Analytical solution of the ideal model of chromatography for a bi-Langmuir adsorption isotherm

The (closed-form) analytical solution of the ideal model of chromatography was derived in the case of a bi-Langmuir adsorption isotherm model. This explicit solution provides the sample concentration as a function of the elution time as the unique real and positive root of a quartic polynomial equation. Using this analytical solution, we could calculate the parameters of the bi-Langmuir adsorption model for phenol on a column packed with Poroshell 120 C18 core-shell particles, eluted with a mixture of methanol and water, and operated under high inlet pressure using the retention time method. The method permitted a rapid, economical determination of the best adsorption isotherm parameters for phenol. It required less than 32 mg of sample, 50 mL of eluent, and less than 4 h to complete the measurement of 20 adsorption data points. The relative precisions of the isotherm parameters were 5% for the equilibrium constant and 9% for the saturation capacity of the strong adsorption sites and 24% for the equilibrium constant and 5% for the saturation capacity of the weak adsorption sites. Compared with determinations made with the standard HPLC technology and frontal analysis experiments, the working time, eluent volume consumed, and mass of compound used were reduced by factors of 4, 20, and 400, respectively.