## Free energy composition curves

In the free energy formula of Eq (7.36), nonideality is expressed by the general form gex, the excess free energy. The simplifications used in the prior analyses of ideal melting and phase separation, namely neglecting sex and confining hex to the regular-solution model, are not valid for most binary systems. In order to construct phase diagrams by the common-tangent technique, more elaborate solution models are needed to relate free energy to composition for all likely phases. Figure 8.9 shows plots of g Vs xB for the three phases at six temperatures, with T6 the highest and T1 the lowest. In the six graphs, the curves for each phase keep approximately the same shape but shift relative

5 "eutectic" is the Greek word for low-melting, indicating a melting temperature lower than that of either of the pure components.

to each other. Examination of the six plots shows that the liquid phase curve shifts upward as the temperature is reduced more rapidly than do the curves for the two solid phases. This feature arises from the thermodynamic relation of Eq (6.16):

Since the entropy of a liquid is always larger than that of a solid, gL moves up with decreasing temperature faster than ga or gp. The relative movement with T of each of the phases and the shape of their g Vs xB curves determines the phase diagram.

Fig. 8.9 Free energy — composition curves for an A-B binary system with two solid phases (a and P) and a liquid phase (From Ref. 1)

Fig. 8.9 Free energy — composition curves for an A-B binary system with two solid phases (a and P) and a liquid phase (From Ref. 1)

Consider first the composition-dependence of the two solid phases. The free energies of the a and P solids at first decrease from the pure-component value then increase rapidly as the other component is added. The initial decreases are due chiefly to stabilization of the dilute solutions by the entropy of mixing. The shape of the curve is also affected by the composition-dependence of the nonideality term gex.

The subsequent rise in free energy of the two solid phases is due for the most part to crystal structure effects. The intercept of the a curve with the left-hand axis (e.g., point a in the T6 plot) represents the free energy of pure A in the a crystal structure. If the a phase curve were extended to intersect the right-hand axis, the free energy here would represent that of pure B in the a crystal structure. That this intercept is higher than the intercept of the P curve with the right-hand axis simply reflects that fact that pure B is more stable in the P crystal structure than it is in the structure of the a phase.

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