Solution, regular

We shall in section 4.2 deal with regular solutions of small-molecule substances. The construction of phase diagrams from the derived equations is demonstrated. The Flory—Huggins mean-field theory derived for mixtures of polymers and small-molecule solvents is dealt with in section 4.3. It turns out that the simple Flory—Huggins theory is inadequate in many cases. The scaling laws for dilute and semi-dilute solutions are briefly presented. The inadequacy of the Flory-Huggins approach has led to the development of the equation-of-state theories which is the fourth topic (section 4.6) Polymer-polymer mixtures are particularly complex and they are dealt with in section 4.7. 

The volume of the regular solution is equal to the sum of the volumes of its components, i.e. there is no change in volume on mixing the components. It is convenient to think about regular solution in terms of a regular lattice with positions that can be occupied by either of the components. The entropy and enthalpy changes on mixing are calculated separately. 

Let us now consider a binary mixture. The mixing entropy is determined by the number of possible ways (P) of arranging the mixture of two low molar mass components, denoted 1 and 2, it being assumed that each molecule occupies only one lattice position, as shown in Fig. 4.1. P is given by  

The enthalpy of mixing (AH ,ix) can be calculated from the interaction energies of the contacting atoms (1-1, 2-2 and 1-2 contacts)  

The free energy of mixing (AG ,ix) for a regular solution is obtained by inserting the entropic and enthalpic components into the equation AGj x = AH ,x - TAS  

Hildebrand has found experimentally that a large number of binary mixtures show a behavior which can be represented quite well by the laws of regular solutions. A regular solution is, by definition, one in which the partial entropies of the various components have ideal forms. In this section, we discuss some of the properties of regular solutions. 

The partial molal entropy of component i is given by The entropy of mixing of the ideal mixture is 

A solution in which the partial entropies are given by Eq. (11-115) is a regular solution. 

Thus the partial molal heat capacity at constant pressure of component i in the regular solution is the same as the molal heat capacity of pure liquid /. Differentiation of Eq. (11-118) with respect to p at constant T and X yields 

Equations (11-121) and (11-124) are statements of two of the many properties of regular solutions that are identical with the properties of ideal solutions. 

This new, non-ideal model is only one step removed from an ideal solution. All restrictions remain the same except that the intermolecular forces are no longer uniform (hence G and H will be non-ideal, but S should remain ideal, and V nearly so). The special conditions we have just described define what is called a regular solution. 

Here the subscript reg dissol n refers to the difference between the property of a regular solution and a mechanical mixture of its components. Thus 

At this point we can write out the defining equations of a truly non-ideal solution (equations (15.3)). This would be one for which all properties in addition to the enthalpy differ from the ideal values. Here again, the subscript non-ideal dissol n refers to the difference between a non-ideal solution and a mechanical mixture of its pure components  

Consider a mixture of two molecules A and B which are both approximately spherical in shape. As perfect spheres they can pack together in a face-centered cubic lattice to form a liquid with a coordination number c of 12 in each pure liquid. The mixture packs in the same way provided the molecular sizes are not too different, more specifically, provided the molecular volumes do not differ by more than a factor of two [4]. If the free volume between molecules in the mixture does not differ from the sum of those in the two pure liquids used to form the mixture, then the volume of mixing is effectively zero, and the interaction energy experienced by a given molecule in the mixture may be calculated by summing the contributions from nearest neighbors. Mixtures with these properties are strictly regular. 

Let us now consider how one can estimate the enthalpy of mixing given the enthalpies associated with A-A, A-B, and B-B interactions at the molecular level. If each molecule has c nearest neighbors, then the number of interactions experienced by type A molecules is c a/2, and the number of B molecules, cn jl, where the factor of two appears in order to avoid counting the interactions twice. If one defines the number of A-A interactions as the number of B-B interactions as bB) and the number of A-B interactions as it follows that 

By adding these equations, one obtains an expression for twice the total number of interactions in the solution. 

