Regular Solutions Solubility Parameter

If A Wp, is not required to be zero, a so-called regular solution is obtained. All deviations from ideality are ascribed to enthalpic effects. Theheatof mixing AW, can be formulated in terms of relative numbers of intermolecular contacts between like and unlike molecules. Nonzero AWp, values are assumed to be caused by the net results of breaking solvent (1-1) contacts and polymer (2-2) contacts and making polymer-solvent (1-2) contacts [1,2], 

In effect, this takes W12 to be equal to the geometric mean of Wi and 1022- Then 

The first two terms on the right-hand side of Eq. (12-7) represent the interaction energies of the isolated components, and the last term is the change in internal energy AUm of the system when the species are mixed. If the contact energies can be assumed to be independent of temperature, the enthalpy change on mixing, AHm, is then 

The terms in (cium jlviYare solubility parameters and are given the symbol 5,. It is convenient to recast Eq. (12-8) in the form 

Miscibility occurs only if AGm 0 Eq. (12-1), Since ASm in Eq. (12-3) is always positive (the In of a fraction is negative), the components of a mixture are assumed to be compatible only if A TAS,. Thus solution depends in this analysis on the existence of a zero or small value of A W i. Note that this theory allows only positive (endothermic) heats of mixing, as in Eq. (12-10). In general, then, miscibility is predicted if the absolute value of the (i5i — S2) difference is zero or small. 

When an ideal solution is fonned from its components, there occurs no enthalpy change = 0) and the entropy change (A mix) is ideal, the latter being 

The requirement that a regular solution has ideal A5n,ix, d hence a random molecular distribution, despite the existence of A-B interactions that lead to a nonideal effectively restricts regular solutions to those systems in which only 

Interaction energy of B-B contacts = Interaction energy of A-B contacts = 

In effect, this takes wab to be equal to the geometric mean of waa nd wbb- The geometric mean rule, based partly on theoretical principles and partly on observation, is expected to hold, however, only in situations where dispersion forces provide the only significant interaction energy. 

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Actually, perfluorocarbons behave in many ways quite differently from hydrocarbons, as already established in the literature. For example, then-behaviour in the context of thermodynamic models differs significantly from all other compounds, including hydrocarbons (Figure 3.12). Moreover, in the context of the regular solution/solubility parameter theory (see below), solutions containing fluorocarbons show deviations not shared by other mixtures of non-polar compounds, which constitutes the main area of applicability of regular solution/ solubility parameter theory (Prausnitz, Lichtenthaler and de Azevedo, 1999). 

Similarities in the solvent-solute solubility parameters allow a more negative Gibbs energy of mixing. If it is assumed that the solution is regular (A S ideal)... 

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). 

Regular solutions, the solubility parameter and Scatchard-Hildebrand theory... 

Solubility parameters are generally tabulated, together with the corresponding liquid molar volumes, only at 25°C. Although solubility parameters are themselves temperature-dependent, the combination of quantities in Eq. 70 is not. Differentiating Eq. 70 with respect to temperature gives — the excess entropy, a quantity which has been assumed to be zero in accord with the definition of a regular solution. Thus only data at 25°C are needed. Solubility parameters may be... 

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. 

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. 

The solubility parameter 5 of a pure solvent defined initially by Hildebrand and Scott based on a thermodynamic model of regular solution theory is given by Equation 4.4 [13] ... 

Still, the strain enthalpy is of particular importance. An elastic continuum model for this size mismatch enthalpy shows that, within the limitations of the model, this enthalpy contribution correlates with the square of the volume difference [41,42], The model furthermore predicts what is often observed experimentally for a given size difference it is easier to put a smaller atom in a larger host than vice versa. Both the excess enthalpy of mixing and the solubility limits are often asymmetric with regard to composition. This elastic contribution to the enthalpy of mixing scales with the two-parameter sub-regular solution model described in Chapter 3 (see eq. 3.74) ... 

The solubility parameter concept was established in the 1930s by the work of Hildebrand and Scatchard. The original concept covers regular solutions, i.e., solutions that do not show an excess entropy effect on mixing. The solubility parameter concept offers the following interesting features ... 

If Z9b(ai) can be equated with P calculated from the entries in Table 2.5, then Z9b(a2) in any other solvent Ab can be estimated from Eq. (2.62). Equation (2.62) is actually a combination of four expressions of the form of Eq. (2.8) (see section 2.2.2), two for water and solvent Ai and two for water and solvent A2, presuming them to be immiscible pairs of liquids. It employs concentrations on the mole fraction scale, and assumes that the systems behave as regular solutions (which they hardly do). This eliminates the use of the solubility parameter 8 of water, which is a troublesome quantity (see Table 2.1). Solvent Ai need not, of course, be 1-octanol for Eq. (2.62) to be employed, and it suggests the general trends encountered if different solvents are used in solvent extraction. 

Thus from solubility parameters, which are specific for the various solutes and solvents, and molar volumes, values for can be estimated, or deviations from regularity can be assessed. These deviations can be estimated quantitatively and, in individual systems, can be ascribed to specific reactions in either of the phases, e.g., hydration, solvation, adduct formation, etc. 

Fig. 4. 23 Application of the regular solution theory for correlation of distribution constants for ZnA2 and CuA2 with solvent properties (solubility parameters) the numbers refer to the solvents listed in Table 4.10. (From Ref. 22.)...

