New Sorption Isotherms Derived from a Gamma Distribution of Binding Constants.

New sorption isotherms for heterogeneous sorbents are derived by combining a Gamma distribution of binding constants with a local isotherm defined by a Langmuir or Hill equation. The new "Gamma isotherms" are expressed as Stieltjes transforms of the distribution and involve generalized exponential integrals. The related energy distributions are asymmetric and present a peak corresponding to the mean binding constant. The advantages of the new isotherms are (1) at low pressures or concentrations, with a Langmuir local isotherm, the global "Gamma-Langmuir" isotherm retrieves Henry's law; (2) contrary to the power Freundlich or hypergeometric Freundlich global isotherms, these Gamma isotherms do not need a redefinition of the standard state; (3) with a Hill local isotherm, the global "Gamma-Hill" isotherm allows a separate estimation of the cooperativity and heterogeneity parameters; and (4) the condensation approximation is a good approximation if the local isotherm is Hill and displays a high degree of cooperativity. The Gamma-Langmuir model is applied to three examples from the literature, with rather different Gamma distributions.

Rooting out faulty adsorption models. Comment on "Magnetic Prussian blue nanoshells are controllable (sic) anchored on the surface of molybdenum disulfide nanosheets for efficient separation of radioactive cesium from water".

This correspondence critically examines and rectifies modeling deficiencies identified in a recent article published in this journal. Our analysis covers a range of models and issues, including the Temkin isotherm, the Flory-Huggins isotherm, the pseudo-first-order kinetic model, the pseudo-second-order kinetic model, the intraparticle diffusion model, the Elovich kinetic model, and the computation of thermodynamic parameters. The elucidation and correction of these modeling issues contribute to a more accurate and reliable understanding of the studied phenomena, thereby enhancing the scientific rigor of the subject paper.

Comparing different versions of the Yoon-Nelson model in describing organic micropollutant adsorption within fixed bed adsorbers.

The Yoon-Nelson model serves as a widely used tool for describing the breakthrough behavior of organic micropollutants within fixed bed adsorbers. This study aims to augment its modeling efficacy through two proposed refinements found in the literature: a logarithmic transformation and the incorporation of steric hindrance effects. We systematically evaluated the original Yoon-Nelson model alongside the modified versions, using breakthrough data associated with micropollutant adsorption on solid materials. Three distinct cases were scrutinized: (1) caffeine adsorption on activated carbon; (2) tetracycline adsorption on hierarchical porous carbon; and (3) diclofenac adsorption on organoclay. While all three models demonstrated comparable performance with highly symmetric breakthrough data in case 1, their efficacy diverged significantly when confronted with strongly asymmetric breakthrough data in cases 2 and 3. The original Yoon-Nelson model and the logarithmically modified version fell short in accurately representing these intricate breakthrough curves. In contrast, the version incorporating steric hindrance effects showcased substantial accuracy, outperforming other models in capturing the complexities of asymmetric breakthrough data. This advancement markedly enhances the modeling accuracy and versatility of the Yoon-Nelson model, particularly in assessing the dynamic behavior of organic micropollutants within fixed bed adsorbers.

Comment on "Algal mediated intervention for the retrieval of emerging pollutants from aqueous media".

Addressing inaccuracies in review articles is essential to prevent the proliferation of misinformation. This communication is dedicated to rectifying factual errors identified in a recent review article featured in this journal, with a specific emphasis on addressing errors related to the Temkin, Flory-Huggins, Sips, and Baudu isotherm models. By elucidating and clarifying these inaccuracies, we aim to uphold the integrity of scientific discourse and ensure the accurate dissemination of information within the scholarly community.

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.

Comments on "Breakthrough analysis of continuous fixed-bed adsorption of sevoflurane using activated carbons".

This communication discusses the fixed bed modeling results of the recent paper published by Ang et al. (2020) in this journal.

Extracting surface diffusion coefficients from batch adsorption measurement data: application of the classic Langmuir kinetics model.

