How to avoid mistakes in treating adsorption isotherm data (liquid and solid phases): Some comments about correctly using Radke-Prausnitz nonlinear model and Langmuir equilibrium constant

Two flaws in concepts were identified and discussed in the paper ("Removal of Pb(II) from contaminated waters using cellulose sulfate/chitosan aerogel: Equilibrium, kinetics, and thermodynamic studies". J. Environ. Manag. 286, 112167; https://doi.org/10.1016/j.jenvman.2021.112167). In the literature, the Radke-Prausnitz model is expressed in different forms, but some of them are incorrect. The first flaw is related to the nonlinear form of the Radke-Prausnitz model. The nonlinear form of this three parameters model is expressed correctly as [Formula: see text] . The units of two parameters are A_RP (L/kg) and B_RP [(mol/kg)/(mol/L)^β] by considering q_e (mol/kg) and C_e (mol/L). The limitation for its exponent is 0≤ β ≤ 1. This model is developed by two authors (Radke and Prausnitz). The correct paper (DOI: 10.1021/i160044a003) cited as reference of this model is "Radke, C.J., Prausnitz, J.M., 1972. Adsorption of organic solutes from dilute aqueous solution of activated carbon. Ind. Eng. Chem. 11, 445-451". The second is the misconception about the unit of the Langmuir constant (K_L; L/mg). The correct unit of K_L is litre per milligram of adsorbate (i.e., Pb ions), not litre per milligram of adsorbent (the cellulose sulfate/chitosan aerogel material as reported by Najaflou and co-workers. They proposed a new equation [K_L (L/mg) × m/V (mg/L)] to convert the Langmuir constant and then applied it to calculate the thermodynamic parameters of the adsorption process. The m/V is a solid/liquid ratio (g/L or kg/L). However, this conversion and application are mistakes that were thoroughly discussed in this paper. The correction is K_Eq^o=1γ_Adsorbate×K_LLmol×C^omolL, with C° (1 mol/L by definition) being the standard state of solute and γ_Adsorbate (dimensionless) being the activity coefficient of adsorbate in solution. To avoid unexpected mistakes, the present authors suggest that researchers should have a correct citation (citing the original reference instead of using secondary references) and check the consistency of units (i.e., the constants of adsorption models) carefully.