## 3 Results and Discussion
### Adsorption Studies  
#### 3.5.1 Equilibrium Study  
Typical Langmuir and Freundlich models for adsorption isotherms are used to illustrate the equilibrium between the dye solution and the adsorbents, as shown in Figure 7.  
For the Langmuir isotherm model, the adsorption occurs at a specific uniform location within the adsorbent and a monolayer is formed on the solid surface, as indicated in the literature [12; 45]. The equilibrium considerations on the isotherm equation are expressed as:  
\[q_{e}=\frac{q_{o}\cdot b\cdot C_{e}}{(1+bC_{e})} \tag{2}\]  
where \(q_{e}\) (mg\(\cdot\)g\({}^{-1}\)) and \(C_{e}\) (mg\(\cdot\)L\({}^{-1}\)) are the equilibrium adsorption capacity of the adsorbate and the equilibrium concentrations of the adsorbent, respectively. \(q_{o}\) is the maximum amount of the adsorbent per unit mass of sorbent to form a complete monolayer on the surface bound at high \(C_{e}\). \(b\) is the Langmuir adsorption equilibrium constant, which is related to the affinity of the binding sites (L-mg\({}^{-1}\)) [46; 47]. The Langmuir isotherms show that the temperature increment is not favorable to the adsorption process (Figure 7), as an intrinsic characteristic of the exothermic adsorption process. Thus, the dye adsorption at ambient temperature (25 \({}^{\circ}\)C) conditions is more favorable to occur. Accordingly, the values of the adsorption of Acid Red 66 are 8.61, 13.92 and 15.53 mg\(\cdot\)g\({}^{-1}\) for Fe, FeC and Fe/FeC samples, respectively (Table 4). Moreover, FeC and Fe/FeC have best results for \(b\) values at room temperature, which further demonstrates an excellent selective adsorption capacities of the two composites. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Adsoorbent**} & \multicolumn{3}{c}{**Temperature**} & \multicolumn{3}{c}{**Langmuir**} & \multicolumn{3}{c}{**Freundlich**} \\ \cline{2-10}  & **(\({}^{\circ}\)C)** & \(\mathbf{q_{o}}\) & \(\mathbf{b}\) & \(\mathbf{K_{\rm F}}\) & \(\mathbf{K_{\rm F}}\) & \(\mathbf{I/n}\) & \(\mathbf{\rm R^{2}}\) & \(\mathbf{SD}\) \\ \hline \multirow{3}{*}{Fe} & 25 & 8.616 & 0.049 & 0.989 & 0.065 & 1.512 & 0.333 & 0.981 & 0.086 \\  & 50 & 6.030 & 0.043 & 0.991 & 0.034 & 0.954 & 0.345 & 0.951 & 0.091 \\  & 70 & 4.080 & 0.022 & 0.995 & 0.054 & 0.361 & 0.435 & 0.981 & 0.061 \\ \hline \multirow{3}{*}{FeC} & 25 & 13.92 & 1.125 & 0.968 & 0.009 & 7.394 & 0.011 & 0.975 & 0.011 \\  & 50 & 10.54 & 0.096 & 0.999 & 0.009 & 2.661 & 0.270 & 0.923 & 0.083 \\  & 70 & 8.007 & 0.072 & 0.994 & 0.022 & 1.732 & 0.294 & 0.973 & 0.051 \\ \hline \multirow{3}{*}{Fe/FeC} & 25 & 15.53 & 0.404 & 0.919 & 0.011 & 8.025 & 0.145 & 0.972 & 0.066 \\  & 50 & 11.46 & 0.114 & 0.993 & 0.014 & 3.174 & 0.263 & 0.914 & 0.091 \\ \cline{1-1}  & 70 & 8.901 & 0.067 & 0.999 & 0.009 & 1.972 & 0.294 & 0.949 & 0.067 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of the Langmuir and Freundlich isotherm models for the adsorption the Acid Red 66 dye over the samples at distinct temperatures. Figure 7: Langmuir and Freundlich plots at different temperatures for Acid Red 66 adsorption on (**a**) Fe, (**b**) Fe/C and (**c**) Fe/FeC. The empirical equation describes heterogeneous systems for Freundlich model isotherm analysis, being expressed as:  
\[q_{e}=K_{F}\text{-C}_{e}^{1/n} \tag{3}\]  
where \(q_{e}\) is the maximum adsorption capacity amount (mg g\({}^{-1}\)); \(C_{e}\) is the equilibrium concentration of the adsorbate (mg\(\cdot\)L\({}^{-1}\)). \(K_{F}\) is the Freundlich adsorption equilibrium constant. \(n\) is the constant adsorption strength, where a favorable range of 1/\(n\) is between 0.01 and 0.435, well described in references [46; 47; 48]. Both the [PERSON] and Freundlich adsorption equilibrium models use a non-linear regression model, according to the Levenberg-Marquardt algorithm [49].