## 3 Results and Discussion
### Adsorption Kinetics  
To design and evaluate the adsorption system, it is necessary to estimate not only the adsorption rate but also gain an insight into the mechanism of the adsorption process. To determine the rate, two surface-reaction kinetic models were used; the pseudo-first-order and the pseudo-second-order, as shown in Equations (9) and (10), respectively:  
\[\mathrm{q_{t}}\mathrm{=}\mathrm{q_{e}(1-e^{-k_{1}t})}, \tag{9}\]  
\[\mathrm{q_{t}}\mathrm{=}\mathrm{(q_{e}^{2}k_{2}t)/(1+q_{e}k_{2}t)}, \tag{10}\]  
where \(\mathrm{q_{t}}\) is the experimental adsorption capacity at time t (min) and \(\mathrm{k_{1}}\) and \(\mathrm{k_{2}}\) are the adsorption rate constant of the pseudo-first-order (min\({}^{-1}\)) and pseudo-second-order (g mg\({}^{-1}\) min\({}^{-1}\)), respectively. The meaning of all other symbols is the same as mentioned above. The kinetic parameters in each model, and its validity in the form of the R\({}^{2}\) and ERR values, are summarized in Table 4. From the R\({}^{2}\) value, which was close to unity in both cases, both kinetic models matched the experimental data, although the pseudo-second-order model had slightly higher R\({}^{2}\) values and the calculated equilibrium adsorption capacity (q\({}_{e,\mathrm{cal}}\)) at different levels of initial dye concentration was closer to the experimental one, resulting in a minimal ERR. Thus, the pseudo-second-order model was more suitable to explain the kinetics of the dye adsorption on the MCC with functional groups on its surface and a gap gallery of cations. To gain deeper insight into its mechanism, it involves the four sequential stages of the bulk diffusion, film or external diffusion, internal or intra-particle diffusion, and adsorption. One or more of these stages might be the rate-controlling step in the adsorption system. Referring to the bulk diffusion stage, even though the dye solution was not stirred during the adsorption process, the bulk diffusion was likely to be negligible due to the high water solubility of the dye (>140 g L\({}^{-1}\)). For the film or external diffusion stage, differences in the concentration of dye between the bulk solution and the surface of the liquid film were driving forces for dye migration. The dye continued to migrate to the interior part of the pore. Finally, dyes were adsorbed on the active sites of the MCC. To interpret the experimental data from the viewpoint of this mechanism, the Weber and [PERSON] equation [35; 40; 41], as expressed in Equation (11), was then used:  
\[\mathrm{q_{t}}\mathrm{=}\mathrm{k_{i}t^{0.5}+C}, \tag{11}\]  
where \(\mathrm{k_{i}}\) is the rate constant at stage i (mg g\({}^{-1}\) min\({}^{0.5}\)) and C is the film thickness. All other symbols are the same as mentioned above. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Kinetic Model**} & \multirow{2}{*}{**Parameters**} & \multicolumn{4}{c}{**Concentration of RR120 Solution (mg L\({}^{-1}\))**} \\ \cline{3-6}  & & **10** & **30** & **50** & **100** \\ \hline \multirow{4}{*}{Pseudo-first-order} & \(\mathrm{k_{1}}\) (min\({}^{-1}\)) & 0.0536 & 0.046 & 0.041 & 0.025 \\  & \(\mathrm{q_{e,cal}}\) (mg g\({}^{-1}\)) & 1.59 & 5.09 & 9.36 & 15.71 \\  & \(\mathrm{q_{e,exp}}\) (mg g\({}^{-1}\)) & 1.79 & 5.52 & 9.05 & 16.32 \\  & \(\mathrm{R^{2}}\) & 0.983 & 0.988 & 0.995 & 0.994 \\  & ERR (\%) & 11.73 & 7.79 & 3.47 & 3.79 \\ \hline \multirow{4}{*}{Pseudo-second-order} & \(\mathrm{k_{2}}\) (g mg\({}^{-1}\) min\({}^{-1}\)) & 0.116 & 0.022 & 0.012 & 0.003 \\  & \(\mathrm{q_{e,cal}}\) (mg g\({}^{-1}\)) & 1.83 & 5.75 & 9.28 & 16.69 \\ \cline{1-1}  & \(\mathrm{q_{e,exp}}\) (mg g\({}^{-1}\)) & 1.79 & 5.52 & 9.05 & 16.32 \\ \cline{1-1}  & \(\mathrm{R^{2}}\) & 0.999 & 0.998 & 0.999 & 0.997 \\ \cline{1-1}  & ERR (\%) & 1.78 & 4.18 & 2.54 & 2.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Kinetic parameter for the adsorption of RR120 on the MCC. The plot of \(\rm q_{t}\) as a function of \(\rm t^{0.5}\) (Figure 7) gave three straight lines with three different slopes.