Non-ideal entropy of mixing

The entropy of mixing of many real solutions will deviate considerably from the ideal entropy of mixing. However, accurate data are available only in a few cases. The simplest model to account for a non-ideal entropy of mixing is the quasi-regular model, where the excess Gibbs energy of mixing is expressed as... 

Deviations from the classical ideal solution in the Temkin s model are thus ascribed to the non-ideal entropy of mixing. 

Thus, deviations from the ideal Langmuir isotherm can be caused both by intermolecular interactions, which result in an enthalpy of mixing, and by area differences between molecules, which produce a non-ideal entropy of mixing [18]. For a simple case where the interactions are of the Frumkin type and the partial molar areas of solvent and surfactant are constant the entropic effect of area differences results in typical features of macromolecular adsorption, e.g., a steep initial increase of adsorption ( high affinity adsorption) and a very slow rise once the surface is approximately half filled [18]. 

Although this theory accounts for a very important phenomenon in the physical chemistry of macromolecules, it fails utterly in those cases where dissolution occurs in spite of a positive E-value. Such cases are reported in the physical chemistry of rubbers and can only be explained by the non-ideal entropy of mixing (see section 6. d). Moreover, the experiments show that the polymer usually is not really insoluble in those cases where there does not exist complete miscibility the equilibrium attained is not an equilibrium between polymer and extremely dilute solution, but between a concentrated and a dilute phase. The composition of the concentrated phase is independent of the degree of polymeru ation, while that of the dilute phase is the smaller the larger the molecular weight. This will be explained on p. 78. 

If a solution possesses an ideal entropy of mixing, the second derivative is always negative and the system can only exhibit an upper consolute temperature. For a non-ideal solution to exhibit a lower consolute temperature, the deviations from ideality must be such that both the sign and the curvature of 5(0 0) is changed (cf, chapter XXIV, 6 chapter XXVI, 7). 

Hildebrand [16] has demonstrated that the solubility behavior of many non-polar solutes in non-polar solvents can be predicted satisfactorily from Eq. (11) for the enthalpy of mixing and Eq. (1) for the ideal entropy of mixing provided that volume fractions instead of mole fractions are used whenever the molar volumes of the components are not equal. 

This is the entropy change involved in mixing the two pure end-member components (FeMiFeM2)Si206 and (MgMiMgM2)Si20g to make a pyroxene solution of some intermediate composition. We will use equations (15.3) for non-ideal entropy and (15.7) for activities of the individual ions on each site, thus... 

Finally it should be noted that the relationship between and is independent of the temperature. Thus all the non-ideal behavior of the mixture resides in the entropy of mixing, the heat of mixing being zero. 

A discrepancy in free enthalpy between the perfect solution and the non-ideal solution, if the reference system is symmetrical, is generally expressed by the excess free enthalpy GE, which consists of the enthalpy term HE and the entropy term -TSE i.e. GE = HE - TSE. Two situations arise accordingly in non-ideal solutions depending on which of the two terms, He and - TSE, is dominant The non-ideal solution is called regular, if its deviation from the perfect solution is caused mostly by the excess enthalpy (heat of mixing) HE ... 

Most real solutions are neither ideal nor regular. As a result a realistic description of their thermodynamic properties must consider the fact that both the excess enthalpy of mixing, and excess entropy, are non- zero. Wilson... 

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