phase diagram of ideal solution

Equilibrium phase diagram in ideal system:-
For a system in which ideal solutions are formed in both the liquid and solid states, the free energy curves for both states will be the form in fig.(1).






At a very high temperature such as T6 in fig.(2),the liquid free energy curve lies below that of the solid at all compositions .with decreasing temperature the free energies of both liquid and solid state increasing in accordance with (dG/dT)_P=-S . Since ,however the entropy of substance at a given temperature is generally higher in liquid than in the solid state, the rate of increase of the liquid free energy7 is greater than that of solid. Thus the free energy curve of liquid in fig.(2)moves up relative to that of the solid as the temperature decreases and ultimately the two intersect. It is assumed in fig.(2)that this happens first at pure A at temperature T5.The free energies of pure solid A and pure liquid A are equal. This is the melting temperature of A. the temperature at which pure liquid A and pure solid A may coexist in equilibrium .this is indicated by the point marked a at the temperature T5 and the composition pure A in the temperature –composition phase diagram of fig.(3).At all other composition at T5 the stable condition is a single liquid solution.
At T4 the liquid and solid free energy curves intersect at some intermediate composition. Under these circumstances it may be demonstrated that within a certain composition range a mixture of two solutions, one liquid and one solid, is more stable than either a liquid solution or a solid solution alone.






Consider , for example, the alloy of composition X1in the free energy diagram of fig.(4).if this alloy exist as a single solid solution, it has the free energy given by the point a in the diagram. It has the lower free energy corresponding to point b if it exist as a single liquid solution, but the lowest free energy it may have is given at point M, representing the free energy of the alloy when it exists as a mixture of solid and liquid solutions with compositions given at point d and, the two points of common tangency to the free energy curves. That this latter condition does the fact correspond to equilibrium may be seen in the following way. Suppose the alloy X1 exists as a mixture of liquid and solid solutions of the general compositions XL,XSas shown in fig(4) we write for the liquid phase:-
dG^L=?_A^L dX_A^L+?_B^L dX_B^L (1)
Since the partial molal free energy ? is by definition the free energy per mole of a component in solution , the free energy of the liquid solution G^L is
G^L=?_A^L X_A^L+?_B^L X_B^L (2)
Upon substituting XA=1-XB and dXA=-dXB, combining eq.(1)&(2),and suitably rearranging ,we obtain:?_B^L=G^L+(1-X_B^L)dG/(dX_B^L ) (3)
?_A^L=G^L-X_B^L dG/(dX_B^(L ) ) (4)
Now comparing eq.(4) with the diagram in fig .(4),it is seen that G^L=(HF) ? and (dG/(dX_B ))^L=-(CH) ?/(X_B^L )
So that ?_A^L=(HF) ?+X_B^L (CH) ?/(X_B^L )=(HF) ?+(CH) ?=(CF) ?
In a similar way it may be shown that ?_A^S=(JF) ? ,but at equilibrium the two chemical potentials are equal and then (CF) ?=(JF) ?
Which means that for equilibrium C and J represent the same point .This condition of equilibrium and that expressed in:
(dG/(dX_B ))^L=(dG/(dX_B ))^S