Theory of conformal solutions

The main interest of the theory of conformal solutions is that it is independent of any particular model such as the lattice model. It applies equally well to liquid, gaseous or solid solutions. For this reason, it gives a kind of test for the consistency of any theory starting from a particular model. The first order terms (that is terms in dr must always reduce to the expressions discussed here. 

This ows that the assumptions used in the simplified lattice model of Ch. Ill cannot be consistent. Indeed this model predicts in its zeroth approximation ( 2) a vanishing excess entropy (cf. 3.2.15), instead of an exce.ss entropy proportional to the excess free energy. This is clearly due to the assumption that the excess free energy is only related to changes in the lattice partition functions and that in spite of changes in the interaction, the excess volume is zero. 

We also see from the theory of conformal solutions that the excess entropy may certainly arise for other reasons than the changes in lattice disorder due to non-randomness of mixing. 

Does first order perturbation theory, by itself, already offer a basis for the soimd understanding of the thennod3mamic properties of mixtures The main qualitative conclusion of 3 is that the excess fimctions are proportional to each other. Positive deviations from Raoult s law (g 0) should always correspond to heat absorption on mixing (A 0) and expansion on mixing (o 0). However, many systems have been reported recently which present positive deviations from Raoult s law together with contraction on mixing (cf. Ch. XI). This occurs even for very simple mixtures like CO—CH4, CCI4—C(CH )4 for which the basic assumption of the theory of conformal solutions should be satisfied. 

Even if we ignore these systems and concentrate on systems for which aU excess functions have the same sign, the validity of the theory of conformal solutions is very doubtful. Let us consider the systems beilzene/cydohexane, benzene/carbon tetrachloride, cyclo-hexane/carbone tetrachloride studied by Scatchard, Wood and Mockel [1939, 1939. 1940]. 

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A great many of the difficulties (and sometimes the misunderstandings) arise from point (c). It is however important to notice that the APM describes the properties of solutions as finite differences between suitable composition-dependent averages and the properties of the pure components. Series expansions in powers of 6, p, 6, and a were introduced afterwards for the purpose of qualitative discussion and comparison with other treatments, e.g., the theory of conformal solutions.34>85>36 They introduce artificial difficulties due to their slow convergencef which have nothing to do with the physical ideas of the APM. Therefore expansions of this type should be proscribed for all quantitative applications one should instead use the compact expressions of the excess functions. 

There have been several approaches to the expression of thermodynamic quantities of solutions. Scatchard published a series of papers (see 1801 and previous papers, especially 1802) based on the classical approach via chemical potentizil. Barker (130, 128, 131) applied the theory of conformal solutions (1254) to some H bonding systems after modifying it to allow for dipole attractions or, more generally, molecular orientations. The curves are similar in both cases. 

The phase behavior that is exhibited by a critical or supercritical mixture of several components is usually not simple Street (jO reports six classes of phase behavior diagrams In the simplest classes of systems (classes 1 and 2), the critical lines are continuous between the critical points of pure components Study of reaction equilibrium at SCF conditions requires knowledge of critical properties of the reacting mixture at various levels of conversion Three different approaches to evaluate critical properties are available, viz, empirical correlations, rigorous thermodynamics criteria and the theory of conformal solutions (10) The thermodynamic method is more general and reliable because it is consistent with the calculation of other thermodynamic properties of the reacting mixture (11) ... 

Activities of individual components are calculated on the basis of the theory of conformal solutions (Reiss et al., 1962). This theory was derived for ionic systems without the formation of complex ions, in which both anions and cations have identical charges. This theory was later applied to systems containing ions of various valences (Saboungi and Blander, 1975). It should be emphasized that the application of this theory to silicate melts has only a formal character. 

Let us now discuss more closely the possible range of application of the theory of conformal solutions. 

These considerations and the existence of many solutions for whidi the excess functions g and have not the same sign indicate that presumably no theory limited to first order terms can at all give an adequate description of the thermodynamic properties of mixtmres. This may be taken as a consequence of the fact that the combining rule (2.7.8) seems to be a feiir ai roximation for mixtures of spherical nonpolar molecules. If the interaction i2 is somewhere between the arithmetical and geometrical mean of u and 22. i becomes a second order quantity (cf. 2.7.9) and the theory of conformal solutions would predict no deviations from the laws of perfect solutions. In order to force agreement using only first order terms, this approach has to overemphasize the deviations from the combining rules (2.7.8) or (2.7.10). This IS probably the reason of the discrepancy between (4.4.1) and (4,4.2). 

If we only retain the first order term all excess functions become proportional to B and have the same sign. This is in agreement with the theory of conformal solutions (Ch. IV, 3-4). However, as we have already seen in the case of the one dimensional model, the second order terms destroy this ample relation between the excess fnnctions. Let us consider in more detail a few typical cases which may arise (a) Geometrical or arithmetical mean (2.7.9, 2.7.11). In these cases 02... 

First of all we notice that the first order contribution in all the excess functions (9.5.4)-(9.5.8) are identical with the excess functions calculated by the theory of Conformal Solutions (Ch. TV, 3). This may be considered as a test for the consistency of the model. 

We have already pointed out that the first-order terms are exactly those given by the theory of Conformal Solutions. The second order terms are quite similar to those given by the cell model of Ch. VIII but are now derived in a much more elegant and direct way. 

Again we see that the first order term in the excess functions (10.7.4) (10.7.9) agrees with the theory of conformal solutions. 

As long as we retain only first order terms as in the theory of conformal solutions due to Longxtet-Higgins [1951] the treatment of critical phenomena is very simple. In this approximation the mixture obeys the theorem of corresponding states exactly as a single component. For example the compressibility factor at die critical point... 

The theory of conformal solutions due to Longuet-Hi ins has been studied in detail in Ch. TV. It is convenient to write the free enei of a binary mixture in the form (cf. 4.3.10)... 

Neglecting all second order terms we obtain again the results (12.4.16) deduced from the theory of conformal solutions. 

As we have seen in (12.4.15) the theory of conformal solutions predicts... 

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