Yesterday, Today, and Tomorrow. Evolution of a Sleeping Beauty: The Freundlich Isotherm

The name of Herbert Freundlich is commonly associated with a power relationship for adsorbed amount of a substance (Cads) against the concentration in solution (Csln), such that Cads = KCslnn; this isotherm (together with the Langmuir isotherm) is considered to be the model of choice for correlating the experimental adsorption data of micropollutants or contaminants of emerging concern (pesticides, pharmaceuticals, and personal care products), but it also concerns the adsorption of gases on solids. However, Freundlich's 1907 paper was a "sleeping beauty", which only started to attract significant citations from the early 2000s onward; moreover, these citations were too often wrong. In this paper, the main steps in the historical developments of Freundlich isotherm are identified, along with a discussion of several theoretical points: (1) derivation of the Freundlich isotherm from an exponential distribution of energies, leading to a more general equation, based on the Gauss hypergeometric function, of which the power Freundlich equation is an approximation; (2) application of this hypergeometric isotherm to the case of competitive adsorption, when the binding energies are perfectly correlated; and (3) new equations for estimating the Freundlich coefficient KF from physicochemical properties such as the sticking surface or probability. From new data treatment of two examples from the literature, the influence of several parameters is highlighted, and the application of linear free-energy relationships (LFER) to the Freundlich parameters for different series of compounds is evoked, along with its limitations. We also suggest some ideas that may be worth exploring in the future, such as extending the range of applications of the Freundlich isotherm by means of its hypergeometric version, extending the competitive adsorption isotherm in the case of partial correlation, and exploring the interest of the sticking surfaces or probabilities instead of KF for LFER analysis.