12.480 Lecture 7 Notes

Gibbs phase rule - an introduction dG = -SdT + VdP + Σᵢ(μᵢdnᵢ) Tells us the number of phases that coexist at equilibrium. Also provides a method for determining the conditions of formation of a phase assemblage. of variables = C of phases = Φ Each phase can vary in composition, so Φ variables and T, P can vary = CΦ + 2 of constraints: For each phase, you have stoichiometric constraints: Σᵢ(xᵢ) = 1 Here is one equation for each phase, for a total of Φ. Gibbs phase rule continued And we have conditions of equilibrium, e.g., the chemical potentials of component i in each phase is the same at equilibrium. So, in phases A, B, C, etc.: μᵢᴬ = μᵢᴮ = μᵢᶜ There are Φ - 1 equalities, one for each component, for a total of c(Φ - 1). Gibbs phase rule F is the relation between the number of variables and the number of constraints. F = # of variables - # of constraints = CΦ + 2 - [Φ + c(Φ - 1)] = C + 2 - Φ Leads to conditions: F = 0: invariant F = 1: univariant F = 2: divariant Recall the phase rule: F = c + 2 - Φ, or for a two-component system: P = 4 - Φ. So, 4 phases coexist at an invariant point, 3 on a univariant curve, and 2 at an invariant coexistence point. An invariant assemblage will only exist at a unique pressure and temperature. It would be very rare, in general, to find a 4-phase coexistence. A 3-phase (univariant) equilibrium would be much more likely to occur if we consider the phase relations at a fixed pressure and variable temperature. So, we will generally see these 3-phase (univariant) lines on our T-X diagrams.   We will pick a pressure (P₁) and choose 3 arbitrary phases A, B, and L in the system. We assume further that we know the relative positions of these G-curves and the way in which they move with variations in T. Our first G-X section is at r₁. Here, all 3 phases are stable. The minimum free energy is given for single phases or for mixtures of phases. A tangent line drawn to the lowest surface of the G-curve of adjacent phases demonstrates that for some parts of the composition space, the lowest free energy is given by a mixture of two phases rather than one phase with the bulk composition in that phase region.   The next two G-X sections are at successively lower temperatures. The G-curve for L is moving to higher values of G relative to the G-curves for A and B. So, at T₂, L is no longer stable with respect to a mixture of A and B. At T₂, we have a special 1-phase coexistence where L, A, and B all coexist. A T-X diagram summarizes only the compositional information for each of these G-X sections and looks like: We can now project into P-T space, but we will lose some of the compositional information that we had in the T-X sections. The large regions in the P-T diagram are divariant regions, and the letters show the possible thermodynamically stable divariant (two-phase) assemblages that can coexist in that region. Each divariant region is bounded by univariant curves (these 3-phase coexistence lines which we saw in the T-X diagrams) that can occur at different temperatures with varying pressure. Finally, all the univariant lines meet at the invariant point, where all four phases coexist at a special P₁ and T₁.   Now, let's change pressure to P₂ (a lower pressure). Here, it turns out that there is another phase stable (C), in addition to L, A, and B. This is all right, because we can have up to 4 phases coexisting in a two-component system. Let us look at the changes in G-X curves at successively higher temperatures T₄, T₅, and T₆. The T-X diagram for P₂ extracts only the compositional information of which phases coexist, and their compositions are: