## 3 Results and Discussion
### MB Adsorption Kinetics  
To understand the adsorbents kinetic behavior, Figure 10 shows the experimental data. For all the adsorbents studied, the equilibrium time was longer. Due to the AC precursor (the coconut shell) the pore distribution was characterized by a high presence of micropores and a high surface area [35], and therefore, the equilibrium time was longer.  
The experimental kinetic data were modeled using the pseudo-first-order and pseudo-second-order kinetic models, and the kinetic parameters, together with the R\({}^{2}\), are shown in Table 3. According to the value, the model that presented the highest correlation coefficients was the pseudo-second-order model, and the \(q_{e}\) values were similar to those of the experimental data. The pseudo-second-order indicated that the adsorption rate was controlled by chemisorption, where the oxygenated functional groups, formed during the oxidation process, interacted with the MB molecule by sharing or electron exchange by valence forces [51].  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & \multicolumn{3}{c}{**Pseudo-First-Order**} & \multicolumn{3}{c}{**Pseudo-Second-Order**} \\ \cline{2-9}
**adsorbent** & \(q_{e}\) & \(k_{1}\) (mg\(\cdot\)g\({}^{-1}\)) & **(min\({}^{-1}\))** & **R\({}^{2}\)** & **Error (\%)** & \(q_{e}\) (mg\(\cdot\)g\({}^{-1}\)) & **(g\(\cdot\)g\({}^{-1}\).min\({}^{-1}\))** & **R\({}^{2}\)** & **Error (\%)** \\ \hline AC & 0.952 & 0.011 & 0.995 & 0.968 & 1.263 & 0.007 & 0.998 & 0.863 \\ \hline AC-21 & 0.783 & 0.029 & 0.934 & 1.436 & 0.916 & 0.039 & 0.961 & 1.222 \\ \hline AC-HAp & 0.736 & 0.020 & 0.920 & 1.231 & 0.870 & 0.027 & 0.945 & 1.507 \\ \hline AC-1h & 0.779 & 0.032 & 0.932 & 1.392 & 0.893 & 0.048 & 0.957 & 1.689 \\ \hline AC-1h21 & 0.710 & 0.073 & 0.796 & 1.929 & 0.780 & 0.136 & 0.880 & 1.058 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters of kinetic models.  
The pseudo-second-order kinetics' parameters of the \(k_{2}\) were directly related to the adsorption speed [52]. The \(k_{2}\) increased according to the modifications in the AC, with the highest value of the \(k_{2}\) at 0.37119 g\(\cdot\)mg\({}^{-1}\cdot\)min\({}^{-1}\) for the AC-2h21, which also showed MB's highest adsorption capacity. In comparison with this material, the others showed a lower value of the \(k_{2}\), and a longer time was necessary to reach the equilibrium. Based on the parameters \(q_{e}\) and \(k_{2}\) obtained from the kinetic model, the adsorption rate was calculated with the following equation (Equation (7)):  
\[h=k_{2}q_{e}^{2} \tag{7}\]  
Figure 10: MB adsorption kinetics on different carbon-based materials (conditions: 1 ppm, pH 7, 60 mL volume, and 60 mg of the initial dose of each material). (**a**) Samples without oxidation, (**b**) samples with 1 h oxidation, (**c**) samples with 2 h oxidation.  
Table 4 contains the values of \(h\), and the highest value (0.2161 g\(\cdot\)mg\({}^{-1}\cdot\)min\({}^{-1}\)) was for the AC-2h21. The results depended on the initial concentration, providing a higher adsorption rate, and were confirmed with the values of the \(k_{2}\) for how the global adsorption rate was.