Thermodynamic Derivation of Gibb's Phase Rule

Thermodynamic Derivation of Gibb's Phase Rule

Let 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-
µ₁(a), µ₂(a), µ₃(a), µ₁(b), µ₂(b), µ₃(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 δn₁ is transformed under equilibrium conditions from a to b phase, then-
µ₁(a).dn₁ + µ₁(b).dn₁ = 0
or,   µ₁(a) = µ₁(b)
Similarly,   µ₂(a) = µ₂(b)
and   µ₃(a) = µ₃(b)
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