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The kinetic models were widely used to analyze the dynamic adsorption behaviors in a batch system and reveal the mass-transfer mechanisms. The previous review papers were mainly confined to the description of the kinetic models, assessment of the fitting quality, determination of the model parameters and practical application in the field of water and wastewater treatment. However, the curve characteristics of the kinetic models and their mathematical relations were rarely mentioned in the literature. How to select and determine the optimum model remained to be further discussed. Thus, in addition to improving previous work, the main objectives of this review were: (i) to identify the curve characteristics of the kinetic models by control variates; (ii) to reveal their mathematical relations by variable substitution; (iii) to determine the optimum model by error functions and residual plot; and (iv) to correct some common mistakes in the literature. The pseudo-first-order (PFO) and pseudo-second-order (PSO) equations were two special cases of mixed 1,2-order equation (MOE). The PFO and Furusawa–Smith equations were mathematically equivalent. This review is expected to help readers better understand and use the adsorption kinetic models and provide potential ideas for the development of new kinetic models.

In-depth Insights into Mathematical Characteristics, Selection Criteria and Common Mistakes of Adsorption Kinetic Models: A Critical Review