Physical Chemistry Third Edition

232 5 Phase Equilibrium

with a similar equation for phase II. The phases are at equilibrium so thatT,P, and the

μ’s have the same values in all phases and require no superscripts.

Let us subtract Eq. (5.6-2) for phase I and for phase II from the expression fordG

in the version of Eq. (5.5-3) that applies to a multicomponent system:

dG−G(I)−G(II)−S−S(I)−S(II)dT+V−V(I)−V(II)dP+γdA

+

∑c

i 1

μid

[

ni−n

(I)

i −n

(II)

i

]

(5.6-3)

which we rewrite as

dG(σ)−S(σ)dT+γdA+

∑c

i 1

μidn(iσ) (5.6-4)

In Eq. (5.6-4), we have used the fact thatV(I)+V(II)V, so that thedPterm vanishes.

The quantityG(σ)is called thesurface Gibbs energy:

G(σ)G−G(I)−G(II) (5.6-5)

andS(σ)is called thesurface entropy:

S(σ)S−S(I)−S(II) (5.6-6)

The surface tensionγis an intensive variable, depending only onT,P, and the

composition of the phases of the system. AlthoughAis not proportional to the size

of the system, we assume that there is a contribution toGequal toγAso that Euler’s

theorem, instead of the version in Eq. (4.6-4), is

GγA+

∑c

i 1

μini (5.6-7)

Each phase obeys Euler’s theorem without a surface term, so that

G(I)

∑c

i 1

μin(I)i (5.6-8)

with an analogous equation for phase II. When Eq. (5.6-8) and its analogue for phase

II are subtracted from Eq. (5.6-7), we obtain

G(σ)γA+

∑c

i 1

μin(iσ) (5.6-9)

We can write an expression fordG(σ)from Eq. (5.6-9):

dG(σ)γdA+Adγ+

∑c

i 1

n(iσ)dui+

∑c

i 1

μidn(iσ) (5.6-10)