Eqs.(2.178), which specify the condition of phase equilibrium are called the equations of phase equilibrium. There are k(φ-1) numbers of such equations. Now the composition of each phase containing k constituents is fixed if k-1 constituents are known, since sum of the mole fractions of each constituent in the phase must equal unity. Therefore, for φ phases, there are a total of φ(k-1) variables, in addition to temperature and pressure, which must be specified. There are, then, φ(k-1)+2 variables altogether.

If the number of variables is equal to the number of equations, then whether or not we can actually solve the equations, the temperature, pressure, and composition of each phase is determined. The system is then called nonovariant and said to have zero variance.

If the number of variables is one greater than the number of equations, an arbitrary value can be assigned to one of the variables and the remainder are completely determined. The system is then called monovariant and said to have a variance of 1.

In general, the variance ƒ is defined as the excess of the number of variables over the numbers of equations.

ƒ = [φ(k-1)+2] - [k(φ-1)] = k-φ+2

(2.179)

This equation is called the Gibbs phase rule.

For chemical reactions, Gibbs phase rule is modified as

ƒ = (c-r) - φ+2

(2.180)

where r = number of independent reversible chemical reaction
          φ = number of phases
          k = constituent
          ƒ = variance or degree of freedom

Consider the T - s diagram for water. At any point on the  saturated vapor line, c=1, φ=1. Hence, ƒ=2. This indicates that 2 (two) independent thermodynamic properties are needed to fix up the state of the system at equilibrium. Similarly, at a location within the saturated liquid and vapor line, c=1,φ=1 ⇒ ƒ=0 . Hence, only one thermodynamic property is sufficient to fix up the state. At the triple point, it is a unique state where all the three phases exist in equilibrium.