ReliabilityAssessment of Adsorption Parameter Prediction from Graphical Optimization and Statistical Validation of Isotherm Models: A Case-Based Methodological Counterexample Study

Abstract

Many environmental problems involve adsorption, making its description essential. This study examines the validity of graphical optimization through adsorption isotherm modeling combined with statistical validation for determining adsorption parameters. A published experimental dataset was selected as a case-based methodological counterexample and analyzed using this approach. The Langmuir and Freundlich models as well as their combined forms (Langmuir–Freundlich, Sips, Khan, Redlich–Peterson, Toth, and Radke–Prausnitz) were investigated. All combined models provided similar fits to the experimental data, with coefficients of determination close to 0.989 and average relative errors ranging from 24.368% to 28.611%. Langmuir–Freundlich, Sips, and Toth models generated similar curves but predicted different maximum adsorption capacities of 165.683, 164.930, and 164.680 mg/g, respectively. Redlich–Peterson, Radke–Prausnitz, and Khan isotherms also showed comparable fits despite differing parameter estimates. Notably, the Khan model substantially overestimated the maximum adsorption capacity (206.011 mg/g). The predicted adsorption equilibrium constants were 0.068 L/g for Langmuir–Freundlich and Sips models and about 0.05 L/g for other models. These results revealed that several adsorption models can provide statistically validated fits of similar quality while predicting different values for the adsorption parameters. These findings indicate that satisfactory graphical and statistical agreement does not necessarily ensure accurate quantitative parameter estimates for the investigated system. Therefore, graphical optimization, even when combined with statistical validation, should be interpreted cautiously when the predicted parameters are used for quantitative physical interpretation. Independent experimental validation is recommended whenever suitable data are available. This study does not aim to demonstrate the generalizability of the identified limitation or to represent a specific range of adsorbate–adsorbent systems. Rather, it serves as a methodological warning and proof of concept, providing an early indication of a potential issue and a basis for further investigation by the research community.