Hildebrand-Scatchard solution theory

When X values are determined for a given polymer or other nonvolatile component of a polymer system, and for a series of vapors for which solubility parameter values are known, the IGC method provides a unique approach to determining the solubility parameter, dr, for the polymer phase. The method is based on the principle that the Flory-Hu ins interaction parameter, x> can be related to dr by combining the Hildebrand-Scatchard solution theory with the Flory-Huggins theory [21] ... 

Equation 8.3 is based on the Scatchard-Hildebrand Regular solution theory that considers interaction due only to dispersion forces. This was extended to include hydrogen bonding and polar forces through a 3D solubility parameter by Hansen. In this form. Equation 8.3 becomes... 

A better estimate of all attractive forces surrounding a molecule was found in the use of the solubility parameter [32,33], Hancock et al. [34] has reviewed the use of solubility parameters in pharmaceutical dosage form design. The solubility parameter is used as a measure ofthe internal pressures ofthe solvent and solute in nonideal solutions. Cosolvents that are more polar have larger solubility parameters. The square root ofthe cohesive energy density, that is, the square root of the energy of vaporization per unit volume of substance, is known as the solubility parameter and was developed from Hildebrand s Regular Solution Theory in the Scatchard-Hildebrand... 

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. 

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. 

Option 2 - the regular solution theory by Scatchard-Hildebrand works best for... 

This is the case for the C6H6 - CC14 system, but not for the C6H6 - CS2 system. Therefore, regular solution theory is not applicable to the C6H6-CS2 system. To test the Hildebrand-Scatchard model we use... 

Generally, the activity coefficients are < 1 when polar interactions are important, with a resulting increase in solubility of compounds compared with the ideal solubility. The opposite is often true in nonpolar systems where dispersion forces are important, with the activity coefficients being > 1. A variety of methods are used to calculate activity coefficients of solid solutes in solution. A frequently used method is that of Scatchard-Hildebrand, which is also known as regular solution theory (Prausnitz et al. 1999). 

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 ... 

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. 

For nonpolar systems the activity coefficient can be estimated using the Hildebrand-Scatchard theory of regular solutions. To calculate the activity coefficient of a dissolved solute using regular solution theory, solubility parameters must be available for the components. For ntany materials these paraiiteters can be calculate and/or are available in standard engineering references. 

The solubility parameter defined by eqn (5.39) is not identical to that conventionally measured by swelling, solubility or surface tension data, since in the first case the solute is at infinite dilution in the polymer, while in the other cases its concentration is high. Because it is possible that at high dilutions the behaviour as a regular solution, inherent to the development of the Hildebrand-Scatchard theory, is more rigorously obeyed, 8" may be more meaningful than 8j. 

Note A more complete table of this type, showing more details is in Walas [ 1, Chapter 4]. Most of the coefficients in these tables, A B, etc. are data-fitdng values, obtained from experimental VLE measurements. Some have semi-theoretical bases. The constants in the Scatchard-Hildebrand equation are based on the Regular Solution theory, and are calculable from pure species properties, without any data for tha mixture. Many authors replace all the symbols for the coefficients with a universal set, A j and Aba, etc. Here the originally used symbols are shown. 

They calculated y,-, by a slightly modified version of Equation 9.24, based on Scatchard and Hildebrand s regular solution theory, and found (< i)pure liquid t by a Pitzer-type equation... 

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 ... 

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... 

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). 

As early as 1916 Hildebrand pointed out that the order of solubility of a given solute in a series of solvents is determined by the internal pressures of the solvents. Later Scatchard (1931) introduced the concept of "cohesive energy density" into Hildebrand s theories, identifying this quantity with the cohesive energy per unit volume. Finally Hildebrand (1936) gave a comprehensive treatment of this concept and proposed the square root of the cohesive energy density as a parameter identifying the behaviour of specific solvents. In 1949 he proposed the term solubility parameter and the symbol S. 

The groups contribution methods can also be used to calculate solubility in binary (solute-solvent) systems. A comparison of solubilities calculated employing the UNIFAC method with experimental values and values obtained from the Scatchard-Hildebrand theory is given in Table 1.9. 

Finding an appropriate mixed solvent system should not be done on a strictly trial and error basis. It should be examined systematically based on the binary solubility behavior of the solute in solvents of interest. It is important to remember that the mixed solvent system with the solute present must be miscible at the conditions of interest. The observed maximum in the solubility of solutes in mixtures is predicted by Scatchard-Hildebrand theory. Looking at Eq. (1.50) we see that when the solubility parameter of the solvent is the same as that of the subcooled liquid solute, the activity coefficient will be 1. This is the minimum value of the activity coefficient possible employing this relation. When the activity coefficient is equal to 1, the solubility of the solute is at a maximum. This then tells us that by picking two solvents with solubility parameters that are greater than and less than the solubility parameter of the solute, we can prepare a solvent mixture in which the solubility will be a maximum. As an example, let us look at the solute anthracene. Its solubility parameter is 9.9 (cal/cm ). Looking at Table 1.8, which lists solubility parameters for a number of common solvents, we see that ethanol and toluene have solubility parameters that bracket the value of anthracene. If we define a mean solubility parameter by the relation... 

LIQUID-LIQUID CURVE AND CRITICAL SOLUTION POINT. In the now classical theory of regular solutions developed by Scatchard (10) and Hildebrand (5) with the nonideal entropy correction given by Flory and Huggins (5), the activities of the components of a binary system are given by... 

---