## Solubility in Regular Solutions

One rarely encounters ideal solutions in practice, and practically all solutions of pharmaceutical interest are non-ideal in character. For such non-ideal solutions, the activity coefficient (y B) of the solute does not equal one because the range of solute-solute, solvent-solvent, and solute-solvent interactions are significant. Therefore, one must consider the effect of the activity coefficient in order to predict the properties of non-ideal solutions:

In equations (28) and (29), aB is the activity of the dissolved solute and the undissolved solid, which may be evaluated using the hypothetical supercooled liquid as the standard state of unit activity. ln(aB) may be expressed by equation

(17), as was the case for ideal solutions. Therefore:

ln Yb ln Yb

The value of the activity coefficient depends on many factors, and for non-ideal solutions the activity coefficient may be predicted from knowledge of the nature of the solute and the solvent.

For the sake of simplicity, the prediction of activity coefficients in regular solutions, the simplest non-ideal solution, will be discussed. For a regular solution, the energy of mixing and the enthalpy of mixing are not negligible because the intermolecular solute-solute, solvent-solvent, and solute-solvent interactions are different. However, the total volume is still assumed to be unchanged during mixing.

The activity coefficient in a regular solution can be estimated by considering the changes in intermolecular interaction energies that accompany the mixing of solute and solvent. For this purpose, the solution process may be divided into the three steps illustrated in Figure 2. The first step would consist of the removal of a solute molecule from its pure solute phase into the vapor phase, the second step would be the creation of a hole in the solvent for incorporation of the solute molecule, and the third step is the process where the free solute molecule fills the hole created in the solvent (Higuchi, 1949; Hildebrand and Scott, 1950; Martin, 1993).

To begin the analysis, the potential energy of solute-solute, solvent-solvent, and solute-solvent pairs is identified as wBB, wAA, and wAB. In the first step, an energy equal to 2wBB must be absorbed to break the solute-solute interaction between two adjacent solute molecules in the solid. After the solute molecule

Step 1. Free a molecule from the solute

Step 2. Create a hole in solvent ooo ooo ooo ooo * o o ooo

Step 3. Free solute molecule fills the hole in the solvent ooo o o + •

Figure 2. Hypothetical steps in solution process.

is removed to the vapor phase, the hole created in the solute closes, which releases an energy equal to wBB, making the net energy change associated with liberation of a solute molecule equal to wBB. In the second step, energy equal to wAA is absorbed to separate a pair of solvent molecules, and to produce a hole in the solvent which the solute molecule may occupy. Finally, the solute molecule liberated from its solid phase is inserted in the hole in the solvent, forming two solute-solvent interactions and releasing an energy equal to 2wAB. The overall potential energy change, Au, is therefore:

Using this simplified model, Hildebrand and Wood (1933) proposed