## 3.B. Coexistence of Phases: Gibbs Phase Rule

Coexisting phases are in thermal &
mechanical equilibrium.

They can also exchange matter.

Hence, they have the same *T*, *Y*,
and *μ*.

*YXT* System of 1 Kind of Particles

The condition for the coexistence of 2
phases *I* and *II* is

μ
I
(
Y,T
)=
μ
II
(
Y,T
)
(3.1)

where, according to the Gibbs-Duhem
equation (2.62), *μ* is a function of
intensive variables only.

Eq(3.1) can be solved to give the **coexistence curve **of the 2 phases

Y=Y(
T
)
(3.2)

in the *Y-T*
plane.

The conditions for the coexistence of 3
phases *I* , *II* and *III* are

μ
I
(
Y,T
)=
μ
II
(
Y,T
)=
μ
III
(
Y,T
)
(3.3)

which can be solved to give the locations
of **triple points** in the *Y-T* plane.

There cannot be a coexistence of more than
3 phases because the coexistence conditions will be over-determined.

*YXT* System of *m* Kinds of
Particles

For a mixture of *m* different types of particles in a *YXT* system, the independent variables can be chosen as
Y,T,
n
i
,
where
n
i
is the number of moles of the *i*th type particles, and
i=1,⋯,m
. For a closed system, the total number *n* of moles is a constant and we can
write

∑
i=1
m
n
i
=n
→
∑
i=1
m
x
i
=1

where
x
i
=
n
i
n
is the **molar
fraction **of particles of type *i*.

Thus, only
m−1
of the
x
i
's are independent and the system itself has
only
m+1
independent (intensive) variables
Y,T,x
,
where
x=(
x
1
,⋯,
x
m−1
)
.

The conditions for the coexistence of *r* phases are

μ
1
I
(
Y,T,
x
I
)=
μ
1
II
(
Y,T,
x
II
)=⋯=
μ
1
r
(
Y,T,
x
r
)
(3.4)

… … …

μ
i
I
(
Y,T,
x
I
)=
μ
i
II
(
Y,T,
x
II
)=⋯=
μ
i
r
(
Y,T,
x
r
)
(3.5)

… … …

μ
m
I
(
Y,T,
x
I
)=
μ
m
II
(
Y,T,
x
II
)=⋯=
μ
m
r
(
Y,T,
x
r
)
(3.6)

where
μ
i
s
is the chemical potential of the *i*-type particles in the *s*th phase whose composition is
x
s
=(
x
1
s
,⋯,
x
m−1
s
)
.

The number of independent
equations in (3.4-6) is
m(
r−1
)
.

The number of unknowns is
2+r(
m−1
)
.

The maximum number *R* of phases that can coexist simultaneously is therefore given by

m(
R−1
)=2+R(
m−1
)

i.e.,

R=m+2