Phase rule, Building Blocks in binary system diagrams, Invariant equilibria

The description of a thermodynamic system formed by C components present in P phases needs the specification of temperature, pressure and composition of all the P phases. 

The composition of each phase is known when the concentrations of C — 1 components in the phase have been defined. Thus, for all the phases, there are P(C — 1) concentration variables. In principle, we have to add P values for the temperature of the different phases and P values of pressure to obtain the total number, 

P + P + P(C - 1) = P(C + 1), of variables. For equilibrium conditions to exist, however, several relationships between the stated variables must occur. 

Each of the P phases must be at the same temperature and pressure so there are P — 1 equalities of temperature and P — 1 of pressure. As for the concentration, the C values of each component in each phase must be related to the corresponding C values in the other phases (technically this means the same activity of each component in all the phases) so C(P — 1) relations are found. The total number of equilibrium conditions, therefore, is given by  

The variance (V, degree of freedom) of the system is defined as the maximum number of variables which may be independently altered without disturbing the equilibrium state of the system. It is given by the difference between the total number of variables and the total number of equations among these variables defined by the equilibrium conditions. Therefore  

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