Physical Chemistry Third Edition

5.1 The Fundamental Fact of Phase Equilibrium 201

IfTandPare constant, the fundamental criterion of equilibrium implies thatdGmust

vanish for an infinitesimal change that maintains equilibrium:

dG

∑c

i 1

μ(I)i dn(I)i +

∑c

i 1

μ(II)i dn(II)i



∑c

i 1

[

μ(I)i −μ(II)i

]

dn(I)i 0 (at equilibrium) (5.1-4)

where we used the fact thatdn(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 componentito pass, but not the

others. The term of the sum for componentimust vanish since all of the otherdn’s

vanish. Sincedniis 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 constantTandP. We call

it thefundamental 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(TandPconstant) (5.1-6)

SincedTanddPvanish and since the system as a whole is closed,

dG

∑c

i 1

μ(I)i dn(I)i +

∑c

i 1

μ(II)i dn(II)i 

∑c

i 1

[

μ(I)i −μ(II)i

]

dn(I)i ≤ 0 (5.1-7)

Each term separately must be negative since the introduction of semipermeable mem-

branes 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 thatdn(I)i ≤ 0 (5.1-8)

μ(I)i <μ(II)i implies thatdn(I)i ≥ 0 (5.1-9)