Nonideal solutions are often described by the following equation for the Gibbs energy:

[1]

where C is a constant describing the pair interaction in the solution. This leads to the following expressions for the chemical potentials of A and B in the solution:

[2]

In a system consisting of two phases a and b, the solution of components A and B may be nonideal in both phases. Each phase is then described by its own nonideality parameter, C_a or C_b. If pure A and pure B undergo the a®b transition at the respective temperatures T_A and T_B, with respective enthalpies of transition D_transH_A and D_transH_B. one can write the following conditions for equilibrium of species A and B between phases a and b at a given temperature T

[3]

Equations [3] can be used to construct a 2-component phase diagram that reproduces many of the qualitative features of azeotropic or eutectic behavior characteristic of nonideal mixing in one or both of the phases. The procedure described here is modeled after that used by E.E. Brumbaugh and C. Huang [Methods in Enzymology, 210:521 (1992)] for analyzing the phase behavior of phospholipids in model biological membranes.

At right are two pictures of a phase diagram that illustrate how the equations may be used. In the upper diagram, the temperature is fixed at T. If T is plugged into equations [3], they can be solved (although they are nonlinear) for the compositions of the two phases, i.e. x_B^a and x_B^b. These are the x_B values where the constant-T line intersects the phase a and the phase b curves, respectively.

For comparison with experimental data, however, it is more reasonable to fix the overall composition at a value x_B and find the temperatures corresponding to the appearance and disappearance of the 2 phase region, as shown in the lower diagram. To find the intersection of the isopleth at x_B with the a phase line, one would set x_B^a = x_B and then find the combination of T and x_B^b that satisfy Equations [3]. Brumbaugh and Huang suggest solving each equation for T, plotting both as a function of x_B^b, and finding the point at which the two curves give the same value of T. (They use a root-finding routine on a computer to do this). The resulting value of T (T_onset) is the temperature at which the isopleth crosses the lower phase boundary. To find the temperature at the upper phase boundary (T_offset), one repeats the procedure for x_B^b = x_B

The class will build three selected phase diagrams for a given mixture assuming the following thermodynamic parameters:

D_transH_A = 22.0 kJ mol^-1; T_A = 300 K

D_transH_B = 32.5 kJ mol^-1; T_B = 290 K

Assignment:

(a) Derive equations [2] from Equation [1].
Hint: . Your result for the chemical potential of B may be derived by analogy.

(b) Derive Equations [3] from Equations [2] assuming phase equilibrium.

(c) Use Equations [3] and the parameters given above to predict T_onset and T_offset for the two particular pairs of (C_a, C_b) values and the x_B value that have been assigned to you (see below)

Point Assignments for Class Phase Diagrams

Name	Ca kJ mol^-1	Cb kJ mol^-1	xB	Name	Ca kJ mol^-1	Cb kJ mol^-1	xB
Bouchard	+2.0, -2.0	0	0.1	Mayer	0	+2.0,-2.0	0.8
Ciampa	+2.0,-2.0	0	0.2	Morais	0	+2.0,-2.0	0.9
Crowe	+2.0,-2.0	0	0.3	Neveu	-2.0, 0	2.0, 0	0.1
Cummings	+2.0,-2.0	0	0.4	Noonan	-2.0, 0	2.0, 0	0.2
Foster	+2.0,-2.0	0	0.5	Porter	-2.0, 0	2.0, 0	0.3
Fowler	+2.0,-2.0	0	0.6	Psichopaidas	-2.0, 0	2.0, 0	0.4
Goldfarb	+2.0,-2.0	0	0.7	Racke	-2.0, 0	2.0, 0	0.5
Howanski	+2.0,-2.0	0	0.8	Rasputnis	-2.0, 0	2.0, 0	0.6
Jarvis	+2.0,-2.0	0	0.9	Renzi	-2.0, 0	2.0, 0	0.7
Lawton	0	+2.0,-2.0	0.1	Sicoli	-2.0, 0	2.0, 0	0.8
LeClerc	0	+2.0,-2.0	0.2	Small	-2.0, 0	2.0, 0	0.9
Lightburn	0	+2.0,-2.0	0.3	Stavropolous	-2.0, 0	-2.0, 0	0.3
Littleke	0	+2.0,-2.0	0.4	Streit	-2.0, 0	-2.0, 0	0.4
Loonan	0	+2.0,-2.0	0.5	Swistak	-2.0, 0	-2.0, 0	0.7
Luke	0	+2.0,-2.0	0.6	Tanagho	-2.0, 0	-2.0, 0	0.8
Malinovskaya	0	+2.0,-2.0	0.7