## Thermodynamics of Solubility

The equilibrium solubility of a substance is defined as the concentration of solute in its saturated solution, where the saturated solution exists in a state of equilibrium with pure solid solute. As solutes and solvents can be gaseous, liquid, or solid, there are nine possibilities for solutions, although liquid-gas, liquid-liquid, and liquid-solid are of particular interest for pharmaceutical applications. Among these, the most frequently encountered solubility behavior involves solid solutes dissolved in liquid solvent, so systems of this type will constitute the examples of the following discussions.

For the particular system of a saturated solution, the dissolved solute in the solution and the undissolved solute of the solid phase are in a state of dynamic equilibrium. Under those conditions, the rate of dissolution must equal the rate of precipitation and hence the concentration of the solute in the solution remains constant (as long as the same temperature is maintained).

For two phases in equilibrium, the chemical potential, / of the component in the two phases must be equal:

The chemical potential, also known as the molar free energy, can be represented by:

where |J.° is the chemical potential of the solute molecule in its reference state, and a is the activity of the solute in the solution. Since both the dissolved solute and the undissolved solid must refer back to the same standard state, it follows that the activities of the dissolved solute and that of the undissolved solid must be identical.

The activity of a component in a solution is defined as the product of its activity coefficient, y, and its mole fraction, X:

For the solute B in a saturated solution:

According to equation (8), the solubility of a substance would be proportional to the activity of the undissolved solid, and inversely proportional to its activity coefficient. Although the activity of a substance in its standard state is defined as unity, the activity of the undissolved solid must depend on reference state. A hypothetical, supercooled liquid state of solute at the temperature of interest is commonly taken as the standard state, making the activity coefficient a more complicated term. The activity coefficient will depend on the nature of both the solute and solvent, as well as on the temperature of the solution.