Thermodynamic Estimate of the Number of Solvent Molecules Displaced by a Solute Molecule for Enthalpy-Driven Adsorption: Phenobarbital and Activated Carbons as the Model System

A Modified Crisp Equation, describing the differential Gibbs free energy of the adsorption process, is being proposed, which considers multiple sites available on the surface for adsorption and their relative fractions. The differential Gibbs free energy can be calculated by the van't Hoff Equation, which depends on the affinity constant in the Langmuir-like equation. To consider the number of solvent molecules displaced by a solute molecule in the adsorption process, a new derivative of the Langmuir-like equation is being proposed as well. By comparing the differential Gibbs free energies obtained from the 2 thermodynamic relationships, it can be determined that a phenobarbital molecule displaces 5 water molecules on the activated carbon surface for site-specific adsorption from solution. For the series of experimental conditions studied, including 4 activated carbons, pH effects, temperature effects, and solvent effects, the corrected differential Gibbs free energies using n₁ = 5 for site-specific adsorption are quite consistent between the 2 thermodynamic relationships. The difference between the estimates of the differential Gibbs free energies by the Modified Crisp Equation and the van't Hoff Equation provides a new experimental method to calculate the number of solvent molecules displaced by an adsorbing solute molecule.