Solutions theories

The vacancy solution theory was developed by Suwanayuen and DanneE as a method of predicting multicomponent adsorption equilibria from singlecomponent isotherms without the assumption of an ideal adsorbed phase. A somewhat different analysis is given here although the essential features of the model are retained. 

Instead of incorporating the surface work term vj/ directly into the definition of the free energy of the adsorbed phase [Eq. (3.26)] one may 

Differentiating at constant temperature and spreading pressure we may define a partial molar free energy or chemical potential by the relation 

As a first-order deviation from ideal behavior one may assume that the adsorbed phase obeys the regular solution model  

Proceeding in the same manner through the Gibbs isotherm Eq. (3.65) leads to 

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Friedman H L 1962 Ionic Solution Theory (New York Interscience)... 

Allnatt A 1964 Integral equations in ionic solution theory Mol. Phys. 8 533... 

Optimized convergence and application to ionic solution theory J. Chem. Phys. 55 1497... 

Hynes J T 1985 The theory of reactions in solution Theory of Chemical Reaction Dynamics ed M Baer (Boca Raton, FL CRC Press) pp 171-234... 

Ideal Adsorbed Solution Theory. Perhaps the most successful approach to the prediction of multicomponent equiUbria from single-component isotherm data is ideal adsorbed solution theory (14). In essence, the theory is based on the assumption that the adsorbed phase is thermodynamically ideal in the sense that the equiUbrium pressure for each component is simply the product of its mole fraction in the adsorbed phase and the equihbrium pressure for the pure component at the same spreadingpressure. The theoretical basis for this assumption and the details of the calculations required to predict the mixture isotherm are given in standard texts on adsorption (7) as well as in the original paper (14). Whereas the theory has been shown to work well for several systems, notably for mixtures of hydrocarbons on carbon adsorbents, there are a number of systems which do not obey this model. Azeotrope formation and selectivity reversal, which are observed quite commonly in real systems, ate not consistent with an ideal adsorbed... 

Isotherm Models for Adsorption of Mixtures. Of the following models, all but the ideal adsorbed solution theory (lAST) and the related heterogeneous ideal adsorbed solution theory (HIAST) have been shown to contain some thermodynamic inconsistencies. References to the limited available Hterature data on the adsorption of gas mixtures on activated carbons and 2eohtes have been compiled, along with a brief summary of approximate percentage differences between data and theory for the various theoretical models (16). In the following the subscripts i and j refer to different adsorbates. 

If the mutual solubilities of the solvents A and B are small, and the systems are dilute in C, the ratio ni can be estimated from the activity coefficients at infinite dilution. The infinite dilution activity coefficients of many organic systems have been correlated in terms of stmctural contributions (24), a method recommended by others (5). In the more general case of nondilute systems where there is significant mutual solubiUty between the two solvents, regular solution theory must be appHed. Several methods of correlation and prediction have been reviewed (23). The universal quasichemical (UNIQUAC) equation has been recommended (25), which uses binary parameters to predict multicomponent equihbria (see Eengineering, chemical DATA correlation). 

The solubihty parameter is therefore a measure of the energy density hoi ding the molecules in the hquid state. Note that regular solution theory can only predict positive AH. Thus, with this approach, prediction of solubihty involves matching the solute and solvent solubihty parameters as closely as possible to minimize AH. As a very rough mle of thumb 1— 21 must be less than 2 Q/cm ) for solubihty. 

Practical Solubility Concepts. Solution theory can provide a convenient, effective framework for solvent selection and blend formulation (3). When a solute dissolves in a solvent, a change in free energy occurs as a result of solvent—solute interactions. The change in free energy of mixing must be negative for dissolution to occur. In equation 1,... 

Regular Solution Theory. The key assumption in regular-solution theory is that the excess entropy, is zero when mixing occurs at constant volume (3,18). This idea of a regular solution (26) leads to the equations ... 

Consider a binary adsorbed mixture for which each pure component obeys the Langmuir equation, Eq. (16-13). Let n = 4 mol/kg, nl =. 3 mol/kg, Kipi = K2P2 = 1. Use the ideal adsorbed-solution theory to determine ni and n. Substituting the pure component Langmuir isotherm... 

This approach to solution chemistry was largely developed by Hildebrand in his regular solution theory. A regular solution is one whose entropy of mixing is ideal and whose enthalpy of mixing is nonideal. Consider a binary solvent of components 1 and 2. Let i and 2 be numbers of moles of 1 and 2, 4>, and 4>2 their volume fractions in the mixture, and Vi, V2 their molar volumes. This treatment follows Shinoda. ... 

Recall that regular solution theory deals with nonpolar solvents, for which the dispersion force is expected to be a major contributor to intermolecular interactions. The dispersion energy, from Eq. (8-15), is for 1-2 interactions... 

We encountered the quantity AE ap/V in Eq. (8-35) it is the cohesive energy density. The square root of this quantity plays an important role in regular solution theory, and Hildebrand named it the solubility parameter, 8. 

