Chapter 10: Gibbs Free Energy & Binary Phase Diagrams

The foundation of this thermodynamic correspondence rests on the condition that any phases coexisting in equilibrium must exhibit identical partial molar Gibbs free energies, also known as chemical potentials, for each component. Graphically, this equilibrium condition is satisfied by drawing a single common tangent connecting the molar Gibbs free energy of mixing curves for the phases involved. For ideal solutions, the molar Gibbs free energy of mixing is negative, determined solely by temperature and the contribution of configurational entropy. Deviations from ideal mixing (Raoultian behavior) are quantified by activity coefficients; these deviations cause the Gibbs free energy curve to shift upwards for positive deviations or downwards for negative deviations. A key outcome of this approach is the rigorous derivation of the liquidus and solidus curves for ideal binary systems, demonstrating that their compositions are determined exclusively by the melting temperatures and molar heats of melting of the pure components. The choice of standard state for defining component activity, whether pure liquid or pure solid, is critical for activity calculations but does not alter the equilibrium compositions determined by the common tangent points. The chapter further explores non-ideal behavior using the regular solution model, showing that a sufficiently large positive interaction parameter can lead to immiscibility gaps (phase separation) in both the liquid and solid states, defining a critical temperature for this separation. Finally, the text explains how various invariant three-phase reactions—including the eutectic (one liquid yields two solids), the monotectic (one liquid yields a solid and a second liquid), and the peritectic (liquid plus one solid yields a second solid)—are characterized by a unique triple common tangent drawn to the three participating phase curves. All phase boundaries must conform to topological rules, such as the Gibbs–Konovalov rule, which dictates that coexistence curves meeting at a maximum or minimum (congruent points) must have zero slope at the intersection, and the Third Law constraint, requiring terminal solid solubility limits to approach infinite slopes near zero Kelvin. The chapter concludes by noting that higher-order transitions, like the magnetic Curie temperature, are displayed on phase diagrams but do not adhere to the same stringent thermodynamic constraints governing first-order phase transformations.