# Thermodynamic Derivation of Gibb's Phase Rule

## Thermodynamic Derivation of Gibb's Phase Rule

Let a system of component 'C' distributed between phase 'P' and let each phase has all components 'C'. The total number of variables for concentration alone is P(C − 1) as there are 'P' phases. Two more variables are added because the process is being carried out at a constant temperature and pressure. Hence the total number of variables = P(C − 1) + 2Let a heterogeneous system of two phases a and b containing three components 1,2 and 3 in equilibrium. The chemical potential (µ) of these three components in two phases can be written as-

µ

_{1}(a), µ

_{2}(a), µ

_{3}(a), µ

_{1}(b), µ

_{2}(b), µ

_{3}(b)

If we consider the closed system in equilibrium at a fixed temperature and pressure, the Gibb's Duhem equation we have-

Σµ dn = 0

Let a small amount δn

_{1}is transformed under equilibrium conditions from a to b phase, then-

µ

_{1}(a).dn

_{1}+ µ

_{1}(b).dn

_{1}= 0

or, µ

_{1}(a) = µ

_{1}(b)

Similarly, µ

_{2}(a) = µ

_{2}(b)

and µ

_{3}(a) = µ

_{3}(b)

Now consider the system having three phases, then-

µ

_{1}(a) = µ

_{1}(b) = µ

_{1}(c)

Therefore, the number of independent equations defining equilibrium among three phases are two only. In general, the above conditions also hold good for phases 'P' and components 'C' to give following relations-

µ

_{1}(a) = µ

_{1}(b) = µ

_{1}(c) = ... = µ

_{1}(P)

µ

_{2}(a) = µ

_{2}(b) = µ

_{2}(c) = ... = µ

_{2}(P)

µ

_{3}(a) = µ

_{3}(b) = µ

_{3}(c) = ... = µ

_{3}(P)

----- ----- ----- -----

----- ----- ----- -----

----- ----- ----- -----

µ

_{c}(a) = µ

_{c}(b) = µ

_{c}(c) = ... = µ

_{c}(P)

These equations constitute C(P − 1) independent equations. Hence, the degree of freedom 'F' must be equal to-

F = [P(C − 1) + 2] = C(P − 1)

or, F = C − P + 1

This equation is known as Gibb's Phase Rule.

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