The regular solution theory

There already exists a substantial literature devoted to the estimation of various material properties with the help of additive structual increments (Reid et. al, 1987, Van Krevelen, 1990). The regular solution theory in combination with additive structural increments has a wide application for estimating the relative solubilities of organic substances in polymers and the solubility of polymers in various solvents (Barton, 1983) and will be described later in this chapter. When estimating partition coefficient values, one is quickly confronted with this method s application limits, particularly with polar and non-polar structures, for example the partitioning of substances between polyolefins and alcohol (Baner and Piringer, 1991). 

A modern set of methods which can be used to estimate partition coefficients is the group contribution method. These methods were developed to allow chemical engineers to estimate activity coefficients in liquid and polymeric systems. Of the numerous methods developed, UNIFAC, the oldest and most thoroughly tested method, is probably the most universally applicable to a wide variety of substances and sytems despite its known weaknesses (Baner, 1999). The use of UNIFAC and example calculations will be described later in this chapter. 

For a solute 1 in a pure liquid 1 a solubility parameter designated as 8/ can be defined  

Where U. = AHv l - RT is the molar interaction energy for particles of solute 1 and can be approximated by the molar enthalpy of vaporization AHV j expressed at temperature T. is the molar volume of the pure liquid substance 1. The cohesive energy density is expressed by cn which describes the interaction between the particles of the pure liquid 1. In a two component system it is assumed that the interaction 

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  

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The micellization and adsorption properties of industrial sulfonate/ ethoxylated nonionic mixtures have been assessed in solution in contact with kaolinite. The related competitive equilibria were computed with a simple model based on the regular solution theory (RST). Starting from this analysis, the advantage of adding a hydrophilic additive or desorbing agent to reduce the overall adsorption is emphasized. 

The corresponding monomer/micelle equilibria can be dealt with by the regular solution theory (RST), as shown in particular by Rubingh in 1979 (1). The application of this theory to numerous binary surfactant systems (2 - 4) has followed and led to a set of coherent results (5). 

Comparison Theoretical Equilibrium Calculations and Results of Circulation Tests in Porous-Media To make this interpretation more quantitative, the regular solution theory (RST) was applied to sulfonate/desorbent dynamic equilibria reached inside porous media by using the approach described above. In so doing, we assumed that the slugs injected were sufficiently large and that a new equilibrium was reached at the rear of micellar slug in the presence of desorbent. 

Other LAS homolog structural effects on wettability and soil removal were found when the data were analyzed using the cohesive energy ratio, R, the regular solution theories of the... 

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.)...

Equation 1 has proved to be a better predictor of the equilibrium which exists between monomer and micelles for mixed surfactant systems than is the regular solution theory model. It also predicts well the mixture CMC and shows the heat of mixing to be smaller than that predicted by the regular solution theory in agreement with the experiment (13). The purpose of this paper is to further explore... 

It was recently ascertained that the behavior of the adsorbed film of two surfactants in equilibrium with their micelle can be explained by assuming both the surface region and the micelle particle to be mixtures of the surfactants (1 - ) - Further, the application of the regular solution theory to the mixtures was shown to be useful to describe the nonideal behavior of ionic surfactants ( - ) However, the above treatments are incomplete from the thermodynamic viewpoint, because they do not consider the dissociation of surfactants and ignore the presence of solvent (T). In addition, it is impossible to suppose that the regular solution theory is applicable to both the adsorbed film and the micelle of ionic surfactants accompanied by the electrical double layer ( ). 

The regular solution theory may be applied to surface adsorption and micelle formation of mixed nonaethoxylated fatty alcohols with Gaussian distribution In hydrophobic chain length. Such a system can be treated as an Ideal mixture,... 

Other attempts at characterizing the deviation from ideal solubility theory have been made. Anderson et al. [50] showed that solubilities that could not be rationalized from the regular solution theory could be rationalized by assuming the formation of speciLc solute-solvent complexes. Yalkowsky et al. [51,52] showed that the deviation from the ideal solubility equation could be expressed in terms of interfacial tension and surface area. In equation form,... 

