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### Gibbs Phase Rule

The Gibbs Phase Rule describes the possible number of degrees of freedom (f) in a closed system at equilibrium, in terms of the maximum number of stable phases (M) and the number of system components (N):

f = N - M + 2

In other words, the number of degrees of freedom for a system at equilibrium is the number of intensive variables (often taken as the pressure, temperature, and composition fraction) that may be arbitrarily specified without changing the number of phases. In a region with M stable phases, the values of N-M+2 state variables can be changed independently and preserving the same set of stable phases.

If the number of degrees of freedom is equal to zero, then an invariant equilibrium is defined (i.e., equilibrium can be attained only for a single set of values of all the state variables). The three-phase equilibrium assemblage in either a unary system with variable temperature and pressure or a binary system at constant pressure are examples of invariant equilibria.

If the number of degrees of freedom is equal to one, then a univariant equilibrium is defined (i.e., the set of stable phases depends on the value of one stable variable only). The two-phase equilibrium assemblage in either a unary system with variable temperature and pressure or a binary system at constant pressure are examples of univariant equilibria.

It is worth noting that a phase diagram is constructed from lines of univariant equilibria and cross-points of invariant equilibria.

Finally, the properties of a phase are independent of the properties of any other phase in the system. Consequently, the Gibbs energy can be defined separately for each individual phase. The sum of each of the stable phases multiplied by the amount of that phase is equal to the Gibbs energy of the whole system.

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