Two phases, a and β, are at the same temperature, T, and pressure, P (any effects due to gravitation, surface forces or semi-permeable membranes are neglected; systems with a semi-permeable membrane are considered in a later section). Phase a has composition zi

concentration (often zi is the mole fraction) and subscript i denotes a component; thus z2

the following variables: T and P, (m - 1) compositions for a and (m - 1) compositions for β. The total number of compositions is:

2 1 2 2( )m m− + = (2.1)

Gibbs’ phase rule tells us that the number of degrees of freedom, D, is given by:

D m P= − +# 2 (2.2)

where #P is the number of phases. For a two-phase system, D is equal to m. The phase-equilibrium problem can then be formulated in this way: for a two-phase system containing m components at equilibrium, we have 2m variables. If we specify m variables, our task is to find the remaining m variables. To make this formulation less abstract, consider vapor-liquid equilibrium in a binary system. There are four variables: T, P, y and x, where y is the mole fraction in the vapor phase and x is the mole fraction in the liquid phase for either component 1 or 2. Two of these four variables can be arbitrarily specified, thus, the task is to find the remaining two.