During the years, adsorption has garnered considerable attention being one of the most cost-effective and efficient methods for separating contaminants out of liquid phase. A comprehensive understanding of adsorption mechanisms entails several crucial steps, including adsorbent characterization, batch and column adsorption tests, fitting of predefined kinetic and isotherm models, and meticulous thermodynamic analysis. These combined efforts serve to provide clarity and insights into the intricate workings of adsorption phenomena. However, the vast amount of literature published in the field each year is riddled with ill-considered model selections and incorrect parameter analyses. Therefore, the aim of this paper is to establish guidelines for the proper employment of these numerous kinetic, isotherm, and fixed-bed models in various applications. A thorough review has been undertaken, encompassing more than 45 kinetic models, 70 isotherm models, and 45 fixed bed models available hitherto, with their classification determined based on the adsorption mechanisms expounded within each of them. Moreover, five general approaches for modifying fixed-bed models were provided. The physical meanings, assumptions, and interconversion relationships of the models were discussed in detail, along with the information criterion used to evaluate their validity. In addition to commonly used activation energy and Gibbs energy analysis, the methods for calculating site energy distribution were also summarized.

Multiwalled carbon nanotubes (MWCNTs) were modified by oxidation and acidification with concentrated HNO3 and H2SO4, and the modified multiwalled carbon nanotubes (M-MWCNTs) and raw MWCNTs were characterized by several analytical techniques. Then the demanganization effects of MWCNTs and M-MWCNTs were well investigated and elucidated. The experimental data demonstrated that the adsorption efficiency of Mn(II) could be greatly promoted by M-MWCNTs from about 20% to 75%, and the optimal adsorption time was 6 h and the optimal pH was 6. The results of the kinetic model studies showed that Mn(II) removal by M-MWCNTs followed the pseudo-second-order model. Isothermal studies were conducted and the results demonstrated that the experimental data fitted well with the three models. The reliability of the experimental results was well verified by PSO-BP simulation, and the present conclusion could be used as a condition for further simulation. The research results provide a potential technology for promoting the removal of manganese from wastewater; at the same time, the application of various mathematical models also provides more scientific ideas for the research of the mechanism of adsorption of heavy metals by nanomaterials.

The quantitative description of the equilibrium data by the isotherm models is an indispensable link in adsorption studies. The previous review papers focus on the underlying assumptions, fitting methods, error functions and practical applications of the isotherm models, usually ignoring their curve characteristics, selection criteria and common controversies. The main contents of this review include: (i) effect of the model parameters on the isotherm curves; (ii) determination of the site energy distribution; (iii) selection criteria of the isotherm models; and (iv) elimination of some common controversies. It is of great significance to reveal the curve characteristics for selecting a proper isotherm model. The site energy distribution is conducive to understanding the physicochemical properties of the adsorbent surface. The complete isotherm is recommended to be correlated with the experimental data. The model parameter qmax should be cautiously adopted for comparison of the adsorbent performance. The residual plot can be used to diagnose the fitting quality of the isotherm models further. This review also addresses some common mistakes and controversies and thereby avoids their propagation in future publications.

Nonylphenol (NP) is considered to be an environmentally toxic, endocrine-disrupting chemical that affects humans and ecosystems. Adsorption is one of the most promising approaches for the removal of nonylphenol contamination from water. Herein, in order to design an adsorbent with high adsorption capacity, magnesium silicate with different Mg/Si ratios was successfully synthesized by a sol-gel method at 60 °C. Magnesium silicate with a Mg/Si ratio of 1:6 was found to possess the best adsorption performance, with maximum 4-NP sorption 30.84 mg/g under 25 °C and 0.2 g/L adsorbent dose. The adsorption was negatively affected by increasing adsorbent dose and temperature. The kinetics and isotherm of 4-NP adsorption by Mg/Si were well described by the pseudo-second-order and Sips model, respectively, and behavior was proven to be physisorption-enhanced by a chemical effect. Detailed characterization by XRD, BET, and SEM confirmed that the magnesium silicate possesses an amorphous, mesoporous structure. The study will contribute to the applicability of cheap magnesium silicate for removal of NP contamination in water.

Calcium and magnesium are the most common sources of water hardness. These divalent ions can react with soap anions decreasing the cleaning efficiency and hence, high consumption of detergents occurred as a result. Development of novel low-cost adsorbents for metals removal has attracted a great attention. In this study, bentonite/γ-alumina nanocomposites were used to remove Mg2+ from water. Effects of process parameters including γ-alumina content, initial ion concentration, adsorbent dosage, contact time and pH on adsorption process were investigated. Increasing the amount of alumina in composite from 1 to 3 and 5 wt%, caused a negative effect on the amount of adsorbed magnesium ions per gram of adsorbent; while increasing the initial ion concentration from 60 ppm to 100 ppm resulted in higher uptake per unit mass of the adsorbent from 2.15 mg/g to 2.80 mg/g, respectively. Langmuir, Freundlich and D-K-R isotherm models were used for data analysis, among which the Langmuir model was found to be more successful (R2 = 0.9955), obtaining the maximum adsorption capacity (Qm) of 3.478 mg/g. Moreover, calculation of the adsorption energy (E) from DKR isotherm model depicted the physical nature of the adsorption of Mg2+ onto bentonite/γ-alumina nanocomposite powder.

This study involved the synthesis of magnetic reduced graphene oxide (M-rGO) using a co-precipitation method and examined its resultant adsorption properties for mixtures containing silver ions and silver nanoparticles (AgNPs). The results indicate that M-rGO preferentially adsorbs silver ions in mixtures containing AgNPs, enabling the size characterization of smaller AgNPs (<60nm) at ultra-trace concentration levels to be more attainable. The sorbents after adsorption could be easily recovered through an external magnet. The AgNPs retained in solution were characterized using single-particle ICPMS (SP-ICPMS). The adsorption behavior of silver ions on M-rGO was well fitted with the pseudo-second-order kinetic model and the Freundlich adsorption isotherm model, with the conclusion that the adsorption of silver ions occurred primarily through the chemical bond effect and the heterogeneous surface of the sorbent. Finally, the application of M-rGO with the approach developed herein to actual environmental water samples was successful.

The sorption data of an anionic dye on cellulose fiber are often correlated with a log-linear model to determine the internal accessible volume of the fiber to the anionic dye (V, L/kg) and as such the standard affinity of the anionic dye to the fiber (-Δμ°, J/mol), but without taking into account the influence of ionized carboxyl groups due to cellulose oxidation ([COO(-)]f, mol/kg). In this study, a nonlinear isotherm model was derived by incorporating [COO(-)]f, V and -Δμ° as three model parameters. A set of classical sorption data of C. I. Direct Blue 1 on bleached cotton was correlated with the nonlinear isotherm model. The nonlinear curve fitting analysis showed that the nonlinear isotherm model was in excellent agreement with the sorption data and robust to determine the values of [COO(-)]f, V and -Δμ° for describing the sorption behaviors of anionic dyes on cellulose fibers.