For a given reaction studied in a series of solvents, (8r- 8 f) is essentially constant, and most of the change in In k will come from the term — AV (8j — 8s)". If AV is positive, an increase in 8s (increase in solvent internal pressure) results in a rate decrease. If AV is negative, the reverse effect is predicted. Thus reactivity is predicted by regular solution theory to respond to internal pressure just as it does to externally applied pressure (Section 6.2). This connection between reactivity and internal pressure was noted long ago," and it has been systematized by Dack. -" ... 

Strictly speaking Eq. (8-51) should be applied only to reacting systems whose molecular properties are consistent with the assumptions of regular solution theory. This essentially restricts the approach to the reactions of nonpolar species in nonpolar solvents. Even in these systems, which we recall do not exhibit a marked solvent dependence, correlations with tend to be poor. - pp Nevertheless, the solubility parameter and its partitioning into dispersion, polar, and H-bonding components provide some insight into solvent behavior that is different from the information given by other properties such as those in Tables 8-2 and 8-3. 

In the present review a description is given of the phase behavior of clathrates on the basis of a solution theory. The treatment is restricted to those cases where the empty host lattice ( solvent") is indeed unstable, although many of the present considerations also apply to the few cases known where the host lattice is stable. An example of the latter is the chroman complex first discovered by Dianin9 and recently examined by Baker and McOmie and Powell and Wett ers.34... 

Recent experiments2 on the equilibrium Hu ice gas at — 3°C in the system HaS-propane-water confirm that these two gases also form mixed hydrates of variable composition, as shown in Fig. 10. In this respect the present system is similar to the system me thane-propane-water of Fig. 7, but unlike the latter it exhibits a minimum pressure (azeotrope). It was further shown that the solution theory of clathrates can account for this interesting phenomenon. For details the reader is referred to ref. 29. 

Configurational energy for clathrates, 12 Configurations, superposition of, 258 Conformal solution theory, 137 Coordination polymerization, 148, 162, 170... 

Lamola, A. A. (1969). Electronic energy transfer in solutions theory and application. In Leermakers, P. A., and Weissberger, A. (eds.), Energy Transfer and Organic Photochemistry, Technique of Organic Chemistry 14 17-132. Interscience Publishers, New York. 

To recapitulate, the Flory version of the Prigogine free-volume or corresponding-states polymer solution theory requires three pure-component parameters (p, v, T ) for each component of the solution and one binary parameter (p ) for each pair of components. 

Many further developments can be expected in the use of corresponding-states polymer solution theory in engineering practice. However, the reliability and versatility of this method is now well demonstrated for engineering use. 

Gee and Orr have pointed out that the deviations from theory of the heat of dilution and of the entropy of dilution are to some extent mutually compensating. Hence the theoretical expression for the free energy affords a considerably better working approximation than either Eq. (29) for the heat of dilution or Eq. (28) for the configurational entropy of dilution. One must not overlook the fact that, in spite of its shortcomings, the theory as given here is a vast improvement over classical ideal solution theory in applications to polymer solutions. 

The general theory of polymer solutions, in which the nonuniformity at high dilutions was disregarded, yielded Eq. (31"), which is of the same form as Eq. (72). The coefficients of the first terms in the two expansions are identical, of course. In view of Eqs. (45) and (46), the second coefficient as given by Eq. (75 ) differs from that in Eq. (31") by the factor Thus the dilute solution theory predicts a slope... 

A number of other attempts have been made to account for the properties of concentrated aqueous solutions of ionic compounds by procedures that represent further improvements on the simple Debye-Huckel approach. However, they lie outside the scope of the present chapter. The important point to emphasize is that the concentrated aqueous solutions that are generally employed in the preparation of AB cements tend to exhibit significant ion-ion interactions such interactions lead to significant deviations from ideality which may be accounted for by substantial extension of the ideas of simple dilute solution theory. 

Of great importance for the development of solution theory were the studies of col-ligative solution properties, detected in the 1870s and 1880s by F. M. Raoult, J. H. van t Hoff, and others. These are properties that depend not on the chemical nature of solutes but on their concentration. Three such colligative properties exist ... 

According to modem views, the basic points of the theory of electrolytic dissociation are correct and were of exceptional importance for the development of solution theory. However, there are a number of defects. The quantitative relations of the theory are applicable only to dilute solutions of weak electrolytes (up to 10 to 10 M). Deviations are observed at higher concentrations the values of a calculated with Eqs. (7.5) and (7.6) do not coincide the dissociation constant calculated with Eq. (7.9) varies with concentration and so on. For strong electrolytes the quantitative relations of the theory are altogether inapplicable, even in extremely dilute solutions. 

Of great importance for the development of solution theory was the work of Gilbert N. Lewis, who introduced the concept of activity in thermodynamics (1907) and in this way greatly eased the analysis of phenomena in nonideal solutions. Substantial information on solution structure was also gathered when the conductivity and activity coefficients (Section 7.3) were analyzed as functions of solution concentration. 

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