## Binary phase diagrams analytical construction

Binary phase diagrams depict the stable condensed phase (or phases) formed by a two-component system as a function of temperature and overall composition. The ordinate of a phase diagram is the temperature and the overall composition is the abscissa1. The phase rule (Eq (1.21)) for a two component system permits F = 4 - P degrees of freedom for a two-component system. Since the diagrams deal only with condensed phases, they are minimally affected by total pressure2. Ignoring the total pressure reduces the number of degrees of freedom by one, thereby allowing 3 - P properties to be independently varied. In a single phase (P = 1) portion of the phase diagram, two degrees of freedom are permitted. These are the temperature T and the composition, represented by the mole fraction of one of the constituents. Single-phase regions appear as areas in the phase diagram.

In two-phase zones (P = 2), only one system property can be specified. Fixing the temperature, for example, determines the compositions of the two coexisting condensed phases. These temperature-composition relationships appear in the phase diagram as lines (or curves) called phase boundaries. A three-phase system (P = 3) has no degrees of freedom and is represented by a point on the phase diagram.

The distinction between overall compositions and the compositions of individual phases is essential to understanding phase diagrams. For single-phase zones, the two are identical. When two phases coexist, their compositions are different from the overall composition. The latter is the quantity-weighted average of the compositions of the two phases (i.e., the lever rule).

The structure of a phase diagram is determined by the condition of chemical equilibrium. As shown in Sect. 8.2, this condition can be expressed in one of two ways: either the total free energy of the system (Eq (8.1)) is minimized or the chemical potentials of the each component (Eq (8.2)) in coexisting phases are equated. The choice of the manner of expressing equilibrium is a matter of convenience and varies with the particular application. Free-energy minimization is usually used with the graphical method and chemical-potential equality is the method of choice for the analytic approach.

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