Regular solution approximation

If the second term in the configurational entropy of mixing, eq. (9.42), is zero, the quasi-chemical model reduces to the regular solution approximation. Here, Aab is given by (eq. (9.21). If in addition yAB =0the ideal solution model results. 

The first term is the ideal entropy of mixing while the second term is the enthalpy of mixing in the regular solution approximation ... 

Among the purposes of this paper is to report the results of calorimetric measurements of the heats of micellar mixing in some nonideal surfactant systems. Here, attention is focused on interactions of alkyl ethoxylate nonionics with alkyl sulfate and alkyl ethoxylate sulfate surfactants. The use of calorimetry as an alternative technique for the determination of the cmc s of mixed surfactant systems is also demonstrated. Besides providing a direct measurement of the effect of the surfactant structure on the heats of micellar mixing, calorimetric results can also be compared with nonideal mixing theory. This allows the appropriateness of the regular solution approximation used in models of mixed micellization to be assessed. 

Consideration of the thermodynamics of nonideal mixing provides a way to determine the appropriate form for the activity coefficients and establish a relationship between the measured enthalpies of mixing and the regular solution approximation. For example, the excess free energy of mixing for a binary mixture can be written as... 

The regular solution approximation is introduced by assuming definition) that the excess entropy of mixing is zero. This requires that the excess free energy equal the excess enthalpy of mixing. For binary mixtures the excess enthalpy of mixing is ordinarily represented by a function of the form... 

This conclusion implies that the excess entropy of mixing is non-zero and that the mixed micelles presumably acquire more internal order than they would by random mixing. An examination of the magnitude of the deviations from the regular solution approximation shows that there must be a large TS contribution to the excess free energy of mixing. 

The finding that the assumptions of the regular solution approximation do not hold for the mixed micellar systems investigated here suggests a re-examination of how the thermodynamics of mixing enter the nonideal mixed micelle model. 

The mixed cmc behavior of these (and many other) mixed surfactant systems can be adequately described by a nonideal mixed micelle model based on the psuedo-phase separation approach and a regular solution approximation with a single net interaction parameter B. However, the heats of micellar mixing measured by calorimetry show that the assumptions of the regular solution approximation do not hold for the systems investigated in this paper. This suggests that in these cases the net interaction parameter in the nonideal mixed micelle model should be interpreted as an excess free energy parameter. 

Ghiorso M. S., Carmichael I. S. E., Rivers M. L., and Sack R. O. (1983). The Gibbs free energy of mixing of natural silicate liquids an expanded regular solution approximation for the calculation of magmatic intensive variables. Contrib. Mineral Petrol, 84 107-145. 

The purpose of this paper will be to develop a generalized treatment extending the earlier mixed micelle model (I4) to nonideal mixed surfactant monolayers in micellar systems. In this work, a thermodynamic model for nonionic surfactant mixtures is developed which can also be applied empirically to mixtures containing ionic surfactants. The form of the model is designed to allow for future generalization to multiple components, other interfaces and the treatment of contact angles. The use of the pseudo-phase separation approach and regular solution approximation are dictated by the requirement that the model be sufficiently tractable to be applied in realistic situations of interest. 

In terms of activity coefficients, the regular-solution approximation can also be expressed as RTIn y% = RTIn yi = iVhx. ) The mixing parameter Cl expresses nonideality... 

The observation that Bib is more exothermic than Bz3 is not surprising when one considers the fact that blends of tetramethyl bisphenol-A polycarbonate with polystyrene are miscible while blends of bisphenol-A polycarbonate with polystyrene are immiscible (43 ). Apparently, substitution of methyl groups on the aromatic rings in the backbones of these materials enhances miscibility with the styrene repeat unit. The observation that Biz is also exothermic is quite surprising, considering the similarity of the trimethyl and dimethyl ether units. Using regular solution approximations, one could... 

In the regular solution approximation, this equality is no longer assumed and the heat of mixing is finite. We define the regular solution parameter as... 

The basic approach described here for ideal solutions can be extended to regular solutions, that is, those where mixing is random but where interaction energies differ significantly for the various species present. For a binary mixture of two species with equal molar volumes it is well known that the regular solution approximation leads to the following expression for the bulk phase chemical potentials ... 

Fig. 3. Calculated and observed nematic-isotropic phase diagrams for binary mixtures of homologous 4,4 -di-n-alkyloxyazoxybenzenes. Solid line is calculated by regular solution approximation. A denotes point calculated assuming ideal solution. denotes experimentally observed point. 3-6 means 4,4 -di-n-propyloxy in mixture with 4,4 -di-n-hexyloxyazoxybenzene. X is mole fraction of ith component.

The evaluation of the interaction parameters is based on the following equations using nonideal or regular solution approximates. 

Rubingh [78], using the regular solution approximation, proposed an interaction parameter fB for the treatment of the cmc of nonideal mixed surfactant solutions ... 

Onabe, K., Unstable regions in III-V quaternary solid solutions composition plane calculated with strictly regular solution approximation. Jpn. J. Appl. Phys., 1982 21 L323-5. 

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