Surface diffusion coefficients may be estimated by fitting solutions of a diffusion model to batch kinetic data. For non-linear systems, a numerical solution of the diffusion model's governing equations is generally required. We report here the application of the classic Langmuir kinetics model to extract surface diffusion coefficients from batch kinetic data. The use of the Langmuir kinetics model in lieu of the conventional surface diffusion model allows derivation of an analytical expression. The parameter estimation procedure requires determining the Langmuir rate coefficient from which the pertinent surface diffusion coefficient is calculated. Surface diffusion coefficients within the 10-9 to 10-6 cm2/s range obtained by fitting the Langmuir kinetics model to experimental kinetic data taken from the literature are found to be consistent with the corresponding values obtained from the traditional surface diffusion model. The virtue of this simplified parameter estimation method is that it reduces the computational complexity as the analytical expression involves only an algebraic equation in closed form which is easily evaluated by spreadsheet computation.

Fixed bed sorption: setting the record straight on the Bohart-Adams and Thomas models.

The Bohart-Adams model and the so-called Thomas model are commonly used in the modeling of fixed bed breakthrough curves in environmental sorption and biosorption research. Some authors fit the two equations to the same set of experimental breakthrough data and compare their modeling capabilities, implying that the two expressions are separate and independent models. This article clarifies unambiguously and demonstrates conclusively that the two models are in reality mathematically equivalent to each other and their parameters interchangeable. The salient features of the Bohart-Adams model and the genuine Thomas model are outlined here and their relationship delineated. The main difference between the Bohart-Adams and actual Thomas models lies in the form of the sorption isotherm assumed. The former assumes a rectangular (irreversible) isotherm while the latter assumes a Langmuir (favorable) isotherm. It is shown that, when the sorption isotherm is highly favorable, the actual Thomas model reduces to the Bohart-Adams model.

Removal of anionic reactive dyes from water using anion exchange membranes as adsorbers.

Two commercial anion exchange membranes, strong basic (SB6407) and weak basic (DE81), were evaluated for the removal of anionic reactive dyes, Cibacron blue 3GA (three sulfonic acid groups per dye molecule) and Cibacron red 3BA (four sulfonic acid groups per dye molecule), from water in this study. The adsorption isotherm results show that the Langmuir maximum adsorption capacities of Cibacron blue 3GA (31.5mg/cm(3) for SB6407 and 25.5mg/cm(3) for DE81) were greater than those of Cibacron red 3BA (24.5mg/cm(3) for SB6407 and 18.5mg/cm(3) for DE81). For each reactive dye, the capacity for SB6407 was higher than DE81 based on the same membrane volume. However, consideration of the number of ion exchange sites interacting with a dye molecule indicates that the DE81 results are close to the theoretical values while the SB6407 membrane had some unused binding sites. In addition, Cibacron red 3BA demonstrated faster and stronger binding with both anion exchange membranes than Cibacron blue 3GA. Both dyes could bind with strong basic SB6407 more quickly and stronger. In the batch desorption process, different desorption solutions were tested and the mixtures of salt, acid, or base in methanol solution (e.g. 1N KSCN in 60% methanol or 1N HCl in 60% methanol) achieved better performance. Finally, in the flow process with one piece of anion exchange membrane (initial dye concentration of 0.05g/L), SB6407 was found superior to DE81 in dye recovery and both membranes retained their original uptake capacities over three cycles of adsorption, washing, and desorption.

Optimization of a fermentation medium using neural networks and genetic algorithms.

Artificial neural networks and genetic algorithms are used to model and optimize a fermentation medium for the production of the enzyme hydantoinase by Agrobacterium radiobacter. Experimental data reported in the literature were used to build two neural network models. The concentrations of four medium components served as inputs to the neural network models, and hydantoinase or cell concentration served as a single output of each model. Genetic algorithms were used to optimize the input space of the neural network models to find the optimum settings for maximum enzyme and cell production. Using this procedure, two artificial intelligence techniques have been effectively integrated to create a powerful tool for process modeling and optimization.