The phase rule is a thermodynamic law that allows one to calculate the number of independent variables that can be fixed in a PVT system at a equilibrium. For instance, consider a closed, fixed volume vessel at constant temperature (T) and pressure (P) filled with an amount of water. At the given temperature, two phases are present; liquid (L) and vapor (V). We're assuming the temperature is above the freezing point at the specified T and P, so there is no solid (S) phase.

            |   (V)   |  Fixed Temperature
            |         |
            |~~~~~~~~~|  Fixed Pressure
            |         |
            |   (L)   |

The two phases in the vessel can only be in equilibrium if the governing intensive thermodynamic properties are the same. Thus, both the temperature and pressures for the two phases must be the same. In general, Thermal equilibrium and pressure equilibrium are given by:

TV = TL = TG

PV = PL = PG

A third requirement is chemical equilibrium. The requirement for chemical equilibrium is that the chemical potentials of all components must be the same in all three phases.

μjV = μjL = μjG (j = 1, 2, 3, ... N)

where μj is the chemical potential of the jth component, and N is the number chemical species.

The chemical potential of a phase is associated with the concentration of a component (although they are not the same). For the given system and any other PVT system, the phase rule by Gibbs now gives the degrees of freedom (F) of the system, i.e. the number of independent variables that must be fixed to establish the equilibrium:

F = 2 - π + N

Where F is the number of degrees of freedom, π is the number of phases, and N is the number of chemical species.

The example mentioned above contains a single chemical species (water); there are two phases present (liquid and vapor), and thus F = 2 - 2 +1 = 1 degree of freedom. This indicates that for a given pressure, water has only one boiling point. Temperature or pressure, but not both may be specified for an equilibrium system containing liquid water and water vapor.

Another example: Consider a liquid solution of alcohol in water in equilibrium with its vapor. In this case N = 2, and π = 2. Thus F = 2 - 2 + 2 = 2 degrees of freedom. In this case the intensive variables are temperature, pressure, and the phase compositions. If we fix one variable, the other variables can be varied independently. If we fix two variables, the equilibrium condition is specified, and the third variable cannot be varied.