## Common tangent rule

The foundation of the graphical method is called the common tangent rule. This is the proof that a graphical construction applied to free energy curves provides the link to the phase diagram. The common tangent rule states that: the compositions of the two coexisting equilibrium phases lie at the points of common tangency of the free energy curves. The two phases may be both solids, both liquids, or one solid and one liquid. For generality, the two phases are labeled I and II. As usual, the components are A and B.

Equation (7.27) for a binary system is:

The equilibrium condition of Eq (8.2) requires that ^AI = ^AII and ^.BI = ^.BII. When these equalities substituted into either of the above equations, the result is the common tangent condition:

Applying this equation to each phase and using dxA = -dxB gives:

This equation states that the equilibrium concentrations of two coexisting phases in a binary system are the points on the free energy curves that are tangent to the same straight line. 