One of the approaches to calculating the solubility of compounds was developed by Hildebrand. In his approach, a regular solution involves no entropy change when a small amount of one of its components is transferred to it from an ideal solution of the same composition when the total volume remains the same. In other words, a regular solution can have a non-ideal enthalpy of formation but must have an ideal entropy of formation. In this theory, a quantity called the Hildebrand parameter is defined as ... 

The solubility of a gas is an integral part for the prediction of the permeation properties. Various models for the prediction of the solubility of gases in elastomeric polymers have been evaluated (57). Only a few models have been found to be suitable for predictive calculations. For this reason, a new model has been developed. This model is based on the entropic free volume activity coefficient model in combination with Hildebrand solubility parameters, which is commonly used for the theory of regular solutions. It has been demonstrated that mostly good results are obtained. An exception... 

The solubility parameter is therefore a measure of the energy density holding the molecules in the liquid state. Note that regular solution theory can only predict positive AH. Thus, with this approach, prediction of solubility involves matching the solute and solvent solubility parameters as closely as possible to minimize AH. As a very rough mle of thumb 61 — 62 must be less than 2 (f/cm3)1 /2 for solubility. 

The factors that cause solutions of iodine to deviate from die behavior of regular solutions are illustrated in Fig. 3. in which values of the left hand member of Eq. (7) are plotted against those of the right for iodine solutions at 25°C ai is tire activity of solidiodine Xy denotes measured solubility Vy is the extrapolated molal volume of liquid iodine. 59 cra3 i is the volume fraction of the solvent, 1.0 62 = 14.1 1 is the solubility parameter of the solvent. Illustrative values of xi and 6] are given in accompanying table. 

A key feature of this model is that no data for mixtures are required to apply the regular-solution equations because the solubility parameters are evaluated from pure-component data. Results based on these equations should be treated as only qualitative. However, mixtures of nonpolar or slightly polar, nonassociating chemicals, can sometimes be modeled adequately (1,3,18). Applications of this model have been limited to hydrocarbons (qv) and a few gases associated with petroleum (qv) and natural gas (see Gas, natural) processing, such as N2, H2, C02, and H2S. Values for 8 and JV can be found in many references (1—3,7). 

A better estimate of all attractive forces surrounding a molecule was found in the use of the solubility parameter [32,33], Hancock et al. [34] has reviewed the use of solubility parameters in pharmaceutical dosage form design. The solubility parameter is used as a measure ofthe internal pressures ofthe solvent and solute in nonideal solutions. Cosolvents that are more polar have larger solubility parameters. The square root ofthe cohesive energy density, that is, the square root of the energy of vaporization per unit volume of substance, is known as the solubility parameter and was developed from Hildebrand s Regular Solution Theory in the Scatchard-Hildebrand... 

The solubility parameter is valid only for regular solutions (where the excess entropy is equal to zero) and mainly for nonpolar classes of substances. Of the numerous suggested improvements that have been made, the one by Hansen is worth mentioning. Here the solubility parameter is the sum of three parts (Barton, 1983) corresponding to a nonpolar or pure dispersive (8,/), polar (8P) and hydrogen bonding (8/,) based interactions ... 

According to the regular solution theory, polymers are as a rule soluble in solvents if their solubility parameters are similar to those of the corresponding solvent. The upper limit for good solubility is defined to he a difference (AS) of 6 units between solubility parameters ... 

Experimental vapor-liquid-equilibrium data for benzene(l)/n-heptane(2) system at 80°C (176°F) are given in Table 1.8. Calculate the vapor compositions in equilibrium with the corresponding liquid compositions, using the Scatchard-Hildebrand regular-solution model for the liquid-phase activity coefficient, and compare the calculated results with the experimentally determined composition. Ignore the nonideality in the vapor phase. Also calculate the solubility parameters for benzene and n-heptane using heat-of-vaporization data. 

Related Calculations. The regular-solution model of Scatchard and Hildebrand gives a fair representation of activity coefficients for many solutions containing nonpolar components. This procedure is suggested for estimating vapor-liquid equilibria if experimental data are not available. The solubility parameters and liquid molar volumes used as characteristic constants may be obtained from Table 1.10. For substances not listed there, the solubility parameters may be calculated from heat of vaporization and liquid molar volume data as shown in step 4. 

Since the Flory interaction parameter, x> was derived by considering only interaction energies between the molecules, it should not contain any entropic contributions and Equation (2D-9) should yield the correct value for the Flory-/ parameter. Unfortunately, x contains not only enthalpic contributions from interaction energies, but also entropic contributions. The solubility parameter includes only interaction energies and by the definition of regular solutions does not include any excess entropy contributions. Blanks and Prausnitz (1964) point out that the Flory / parameter is best calculated from... 

Regular solution theory, the solubility parameter, and the three-dimensional solubility parameters are commonly used in the paints and coatings industry to predict the miscibility of pigments and solvents in polymers. In some applications quantitative predictions have been obtained. Generally, however, the results are only qualitative since entropic effects are not considered, and it is clear that entropic effects are extremely important in polymer solutions. Because of their limited usefulness, a method using solubility parameters is not given in this Handbook. Nevertheless, this approach is still of some use since solubility parameters are reported for a number of groups that are not treated by the more sophisticated models. 

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