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Adsorption has been traditionally modeled using equilibrium models like the Langmuir and the Freundlich isotherm models, and kinetic models such as pseudo-first and pseudo-second order equations. However, most of these equations are not in an Indepedant, and single-equation form. A novel mathematical framework that combines adsorption kinetics and isotherm equations to an independant, single-equation form is presented herin. The adsorption isotherm equations were first solved using the mass balance equations to derive an independant equation, eliminating the intermediate variables. This derivation was done for the commonly used adsorption isotherms such as Langmuir, Freundlich, Langmuir-Freundlich and Linear isotherms. The simplified equations can be used for predicting solid phase adsorption (qe) and aqueous phase concentration (Ce) in terms of equilibrium constants, and was called as Equilibrium Adsorption Solution Equation (EASE). This EASE equation was further integrated into adsorption kinetic equations to give a solution which can also predict kinetic adsorption. This framework was named as Combined Adsorption Kinetic and Equilibrium (CAKE) equation and was further validated with experimental data. The CAKE equation offers the following advantages: By using this combined equation, both the kinetics and equilibrium concentrations of the batch adsorption systems can be predicted. Further, the intermediate variables are eliminated, and the final equation involving only model constants such as maximum adsorption capacity (qmax), affinity constant (ka), initial concentration (Co), and kinetic rate constants (kt) are required to to calculate pollutant concentration at any given time. The model was validated using an experimental dataset of 2,4-Dichlorophenol adsorption studies on activated carbon. The CAKE and EASE models fitted the experimental datasets very well with an R2 of 0.85, and a normalized absolute percentage error (NAPE) of 7.5%., thus validating the developed equations.
Patil et al. (Sun,) studied this question.