232
5 Phase Equilibrium
with a similar equation for phase II. The phases are at equilibrium so that T , 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 for dG
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
i1
µ i d
n i − n
(I)
i − n
(II)
i
(5.6-3)
which we rewrite as
dG
(σ)
−S
(σ) dT + γdA +
c
i1
µ i dn
(σ)
i
(5.6-4)
In Eq. (5.6-4), we have used the fact that V (I) + V (II) V , so that the dP term vanishes.
The quantity G (σ) is called the surface Gibbs energy:
G
(σ)
G − G
(I)
− G
(II)
(5.6-5)
and S (σ) is called the surface entropy:
S
(σ)
S − S
(I)
− S
(II)
(5.6-6)
The surface tension γ is an intensive variable, depending only on T , P, and the
composition of the phases of the system. Although A is not proportional to the size
of the system, we assume that there is a contribution to G equal to γA so that Euler’s
theorem, instead of the version in Eq. (4.6-4), is
G γA +
c
i1
µ i n i
(5.6-7)
Each phase obeys Euler’s theorem without a surface term, so that
G
(I)
c
i1
µ i n
(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
i1
µ i n
(σ)
i
(5.6-9)
We can write an expression for dG (σ) from Eq. (5.6-9):
dG
(σ)
γdA + A dγ +
c
i1
n
(σ)
i du i +
c
i1
µ i dn
(σ)
i
(5.6-10)