The phase diagram for the system C_s-Rb is shown in Figure 10.29. Examine the extent to which the phase diagram can be reproduced, assuming that the liquid solutions are ideal and that the solid solutions are regular.
For C_s :
\Delta G^{\circ }_{m,C_s}=2100-6.95T J
and for Rb:
\Delta G^{\circ }_{m,Rb}= 2200-7.05T
Initially, draw the Gibbs free energy of mixing curves for the solid and liquid solutions at 9.7°C (282.7 K) and determine if some value of α gives Gibbs free energy of mixing curves similar to those shown in Figure 10.25c; that is, the curve for the liquid solutions lies above the curve for the solid solutions, except at a single composition where the two curves touch one another.
Relative to the pure solids as standard states at the temperature T ,
+RT(X_{Rb}\ln X_{Rb}+X_{Cs}\ln X_{Cs}) (i)
which, at 282.7 K, becomes
\Delta G^M(liquid solutions)=205X_{Rb}+133(1-X_{Rb})+8.3144\times 282.7\times \left[X_{Rb}\ln X_{Rb}+(1-X_{Rb})\ln (1-X_{Rb})\right]
Also,
\Delta G^M(solid solutions)=RT(X_{Rb}\ln X_{Rb}+X_{Cs}\ln X_{Cs})+\alpha X_{Rb}X_{Cs} (ii)
Which at 282.7 K becomes
\Delta G^M(solid solutions)=8.3144\times 282.7\times (X_{Rb}\ln X_{Rb}+X_{Cs}\ln X_{Cs})+\alpha X_{Rb}X_{Cs}The Gibbs free energy of mixing curves, drawn with α = 668 J, are shown in
Figure 10.30, which shows that the curves touch one another at X_{Rb}=0.47, in exact agreement with Figure 10.29.
The Gibbs free energies of mixing, given by Equations (i) and (ii), with = 668 J and T = 292 K, are shown in Figure 10.31. The common tangent at the Cs-rich side of the diagram gives the solidus composition as X_{Rb}=0.10 and the liquidus composition as X_{Rb}=0.13, which is in excellent agreement with the phase diagram. At the Rb-rich side of the diagram, the double tangent gives the solidus and liquidus compositions as, respectively, X_{Rb}=0.81, and X_{Rb}=0.75, which are in good agreement with the phase diagram values of 0.80 and 0.77.
It is thus seen that the phase diagram is reproduced by assuming that the liquid solutions are ideal and the solid solutions are regular with α = 668 J.
