The relation between the number of phases, components and the degrees of freedom is known as the phase rule.
One constituent systems: the identification of an area on a P-versus-T with one phase of a component system illustrates the two degrees of freedom that exist, these usually being specified as pressure and temperature.
For a two phase system, the requirement of equality in the molar free energies of the two phases imposes a relation, such as dP/dT = ?S/?V, and thus the pressure and temperature cannot both be arbitrary varied. A two phase component system thus has a single degree of freedom, as shown by the identification of a line on a P-versus-T diagram with two phases in equilibrium.
Finally, for three phases to coexist, the molar energy of the first pair would have to be equal that of the additional phase. One molar restrictive equation then exists, and thus the last degree of freedom is entirely removed. No arbitrary assignment of variable can be made; the system is entirely self determined. The one component P-versus-T diagram feature for three phase is a point.
All this can be assumed by the equation:
∅ = 3 - P [one component]
Multi component systems: rules similar to the above equation can be deduced for systems of more than one component. It is possible, however, to proceed more generally and to obtain the phase rule, which gives the number of degrees of freedom of a system with C components and P phases, this rule was first obtained by J. Willard Gibbs in 1878, but it was published in rather obscure Transactions of the Connecticut Academy and overlooked for 20 years.
Consider the two components to be published in the rather obscure Transaction of the Connecticut academy and overlooked the degrees of freedom of the system can be calculated by first adding the total number of intensive variables required to describe separately each problem and subtracting these variables, whose values are fixed by free energy equilibrium relations between the different phases. To begin, each component is assumed to be present in every phase.
In each phase C - 1 quantity will be define the composition of the phase quantitatively. Thus, if mole fraction are used to measure the concentrations, one needs to be specify the mole fraction of the components, the remaining one being determined because the sum of P (C - 1) such composition variables. In addition the pressure and the temperature if the system is considered phase by phase is denoted by the main composition of phase rule.
The number of degrees of freedom, i.e. of net arbitrary adjustable intensive variables, is therefore:
∅ = P(C - 1) + 2 - (P - 1) = C - P + 2
If a component is not present or is present to a negligible extent in one of the phases of the system, there will be one fewer intensive variable for that phase since the neglible concentration of the species is is of no interest. There will also be one fewer equilibrium relation. The phase rule applies, therefore, to all systems regardless of whether all phases have the same number of components.
The phase rule is an significant generalization. Although it tells us nothing that could not be deduced in any given system, it is a valuable guide for unraveling phase equilibrium in more complex systems.