) equations (7.3) to (7.6) provide c(p 1) simultaneous relations