## 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 ith 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 m1  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 m1 ) .

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 sth phase whose composition is x s =( x 1 s ,, x m1 s ) .

The number of independent equations in (3.4-6) is m( r1 ) .

The number of unknowns is 2+r( m1 ) .

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

m( R1 )=2+R( m1 )

i.e.,

R=m+2