Inversion of loading-implicit adsorption isotherms by means of the Lambert W function

Background Simple isotherm models, such as the Langmuir and Freundlich equations, are frequently used to correlate water contaminant adsorption data. In the conventional data fitting approach, the Langmuir and Freundlich equations are fitted to experimental data by minimizing the difference between the measured and calculated adsorbed phase concentrations. However, some isotherm equations are implicit in adsorbed phase concentration. Such isotherms must be converted to explicit forms to allow the use of standard nonlinear regression methods in data correlation. This paper explores the use of the Lambert W function to invert the loading-implicit isotherms of Elovich, Volmer, and Jossens to solve for adsorbed phase concentration as a function of liquid phase concentration. Results The inverted Elovich, Volmer, and Jossens isotherms were fitted to literature adsorption data of water contaminants using the standard nonlinear regression procedure in Excel. Data fitting was accomplished by making use of a freely available Excel add-in and a highly accurate analytical approximation to evaluate the Lambert W function. The inverted isotherms were tested against the experimental isotherms of cobalt, uranium, and cephalosporin C. In all cases studied, satisfactory fits were obtained, validating the practicability of the proposed data fitting approach. Conclusions The loading-implicit Elovich, Volmer, and Jossens isotherms can be formulated as loading-explicit equations in terms of the Lambert W function, thus allowing the use of conventional least-squares fitting procedures in data correlation. The practical implementation of the Lambert W function in Excel provides a novel method to fit loading-implicit isotherms to experimental data of water contaminants. (c) 2022 Society of Chemical Industry (SCI).