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

Abstract

The name of Herbert Freundlich is commonly associated with a power relationship for adsorbed amount of a substance (C ads ) against the concentration in solution (C sln ), such that C ads = KC sln n ; 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 K F 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 K F for LFER analysis.

A sulfonate ligand-defected Zr-based metal–organic framework for the enhanced selective removal of anionic dyes

“…While the Langmuir isotherm describes monolayer adsorption on a uniform surface, the Freundlich isotherm is associated with the adsorption on a heterogeneous surface with multilayer adsorbate distribution. 29,31 The experimental data was found to have higher determination coefficients ( R 2 ) upon fitting with the Langmuir model (Fig. S6 and S7†).…”

“…Adsorption of adsorbents on heterogeneous surfaces, which is considered to involve multilayer and reversible adsorption on the active sites of the adsorbent, is the foundation of the Freundlich isotherm model. 31 The linear Freundlich equation is expressed by eqn (4).where K F is the Freundlich constant. n is the experimental parameter.…”

New Sorption Isotherms Derived from a Gamma Distribution of Binding Constants

“…The peak energy E 0 = −1.56 kJ/mol can be compared with the mean energy E 0 = −1.03 kJ/mol obtained from the hypergeometric Freundlich isotherm. 2 Example B: Gamma Distribution, a ≈ 5, Low Heterogeneity. In this case, the data of Abd El-Ghaffar et al 15 for the sorption of Ag(I) to chemically modified melamine resin at 298 K are used.…”

“…The Pareto distribution is defined by an exponent β > 0 and a minimum value b min . The choice of b min is arbitrary and corresponds to a redefinition of the standard state. , For b > b min , the probability density declines according to a power law, showing a straight line in log–log coordinates (Figure ).…”

The Jovanović adsorption isotherm in water contaminant research: Unmasking spurious versions and spotlighting the real thing

“…Some other popular models belonging to this category are those of Freundlich (1907), Elovich (Dusart et al, 1991), Temkin (1979), and Dubinin and Radushkevich (Dubinin, 1975). Modern takes on these simple isotherm models are available in the literature (Chu, 2021a;Debord et al, 2022Debord et al, , 2023Puccia and Avena, 2021;Swenson and Stadie, 2019).…”