The Chao-Seader and the Grayson-Streed methods are very similar in that they both use the same mathematical models for each phase. For the vapor, the Redlich-Kwong equation of state is used. This two-parameter generalized pressure-volume-temperature (P-V-T) expression is very convenient because only the critical constants of the mixture components are required for applications. For the liquid phase, both methods used the regular solution theory of Scatchard and Hildebrand (26) for the activity coefficient plus an empirical relationship for the reference liquid fugacity coefficient. Chao-Seader and Grayson-Streed derived different constants for these two liquid equations, however. 

According to a theory, based on the regular solution theory, a deviation from ideal behaviour can be described by the introduction of the activity coefficients / and f2-... 

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 ... 

The scope of application of the Regular Solution Theory is limited only to estimating solute partitioning in PE/ethanol systems. The Retention Indices system is currently only developed for PE/ethanol and PE/water systems. Their better accuracy is not surprising since they are essentially correlations of this experimental data. 

Because of the lack of quantitativeness of the Regular Solution Theory and large amount of effort and computing power required for the UNIFAC method, yet another way will be taken here. This way leads to values using simple means which can adequately estimate values for the most important practical cases. The method described in this section is based on the potential already recognised in gas chromatography that the partition of a substance between a gas and a polymer liquid can be estimated based on its structural increments and these can be used as characteristic quantities for identification. 

Baner, A.L., Piringer, O. Prediction of solute partition coefficients between polyolefins and alcohols using the regular solution theory and group contribution methods. Ind. Eng. Chem. Res. 1991, 30 1506-1515. 

According to the regular solution theory, the dimerization constant of copper(II) decanoate, ditn, described above can be written as follows (142, 151) ... 

For ideal solutions, A// is zero. For real solutions, however, A//m is finite and its value can be estimated by the regular solution theory of Hildebrand as described in the next section. For most vapors, cmut is negative and it dominates the value of A//j. As a result, AA/ is usually negative and the partition coefficient decreases with increasing temperature. 

Another limitation of the regular solution theory is the assumption diat is negligible. While this assumption may be valid for solutions in which all components (solute and solvent) are of similar sizes, it breaks down when the molar voinmes of the components are significantly different, i.e., in the case of high molecular weight (polymeric) solvents and low molecular weight solutes. For such cases, more rigorous models that include entropic considerations, such as... 

Like van der Waals, Cahn and Hilliard also elaborated their theory. Close to the critical point they confirmed van der Waals 3/2 power in 2.5.29, but with a reinterpreted constant. Their profile also obeyed the tanh law 2.5.31 and they applied the regular solution theory (see sec. 1.2.18c) to obtain quantitative values for the surface tension. In table 2.1 some results are collected for simple liquids for which this theory is supposed to apply. These data give some feeling for the achievements. 

However, when the solute or either of the pure solvents is polar, the regular solution theory could no longer provide quantitative agreement regarding the solubility of a solid solute in a mixed solvent (Acree, 1984 Walas, 1985). 

The activity coefficient has to be estimated for nonideal solutions. There is no general method for predicting activity coefficients of solid solutes in liquid solvents. For nonpolar solutes and solvents, however, a reasonable estimate can frequently be made with the regular solution theory, or the Scatchard-Hildebrand relation. 

For the prediction of solubility of polar solutes in polar solvents, the regular solution theory, Eq. (17), has been modified to take into account the additional interactions between the solvent and the solute. Some of these modified methods are discussed by Prausnitz et al. (16). 

For very dilute solutions, hence at low spreading pressiues, the concentrations of the adsorbates are also low, and the activity coefficients in the adsorbed phase approach unity. For real solutions, a suitable model of the activity coefficients must be used. Several such models have been suggested. The following equation, proposed by Gamba et al. [62] and based on the regular solution theory [63], was applied by Kaczmarski et al. [51] to account for the competitive isotherm of 1-indanol on cellulose tribenzoate ... 

Knowing the thermal stability of clathrates permits the prediction of experimental conditions for polymerization (8). A detailed analysis of this problem requires the examination of all the involved phases, particularly the solid and liquid phases. Equations for phase equilibria were derived from within the framework of the regular solution theory they contain an interaction parameter W, (whose value is always positive or zero for ideal solutions), which measures the tendency of host and guest to segregate in the liquid phase. The melting or decomposition point is very sensitive to the value of W, especially when it exceeds 2 RT, i.e. when a miscibility gap is observed in the liquid phase. For this reason the PHTP-hydrocarbon clathrates melt congruently between 115 and 180 C, whereas the urea-hydrocarbon... 

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