To aid in visualizing the interplay of these terms, Eq (8.22) is divided by RT:
rxB(1 - xß) + [xB lnxB + (1 - xB )ln(1 - xb)] = (g - gcomp)/RT
Equation (8.26) is plotted in Fig. 8.5, which shows how the excess enthalpy and the entropy of mixing combine to produce mixing free energy curves. r is varied either by changing the interaction energy Q or by changing the temperature. The development of the curves with two minima for r > 2 is clearly shown. In this range, the positive (repulsive) interaction energy overcomes the negative contribution to the free energy provided by the entropy of mixing (second term) and the solution becomes unstable. The dashed lines represent common tangents to these minima and the points of tangency are the compositions of the two partially-miscible phases.
This is a cumbersome procedure for this particularly simple system, but as shown in the next section, graphical construction of phase diagrams from free energy Vs composition curves becomes the method of choice for complicated systems.
mole fraction of B, xB
Fig. 8.5 Graphical solution of the phase diagram for partially-miscible solution formation
Example: Show (analytically) that the compositions of the two phases at the off-center minima in Fig. 8.5(d) are identical to those obtained by the analytical method for the value of T* corresponding to T = 3 applied to the graphical method.
Analytical Method: Eliminating Q between Eqs (8.18) and (8.20) gives:
Solving Eq (8.21) (numerically) for this value of the relative temperature gives xbi = 0.07 and a second solution at Xbii = 0.93.
Graphical Method: The common-tangent points in the lower-right-hand graph of Fig. 8.5 satisfy the condition of minimum free energy. From Eq (8.26) this is:
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