phase diagram of regular system

The free energy of nonideal solutions:
Real solutions are, of course not ideal. The number of different ways the free energy of real solutions may depart from ideality is large, but fortunately the number of distinctly different types of two component phase equilibria resulting is relatively small. Here only some of the simpler types of departures from ideality and the corresponding equilibrium diagrams. The free energy of a solution is:
G^S=G^M+X_A (H ?_A-H_A )+X_B (H ?_B-H_B )-T[X_A (S ?_A-S_A )+X_B (S ?_B-S_B )]+RT(X_A lnX_A+X_B ?lnX?_B )

Or grouping the various quantities
G^S=G^M+?H^xs-T?S^xs-T?S_m
=G^M+?G^xs-T?S_m (1)
It may be noted that for an ideal solution ?G^xs=?G^(xs.id)=0, and thus ?G^xs for any real solution is the difference between the actual free energy of a solution and the value of the free energy would have if the solution were ideal.?G^xs is consequently designated the excess free energy of solution; similarly ?H^xs is the excess enthalpy ,and ?S^xsthe vexcess entropy. When ?G^xsis positive ,a solution is said to have a positive deviation from ideality ,conversely a negative ?G^xscorresponding to a negative deviation from ideality.
Let us assume first that ?S^xs is negligibly small ,which is equivelant to limiting the discussion to approximately regular solutions,a regular solution being defined as one for which ?S^xs=0 ,eq.(1)becomes
G^S=G^M+?H^xs-T?S_m
This condition is approached in many liquids and in some close-packed solids such as metals.the difference between the free energy curve of such a solution and that of an ideal solution lies entirely in the term ?H^xs.

Quasi-chemical calculation of the excess enthalpy of solution:- :-The excess enthalpy can be calculated from:-
Since

The internal energy is made of the sum of the potential energies between the atoms and their thermal kinetic energies , thus
For condensed phases PV is small at ordinary pressures ,and generally VS-VM is small ,so that PVS-PVM=0.The kinetic energies are also virtually unaffected by composition and so EKE,S=EKE,M.Eq.(2)thus becomes
To account the excess enthalpy a simple model called statistical or quasi-chemical model can be used. In this model the excess enthalpy is only related to the bond energies between adjacent atoms. The assumption is that the interatomic distances and bond energies or interaction energies are independent of composition. An expression for EPE can be obtained by assigning an interaction energy to each atom pair in the system and summing over all atom pairs; that is,.


Where ni is the number of pairs having interaction energy Ein.
The nearest-neighbor approximation is generally applied; it is usually assumed that only nearest-neighbor interactions need be counted. This actually accounts for about 80 to 90 % of the total interaction energy; the other 10 to 20 % is often not important qualitatively but can be very important if the attempt is made to use the theory quantitatively.
To apply this concept to the calculation of potential energy in a two-component solid solution of metals A and B having a crystal structure with the coordination number Z, there are three types of nearest-neighbor pairs, AA, BB, and AB pairs. To find the number of AB pairs consider the surroundings of, say, any A atom. It has Z nearest neighbors, and if the atoms are assumed to be intermixed at random, the fraction XB of these will be B atoms. Thus, each A atom has ZXB B neighbors. In a mole of solution there are XAN (N =Avogadros number)A atoms, so that the total number of AB pairs is:
each A atom also has ZXA A neighbors ,thus
n_AA=ZN/2 X_A^2
The factor 1/2 being introduced since in this case each AA pair was counted twice ,once for each A atom in it .Similarly
n_BB=ZN/2 X_B^2
The potential energy of the solid solution is then:

The potential energy of a mechanical mixture of pure A and pure B is found in like manner:-

Combinig (3), (4), and (5)
Or
?H^xs=ZNX_A X_B E_in (6)
where



An expression similar in form can in principle be developed for liquids; the expression of eq.(6) would differ chiefly in that neither coordination number nor the interatomic distances between nearest neighbors would be exactly identifiable-average values would have to be used.
In regular solutions, the excess enthalpy constitutes the deviation of the solution from ideality; it is now seen that is in the quasi-chemical view the direction of this deviation depends on the sign of Ein, which in turn depends on the relative magnitudes of Ein,AA, Ein,BB and Ein,AB.If Ein,AB is equal to the average of Ein,AA and Ein,BB, that is if the attraction between unlike atoms is the same as that between like atoms, then ?Hxs=0, and the solution is ideal. If like atoms attract more strongly than unlike atoms (Ein,AB greater than the average of Ein,AAand Ein,BB*),