# Derivation of Phase Rule

The phase rule describes the possible number of degrees of freedom in an enclosed system at equilibrium, in terms of the number of separate phases and the number of chemical constituents in the system. It was deduced by J.W Gibbs in the 1870s. Today, the phase rule is popularly known as the Gibbs phase rule all over the world. Here, in the article, we will be discussing the derivation of phase rule.

## Gibbs Phase Rule

Gibbs phase rule states that if the equilibrium in a heterogeneous system is not affected by gravity or by electrical and magnetic forces, the number of degree of freedom is given by the equation

F=C-P+2

where C is the number of chemical components

P is the number of phases

Basically, it describes the mathematical relationship for determining the stability of phases present in the material at equilibrium condition.

In the next section, let us look at the phase rule derivation.

### Phase Rule Derivation

The Gibb’s phase rule on the basis of the thermodynamic rule can be derived as follows:

First, let us consider a heterogeneous system consisting of Pn number of phases and Cn number of components in equilibrium. Let us assume that the passage of a component from one phase to another doesn’t involve any chemical reaction. When the system is in equilibrium, it can be described by the following parameters:

• Temperature
• Pressure
• The composition of each phase

a. The total number of variables required to specify the state of the system is:

• Pressure: same for all phases
• Temperature: same for all phases
• Concentration

The independent concentration variables for one phase with respect to the C components is C – 1. Therefore, the independent concentration variables for P phases with respect to C components is P (C – 1).

Total number of variables = P (C – 1) + 2 (1)

b. The total number of equilibria:

The various phases present in the system can only remain in equilibrium when the chemical potential (µ) of each of the component is the same in all phases, i.e.

 µ1, P1 = µ1, P2 = µ1, P3 = … = µ1, P µ2, P1 = µ2, P2 = µ2, P3 = … = µ2, P : : : : : : : : : : : : µC, P1 = µC, P2 = µC, P3 = … = µC, P

The number of equilibria for each P phases for each component is P – 1.

For C components, the number of equilibria for P phases is P ( C – 1).

Hence, the total number of equilibria involved is E = C (P – 1). (2)

Equating eq (1) and (2), we get

$F=[P(C-1)+2-C]-[C(P-1)]$ $F=[CP-P+2-CP+C]$ $F=C-P+2$

The obtained formula is the Gibbs phase rule. Stay tuned to BYJU’S to learn more physics derivations.
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