Derivation of the Phase Rule

Derivation of the Phase Rule:

                                       According to this rule, for any equilibrium thermodynamic system, the sum of the number of degrees of freedom(F) and the number of phases(P) are equal to the sum of number of independent component ‘C’ and the number of external factors n(T and P) affecting the physical state of the system.

          F+P=C+n

 Or,    F+P= C+2       where (n=2)

 Or ,   F= C-P+2   (Gibbs phase rule)

Let us consider a heterogeneous system in equilibrium having C components (C₁,C₂,----C_c) distributed in P phase (P₁, P₂, ----P_p).

The number of degree of freedom = Total number of variables  - The number of factor define by the system

Step-I :  Determination of the total number of variables:

                                                 Let us consider, a system which is having two component C₁ and C₂. If the molar concentration of one component is known than that of other can be calculated by difference.

         C= C₁­+C₂

         C₂=C-C₁

Then the phase of two component can be define completely by one concentration term. Similarly if a phase has three component’s in order to define it completely only two concentration terms should be known. In general, If a system has C component, Number of concentration term required (C-1). All the P phase, the total number of concentration term of variable required will be P(C-1). Besides these concentration variables, the temperature and pressure of the system, which are for all the phase are two variables.

  Total number of variables= P(C-1)+2

Step-II: Determination of number of factor define by the system:

                                           According to thermodynamics, when the system is in equilibrium the chemical potential of a component (µ) is the same in all the phase. If a system has three phase α, β and γ then.

                     µ_α = µ_β = µ_γ

Now,    µ_α = µ_β

             µ_β = µ_γ

Thus, for a system of three phase two equation are known for each component so, number of equation given are (P-1). In general if a system of P phase and C component the number of equation be known C(P-1). Hence, the degree of freedom of the system will be given by:

   F= P(C-1)+2-C(P-1)

  F= -P+2+C ,   Or   F=C-P+2 àThis is the statement of phase rule.