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 **T**emperature
| |
|~~~~~~~~~| Fixed **P**ressure
| |
| (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:

T^{V} = T^{L} = T^{G}

P^{V} = P^{L} = P^{G}

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.

μ_{j}^{V} =
μ_{j}^{L} = μ_{j}^{G}
(`j` = 1, 2, 3, ... N)

where μ_{j} is the chemical potential of the
`j`th 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.