Derivation expansion of the general solubility equation (GSE)

The general solubility equation (GSE) is widely used in pharmaceutical and environmental sciences to estimate the aqueous solubility of molecules that are nonionizable over the pH range of 2-13 from their melting point and octanol-water partition coefficient. However, the empirical and thermodynamic basis underlying its deceptively simple form is rarely presented in a unified manner. This paper expands the derivation of the GSE by explicitly separating solubility into a crystal term and a mixing term and tracing how successive approximations lead to the final equation. Starting with the Clausius-Clapeyron expression for the crystal-liquid fugacity ratio and invoking the van't Hoff approximation together with Walden's rule for the entropy of melting, a simple expression is obtained for the ideal solubility of an organic crystal. The mixing contribution is then developed within Scatchard-Hildebrand regular solution theory using solubility parameters and cohesive energy densities, with octanol chosen as the reference solvent. Treating nonionizable solutes in octanol as regular solutions and redefining complete miscibility as a solute mole fraction of 0.5, rather than 1.0, lead to a limiting octanol solubility of 3.18 mol/L (log S = 0.5) and directly yields the revised GSE: [Formula: see text] This paper also compares the revised and original forms of the GSE, showing that replacing the implicit miscibility criterion of X = 1.0 and the log S intercept of 0.8 with X = 0.5 and an intercept of 0.5 is both thermodynamically more consistent and better aligned with Hildebrand's treatment of ideal solutions. Regression analyses of multiple large datasets (167-1450 compounds) demonstrate that the fitted coefficients for the melting point term and log K_ow remain close to -0.01 and -1.0, respectively, while the intercept converges toward 0.5, thereby validating the revised GSE and delineating its domain of applicability.