## 3 Results
### Adsorption Studies
#### 3.2.3 Effect of Adsorption Time and Adsorption Kinetics  
A kinetic study of the removal of studied analytes is crucial to understand and describe the kinetic mechanism of Fe\({}_{3}\)O\({}_{4}\)@MSN-QPDMAEMA. Each dye (14 mL) was placed in a 15 mL centrifuge tube of fixed initial concentrations, followed by the addition of 10 mg of Fe\({}_{3}\)O\({}_{4}\)@MSN-QPDMAEMA, and the mixture was shaken at 25 \({}^{\circ}\)C for different time intervals. The data were then fitted with pseudo-first-order [65; 66; 67] and pseudo-second-order kinetic models [68] (Figure 9a-d). Along with these two models, the intraparticle diffusion model was also studied to investigate the process of the sequestration of analytes from aqueous solution onto SNPs, (Figure 9e,f). It can be shown that the uptake data of both dyes by the adsorbent fitted well with the pseudo-second-order model, indicating an inclination toward chemisorption. This implies that the rate of the analyte's uptake depends upon the adsorption capacity and not the concentration of adsorbate [69].  
The results obtained by the intraparticle diffusion model showed that rapid removal of the MO dye at the initial stage was caused by the presence of a considerable number of free active sites on the surface of Fe\({}_{3}\)O\({}_{4}\)@MSN-QPDMAEMA that analyte molecules could bind to (boundary layer diffusion effects). A reduction in the removal rate at the second stage was observed due to a significant drop in the number of the available active sites on the sorbent's surface. This resulted in the diffusion of the MO molecule through the pores (pore diffusion). In the case of E110 dye, a different behavior was observed, as the dye was adsorbed by the sorbent in one kinetic stage, which could be attributed to the multilayer adsorption of E110 dye on the adsorbent surface. The values of the intraparticle diffusion rate constants K\({}_{\text{id}}\) of both dyes are summarized in Table 3. Higher K\({}_{\text{id}}\) value indicates higher intraparticle mass transfer, which means that the kinetics of E110 uptake  
Figure 8: Dyes adsorption at 25 oC by the synthesized Fe\({}_{3}\)O\({}_{4}\)@MSN-QPDMAEMA at equilibrium with different initial analytes concentrations: (**a**,**b**) Freundlich and Langmuir isotherms of MO uptake, respectively; (**c**,**d**) Freundlich and Langmuir isotherms of E110 uptake, respectively.  
by Fe\({}_{3}\)O\({}_{4}\)@MSN-QPDMAEMA may be controlled by film diffusion mechanism, whereas pore diffusion was more active in the adsorption of MO.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Dye**} & \multicolumn{3}{c}{**Intraparticle Diffusion Model**} \\ \cline{2-3}  & \multicolumn{2}{c}{\(\mathbf{K_{id}\ (mg\,g^{-1}\,h^{-0.5})}\)} & \(\mathbf{R^{2}}\) \\ \hline MO & 3.4 & 0.60 \\ E110 & 1.02 & 0.71 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Intraparticle diffusion model parameters for the removal of MO and E110 from aqueous solution by Fe\({}_{3}\)O\({}_{4}\)@MSN-QPDMAEMA.  
Figure 9: (**a,b**) Pseudo-first-order and pseudo-second-order models of MO uptake by Fe\({}_{3}\)O\({}_{4}\)@MSN-QPDMAEMA, respectively; (**c,d**) pseudo-first-order and pseudo-second-order of E110 uptake by Fe\({}_{3}\)O\({}_{4}\)@MSN-QPDMAEMA, respectively; (**e,f**) intraparticle diffusion model of MO and E110 dyes uptake by Fe\({}_{3}\)O\({}_{4}\)@MSN-QPDMAEMA, respectively.