Hildebrand and Scatchard

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 ... 

Equations of this form were first developed by Hildebrand and Scatchard (see ref. 14). 

Equations (25.7) and (25.8) have been generalized by Hildebrand and Scatchard to include the case where the molecules are of different volumes. Their semi-empirical derivation leads to... 

Sis a parameter of intermolecular interaction of an individual liquid. The aim of many studies was to find relationship between energy of mixing of liquids and their 5 The first attempt was made by Hildebrand and Scatchard who proposed the following equation ... 

Hildebrand and Scatchard proposed a relationship between the internal energy of mixing and the solubility parameters of a solvent and a solute ... 

Because of the importance of distillation processes, first it was the objective to develop models only for the prediction of VLE. The first predictive model with a wide range of applicability was developed by Hildebrand and Scatchard [48]. The so-called regular solution theory is based on considerations of van Laar, who was a student of van der Waals and used the van der Waals equation of state to derive an expression for the excess Gibbs energy [49]. Since the two parameters a and b of the van der Waals equation of state can be obtained from critical data, it should be possible to calculate the required activity coefficients using critical data. However, the results were strongly dependent on the mixing rules applied. 

Hildebrand and Scatchard [48] showed that better results are obtained, if instead of the van der Waals constants a and b molar volumes v, and so-called solubility parameters 8i are used instead. For binary systems the following relations are obtained for the activity coefficients ... 

The energy of interaction between molecules was calculated by Hildebrand and Scatchard [22—25] according to van laiar s theory [26] for a Lennard-Jones potential. Let denote the interaction energy between a pair of i molecules then Et, the cohesive energy (potential energy) of one mole of substance becomes ... 

For solubility of solids in liquid, Hildebrand and Scatchard independentiy derived another equation, now called the Hildebrand-Scatchard equation ... 

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

Margules, and Scatchard-Hildebrand) are particular mathematical solutions to Eq. (48) these models do not satisfy Eqs. (45) and (46), except in the limiting case where the right-hand sides of these equations vanish. This limiting case provides a good approximation for mixtures at low pressures but introduces serious error for mixtures at high pressures, especially near critical conditions. 

Assuming that it is only van der Waals forces which are acting in the solute/ solvent system, and that the heat of mixing is responsible for all deviations from ideal behaviour, as well as the fact that the solute/solvent interaction energy is the geometric mean of solute/solute and solvent/solvent interactions, Hildebrand [228, 229] and Scatchard [230] were able to develop the following expression for the activity coefficient f of the nonelectrolyte solute i dissolved in a solvent s (mole fraction basis), referred to a standard state of pure liquid solute (not infinite dilution) ... 

The Hildebrand-Scatchard equation (see Hildebrand and Scott, 1951), which is derived on semiempirical grounds, is... 

TABLE 1.9 Solubility of Napthalene in Various Solvents by UNIFAC and Scatchard-Hildebrand Theory... 

As early as 1906, van Laar (27) modeled the heat of mixing of a binary solutions using van der Waals equation of state. The views of van Laar were subsequently expanded by van Laar and Lorenz (28), Hildebrand (29,30), and Scatchard (31). The latter authors considered a mixtures of two fluids each with isotropic molecules with interaction energies, cn and 022- The interaction energy between the two fluid molecules is C12. Scatchard (31) obtained the form... 

If the interactions are confined to van der Waals ones and the solution conforms to the restrictions of regular solution theory (Hildebrand and Scott (1950)) then the well known Scatchard-Hildebrand solubility parameter expression can be applied ... 

Hildebrand and Scott and Scatchard proposed that the enthalpy of mixing is... 

It will be noted that for all mixtures of Class III, IV, and V liquids, with the exception of Class III with Class IV, no hydrogen bonds are involved. Hildebrand (15) and Scatchard (30) show that in such cases deviations from Raoult s law are a function of the square of the difference in square roots of the internal pressures of the constituents, the importance of this effect being minimized when hydrogen bonding is extensive. Large deviations for such solutions, in other words, result from large differences in internal pressures. The internal pressure, in turn, can be estimated from the heat of vaporization and molar volumes ... 

Hildebrand and Scott [12] first correlated the enthalpy of mixing with the cohesive properties of the permeant, and for a binary mixture of a polymer and a solvent, the Hildebrand-Scatchard equation is expressed as ... 

To evaluate Hildebrand [1] and Scatchard [22] u.sed the approximation of geometric avi rage ... 

Scatchard Hlldebrand regular-solution activity coefficients. Hildebrand (1929) defined a regular solution as the mixture in which components mix with no excess entropy provided there is no volume change on mixing. Scatchard in an independent work arrived at the same conclusion. The definition of regular solutions (Hildebrand and Scott, 1950) is in line with van Laar s assumption that the excess entropy and the excess volume of mixing are negligible. Scatchard and Hildebrand used an approach different from van Laar s to calculate G. They defined parameter C as... 

The concept of solubility parameters to characterize interactions in liquids was introduced by Scatchard [83] and developed further by Hildebrand and Scott [3]. With liquids, the solubility parameter, Sp, was defined as the square root of the cohesive energy density (CED) ... 

The heat of solution for polymer solutions should be similar to that for low molecular weight liquids. Huggins and Flory used a formulation of the heat of solution similar to the regular solution. Hildebrand and Scott [5], thinking in terms of thevanLaar-Scatchard-Hildebrand regular solutions, simply wrote this as in Eq. 3.16... 

One model that has found applicability in the lipids area is the regular solution theory developed by Hildebrand and Scott [9] and Scatchard [10]. Incorporating a partial molar entropy of mixing term [11-14] into the regular solution theory yields the following expression for the activity of a component in a liquid mixture ... 

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... 

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