5.1 The Fundamental Fact of Phase Equilibrium
201
If T and P are constant, the fundamental criterion of equilibrium implies that dG must
vanish for an infinitesimal change that maintains equilibrium:
dG
c
i1
µ
(I)
i dn
(I)
i +
c
i1
µ
(II)
i dn
(II)
i
c
i1
µ
(I)
i − µ
(II)
i
dn
(I)
i 0 (at equilibrium)
(5.1-4)
where we used the fact that dn
(I)
i −dn
(II)
i . We want to show not only that the last
sum in Eq. (5.1-4) vanishes, but that each term vanishes. Assume that we can find a
semipermeable membrane that will selectively allow component i to pass, but not the
others. The term of the sum for component i must vanish since all of the other dn’s
vanish. Since dn i is not necessarily equal to zero, the other factor in the term must
vanish, and we can write
µ
(I)
i µ
(II)
i
(at equilibrium)
(5.1-5a)
If more than two phases are present at equilibrium, we conclude that the chemical
potential of any substance has the same value in every phase in which it occurs. We
write a second version of Eq. (5.1-5a):
µ
(α)
i µ
(β)
i
(system at equilibrium)
(5.1-5b)
where the superscripts (α) and (β) designate any two phases of a multiphase system.
The properties of a system at equilibrium do not depend on how the system arrived
at equilibrium. Therefore, Eq. (5.1-5) is valid for any system at equilibrium, not only
for a system that arrived at equilibrium under conditions of constant T and P. We call
it the fundamental fact of phase equilibrium: In a multiphase system at equilibrium the
chemical potential of any substance has the same value in all phases in which it occurs.
Nonequilibrium Phases
Consider a two-phase simple system that is maintained at constant temperature and
pressure but is not yet at equilibrium. Assuming that the nonequilibrium state of the
system can be treated as a metastable state, the criterion for spontaneous processes is
given by Eq. (4.1-17):
dG ≤ 0 (T and P constant)
(5.1-6)
Since dT and dP vanish and since the system as a whole is closed,
dG
c
i1
µ
(I)
i dn
(I)
i +
c
i1
µ
(II)
i dn
(II)
i
c
i1
µ
(I)
i − µ
(II)
i
dn
(I)
i ≤ 0
(5.1-7)
Each term separately must be negative since the introduction of semipermeable membranes would show each term to obey the inequality separately. The two factors in each
term of the sum in Eq. (5.1-7) must be of opposite signs:
µ
(I)
i > µ
(II)
i
implies that dn
(I)
i ≤ 0
(5.1-8)
µ
(I)
i < µ
(II)
i
implies that dn
(I)
i ≥ 0
(5.1-9)
201
If T and P are constant, the fundamental criterion of equilibrium implies that dG must
vanish for an infinitesimal change that maintains equilibrium:
dG
c
i1
µ
(I)
i dn
(I)
i +
c
i1
µ
(II)
i dn
(II)
i
c
i1
µ
(I)
i − µ
(II)
i
dn
(I)
i 0 (at equilibrium)
(5.1-4)
where we used the fact that dn
(I)
i −dn
(II)
i . We want to show not only that the last
sum in Eq. (5.1-4) vanishes, but that each term vanishes. Assume that we can find a
semipermeable membrane that will selectively allow component i to pass, but not the
others. The term of the sum for component i must vanish since all of the other dn’s
vanish. Since dn i is not necessarily equal to zero, the other factor in the term must
vanish, and we can write
µ
(I)
i µ
(II)
i
(at equilibrium)
(5.1-5a)
If more than two phases are present at equilibrium, we conclude that the chemical
potential of any substance has the same value in every phase in which it occurs. We
write a second version of Eq. (5.1-5a):
µ
(α)
i µ
(β)
i
(system at equilibrium)
(5.1-5b)
where the superscripts (α) and (β) designate any two phases of a multiphase system.
The properties of a system at equilibrium do not depend on how the system arrived
at equilibrium. Therefore, Eq. (5.1-5) is valid for any system at equilibrium, not only
for a system that arrived at equilibrium under conditions of constant T and P. We call
it the fundamental fact of phase equilibrium: In a multiphase system at equilibrium the
chemical potential of any substance has the same value in all phases in which it occurs.
Nonequilibrium Phases
Consider a two-phase simple system that is maintained at constant temperature and
pressure but is not yet at equilibrium. Assuming that the nonequilibrium state of the
system can be treated as a metastable state, the criterion for spontaneous processes is
given by Eq. (4.1-17):
dG ≤ 0 (T and P constant)
(5.1-6)
Since dT and dP vanish and since the system as a whole is closed,
dG
c
i1
µ
(I)
i dn
(I)
i +
c
i1
µ
(II)
i dn
(II)
i
c
i1
µ
(I)
i − µ
(II)
i
dn
(I)
i ≤ 0
(5.1-7)
Each term separately must be negative since the introduction of semipermeable membranes would show each term to obey the inequality separately. The two factors in each
term of the sum in Eq. (5.1-7) must be of opposite signs:
µ
(I)
i > µ
(II)
i
implies that dn
(I)
i ≤ 0
(5.1-8)
µ
(I)
i < µ
(II)
i
implies that dn
(I)
i ≥ 0
(5.1-9)