where SA and SB are the solubility parameters of the solvent and solute, respectively. In the case of a mixed solvent system, the total solubility parameter of the solvent mixture is given by:
where S5 and S2 refer to the respective solvent parameters of pure solvents 1 and 2, and and 02 are the respective volume fractions in the solvent mixture.
Introducing equation (35) into equation (30) yields the Hildebrand solubility equation describing regular solution behavior:
According to equation (37), if the difference between SA and SB is very small, then the second term approaches zero. The implication of this is that a regular solution would behave in an ideal manner when the solute and solvent have similar chemical properties. It may be seen that the Hildebrand solubility equation enables the prediction of solubility in regular solutions, as long as one has knowledge of the solubility parameters of both components in the solution.
Following the introduction of the Hildebrand model, the topic of solubility parameters has been extensively discussed (Hildebrand and Scott, 1962; Hildebrand et al., 1970; Kumar and Prausnitz, 1975; Barton, 1983), and values of S can be found in these reference works. As a general rule, compounds having stronger London forces will be characterized by larger solubility parameters values.
Hildebrand and Scott (1950) proposed that the solubility parameters of similar molecules could be calculated using the enthalpy of vaporization (A H) and the molar volume of the liquid component (V) at the temperature of interest:
Predictions of the solubility of non-polar solutes in non-polar solvents have been successfully achieved using the Hildebrand solubility equation (Davis et al., 1972). These solutions may be classified as regular solutions since the primary intermolecular interactions are London dispersion forces. However, the equation does not provide a good prediction of solubility for solutions involving polar components. When dipole-dipole, dipole-induced-dipole, charge transfer, and/or hydrogen-bonding interactions exist in the solution, w^ = «JwAAwBB, and with the presence of hydrogen bonding the entropy of mixing is no longer ideal. In addition, AVmix will not equal zero if the dimensions of the solute and solvent molecules are very different.
Modifications to the Hildebrand solubility parameter model have been advanced in attempts to achieve better degrees of solubility prediction (Taft et al., 1969; Rohrschneider, 1973). Among these, the three-dimensional solubility parameter introduced by Hansen and Beerbower (1971) showed the most practical application. These workers calculated the total solubility parameter (Stotal) using three partial parameters, SD, SP, and SH:
where the parameters SD, SP, and SH account for dispersion, polar, and hydrogen-bonding interactions, respectively. Some of the values deduced for SD, SP, SH, and Stotal are listed in Table 2. Another modification of Hildebrand solubility parameter considered the effects of polar interaction and hydrogen bonding, and was found to yield good solubility predictions in many cases (Kumar and Prausnitz, 1975). However, the modified Hildebrand solubility equation can only be used empirically in predicting solubility in polar solvents, since the original assumptions associated with regular solutions do not apply in polar solvents (Grant and Higuchi, 1990).
In solvent systems where polar interactions exert a major role, the molecular and group-surface-area (MGSA) approach provides a better quality solubility prediction (Yalkowsky et al., 1972, 1976; Amidon et al., 1974, 1975). Instead of
Solubility parameter (cal/cm3)1/2
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