Suppose that the enthalpy associated with an A-A interaction is /zaa, that with B-B, /zbb, and that with an A-B interaction, /zab- Then the enthalpy of a molecules in pure A associated with intermolecular interactions is 

The enthalpy of the solution formed from these pure liquids, associated with intermolecular interactions is 

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Hildebrand, J. H., and R. L. Scott "Regular Solutions," Prentice-Hall, Englewood Cliffs, N.J., 1962. 

Hydrocarbon mixtures can be assumed to be regular solutions it is thus possible to estimate the activity coefficient using a relation published by Hildebrandt (1950) ... 

Hildebrand, J.H. and R.L. Scott (1962), Regular solutions. Prentice Hall, Engelwood Cliffs, NJ. 

Equations III-10 and III-11 are, of course, approximations, and the situation has been examined in some detail by Cahn and Hilliard [9], who find that Eq. Ill-11 is also approximated by regular solutions not too near their critical temperature. 

Guggenheim [5] extended his treatment to the case of regular solutions, that is, solutions for which... 

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. 

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 ... 

A key feature of this model is that no data for mixtures are required to apply the regular-solution equations because the solubiHty parameters are evaluated from pure-component data. Results based on these equations should be treated as only quaHtative. However, mixtures of nonpolar or slightly polar, nonassociating chemicals, can sometimes be modeled adequately (1,3,18). AppHcations of this model have been limited to hydrocarbons (qv) and a few gases associated with petroleum (qv) and natural gas (see Gas, natural) processiag, such as N2, H2, CO2, and H2S. Values for 5 and H can be found ia many references (1—3,7). 

This model is appropriate for random mixtures of elements in which tire pairwise bonding energies remain constant. In most solutions it is found that these are dependent on composition, leading to departures from regular solution behaviour, and therefore the above equations must be conhned in use to solute concentrations up to about 10 mole per cent. 

An important element that must be recovered from zinc is cadmium, which is separated by distillation. The alloys of zinc with cadmium are regular solutions with a heat of mixing of 8300 Xcd fzn J gram-atom and the vapour pressures of the elements close to the boiling point of zinc (1180K) are... 

In the first category of solutions ( regular solutions ), it is the enthalpic contribution (the heat of mixing) which dominates the non-ideality, i.e. In such solutions, the characteristic intermolecular potentials between unlike species differ significantly from the average of the interactions between Uke species, i.e. 

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

A theory of regular solutions leading to predictions of solution thermodynamic behavior entirely in terms of pure component properties was developed first by van Laar and later greatly improved by Scatchard [109] and Hildebrand [110,1 11 ]. It is Scatchard-Hildebrand theory that will be briefly outlined here. Its point of departure is the statement that It is next assumed that the volume... 

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... 

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. 

The Scatchard-Hildebrand theory of regular solutions is most attractive because of its simplicity, and it is of special interest here because it has been applied to hydrocarbon mixtures at high pressures (PI 3), leading to the correlation of Chao and Seader (Cl). 

P7.2 In a regular solution, the activity coefficients are given by the equations... 

When started with a smooth image, iterative maximum likelihood algorithms can achieve some level of regularization by early stopping of the iterations before convergence (see e.g. Lanteri et al., 1999). In this case, the regularized solution is not the maximum fikelihood one and it also depends on the initial solution and the number of performed iterations. A better solution is to explicitly account for additional regularization constraints in the penalty criterion. This is explained in the next section. 

The regularized solution is easy to obtain in the case of Gaussian white noise if we choose a smoothness prior measured in the Fourier space. In this case, the MAP penalty writes ... 

We briefly recall here a few basic features of the radial equation for hydrogen-like atoms. Then we discuss the energy dependence of the regular solution of the radial equation near the origin in the case of hydrogen-like as well as polyelectronic atoms. This dependence will turn out to be the most significant aspect of the radial equation for the description of the optimum orbitals in molecules. 

It can be demonstrated (2) that two linearly independent solutions of this equation can be chosen in general (i.e. except for some values of e) in such a way that one of them (the so called regular solution) is continuous at the origin and diverges at infinity, and the other one (the so called irregular solution) diverges at the origin and tends to zero at infinity. 

It should be emphasized that we are not interested here specifically by these particular values of e. On the contrary, what is useful here i.e. for the description of optimum orbitals in molecules is to study the variation of the regular solution when e varies continuously. 

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