Question

Why the common tangent construction determines phase equlibrium phase equlibrium in a plot Gibbs free energy versus composition (lever rule)?

Answer 1

The dependence of the Gibbs free energy G from composition at a given temperature T and pressure P is given as

Here we have assumed the binary mixture having mass fractions x1 and x2 while mu are their corresponding chemical potentials. Since the two mass fractions sum to unity, however, let us define

Now simplify Eq. (1) we get

Clearly it is the equation of a line in -X space with intercept ₁ and slope ₂-₁. It infers that in a plot of vs. X at constant T and P, this line is tangent to the (X) curve.  Now, if two or more phases are in equilibrium, the chemical potential of both components must be the same in the coexisting phases, so their (X) curves must share a common tangent line, whose intercept and slope are given by the above equation. Furthermore, the compositions that coexist are given by the tangency points where this "chemical potential line" touches the free energy-composition curves. Hence, the tangent reads the Gibbs free energy versus composition variation. Pictorially it can be shown as follows:

In the above figure red line shows the tangent and superscipts A and B have been used to distinguish the two mixtures.