## 3 Results and Discussion
### Adsorption Isotherms  
For a better understanding of the adsorption process, we carried out equilibrium adsorption studies. The experimental data obtained at equilibrium were correlated with the equations corresponding to the adsorption models--Freundlich (Equation (7)) [65] and Langmuir (Equation (8)) [66]--in order to establish the optimal model for the adsorption of the investigated dyes from the aqueous solutions. \[q_{e}=K_{F}C_{e}{}^{1/n} \tag{7}\]  
\[q_{e}=\frac{q_{m}K_{L}C_{e}}{1+K_{L}C_{e}} \tag{8}\]  
where \(q_{e}\) is the adsorption capacity at equilibrium (mg/g), \(K_{F}\) is the Freundlich constant (mg/g(mg/L)\({}^{-1/n}\)), \(q_{m}\) is the maximum adsorption capacity of the adsorbent (mg/g), \(K_{L}\) is the Langmuir affinity constant (L/mg), \(C_{e}\) is the dye concentration at equilibrium (mg/L), and \(n\) is a constant (dimensionless). The analysis of the experimental data and the determination of the parameters from the equations that describe the theoretical models were performed by non-linear regression  
Figure 13: Pseudo second-order kinetic model fitting for the adsorption of AY 42 (**a**) and AR 213 (**b**) onto PM4. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & & \multicolumn{2}{c}{**Pseudo First-Order Kinetic**} & \multicolumn{2}{c}{**Pseudo Second-Order Kinetic**} \\ \cline{4-7}
**Dye** & **Temp. ** & \(q_{e,exp}\) & \multicolumn{2}{c}{**Model**} & \multicolumn{2}{c}{**Model**} \\ \cline{4-7}  & (°C) & **(mg/g)** & & \(q_{e,calc}\) & \(k_{1}\)\(10^{3}\) & \(q_{e,calc}\) & \(k_{2}\)\(10^{4}\) \\  & & & **(mg/g)** & **(min\({}^{-1}\))** &  
analysis. The statistical criteria for choosing the best model that fits with the experimental data were the chi-square analysis and the squared multiple regression coefficient (R\({}^{2}\)). The dependence \(q_{e}=f(C_{e})\) for the adsorption of anionic dyes on PM4 is presented in Figure 14.  
The analysis of the statistical data performed on the results obtained indicates that the model that best fits the experimental data is the model described by the Langmuir isotherm. According to this model, the adsorption on PM4's surface is localized, takes place on well-defined specific sites, and is monomolecular with a finite saturation limit. The parameters calculated for the isotherms model are presented in Table 9.  
From the Langmuir isotherm model, the maximum adsorption capacities of PM4 were determined to be 62.36 mg/g for AY 42 and 77.99 mg/g for AR 213. These results attest to the high adsorption capacity of the synthesized PM, an important property for the possible application of these sorbents on a large scale. The values of the maximum adsorption capacity reported in the literature for anionic dyes' adsorption on magnetic nanocomposites are presented in Table 10. The results highlight the better performance obtained in this study. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Adsoorbent.** & **Dye** & \(q_{m}\) **(mg/g)** & **Reference** \\ \hline NiFe\({}_{2}\)O\({}_{4}\) magnetic nanoparticles & Eriochrome black-T & 47 & [67] \\ Fe@graphite core–shell nanocomposite & Acid Red 88 & 41.8 & [68] \\ Fe\({}_{3}\)O\({}_{4}\) supported chitosan-graphene oxide composite & Alizarin yellow R & 14.82 & [69] \\ Magnetic Fe\({}_{3}\)O\({}_{4}\)@graphene composite & Congo Red & 33.66 & [70] \\ Activated carbon coated by zinc oxide nanoparticles & Acid Orange 7 & 66.22 & [71] \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Comparison of the maximum adsorption capacity of anionic dyes onto magnetic nanocomposites. Figure 14: Correlation of experimental data with adsorption isotherms for dye adsorption: AY 42 (a) and AR 213 (b).