Explain the Gibbs Phase Rule for Multicomponent Systems?

The Gibbs phase rule for a pure substance was written F = 3 - P. We now consider a system of more than one substance and more than one phase in an equilibrium state. The phase rule assumes the system is at thermal and mechanical equilibrium. We shall assume furthermore that in addition to the temperature and pressure, the only other state functions needed to describe the state are the amounts of the species in each phase; this means for instance that surface effects are ignored. The derivations to follow will show that phase rule may be written either in the form

F = 2 + C - P (13.1.1) or

F = 2 + s - r - P (13.1.2)

Where the symbols have the following meanings:

F = the number of degrees of freedom (or variance)

= the maximum number of intensive variables that can be varied independently while the system remains in an equilibrium state;

C = the number of components

= the minimum number of substances (or fixed-composition mixtures of substances) that could be used to prepare each phase individually;

P = the number of different phases;

s = the number of different species;

r = the number of independent relations among intensive variables of individual phases other than relations needed for thermal, mechanical, and transfer equilibrium.