## 3 Results
### Adsorption Studies
#### 3.2.2 Equilibrium Isotherms  
Detailed knowledge of the equilibrium adsorption isotherm is essential in examining the interactive behavior between the adsorbent and the adsorbate in any adsorption system. Langmuir and Freundlich isotherm models were employed, as shown in Figure 8. In order to study the isotherms, the concentration of the analytes was adjusted in the range of 50-250 mgL\({}^{-1}\) and 300-1000 mgL\({}^{-1}\) for MO and E110, respectively. The experimental parameters were performed using the optimum conditions. Langmuir isotherm assumes that a homogeneous interaction between the adsorbent and the adsorbate at the material's adsorption sites with the formation of a monolayer of the adsorbate on the material's surface [60]. In contrast, the Freundlich isotherm model assumes the possibility of the existence of the multilayer sorption mechanism at the surface of absorbents with a heterogeneous energy distribution of active sites. The correlation coefficient (\(R^{2}\)) is the criterion  
Figure 6: Zeta potential values for Fe\({}_{3}\)O\({}_{4}\)@MSN-PDMAEMA and Fe\({}_{3}\)O\({}_{4}\)@MSN-QPDMAEMA at different pH levels.  
used to assess how well the data are fit with each model. The model with a higher \(R^{2}\) value will be utilized to describe the adsorption mechanism. The equilibrium data of the adsorption of MO (Figure 8b) fitted well with the Langmuir model with R\({}^{2}\) of more than 0.99, indicating the formation of a monolayer of MO molecules on the surface of the adsorbent (homogeneous interaction) with a maximum adsorption capacity of 294 mg g\({}^{-1}\). On the other hand, the equilibrium data of the E110 adsorption fitted well with the Fre-undlich isotherm model, indicating the formation of multilayers of E110 molecules on the surface of the adsorbent (heterogeneous interaction) with a maximum adsorption capacity of 194.8 mg g\({}^{-1}\). The difference between the two dyes in terms of the interaction can be attributed to the presence of two sulfonic moieties in both sides of E110 molecules, making them more electronegative compared with MO molecules, and consequently, more attraction with the positively charged adsorbent surface.  
The extraction of methyl orange and sunset yellow dyes using a variety of adsorbents has been extensively studied. Table 2 demonstrates a comparative study of the maximum adsorption capacities of some adsorbents for MO and E110 along with the current work. It is obvious that the adsorption capacities of our novel material are higher than other competitive adsorbents, indicating that Fe\({}_{3}\)O\({}_{4}\)@MSN-QPDMAEMA can be considered a promising adsorbent for the removal of both dyes from aqueous solutions.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sorbent** & **Capacity (mg/g)** & **Target Analyte** & **Ref.** \\ \hline Chitosan-coated sodium zeolites & 287 & MO & [61] \\ CDs/ZFO & 181.2 & MO & [62] \\ Fe\({}_{3}\)O\({}_{4}\)/PS/US & 67.56 & E110 & [63] \\ Fe\({}_{3}\)O\({}_{4}\)@SiO\({}_{2}\)@Kit-6/NH\({}_{2}\) & 8.41 & E110 & [64] \\ Fe\({}_{3}\)O\({}_{4}\)@MSN-QPDMAEMA & 294 & MO & This Work \\ Fe\({}_{3}\)O\({}_{4}\)@MSN-QPDMAEMA & 194.8 & E110 & This Work \\ \hline \hline \end{tabular}
\end{table}
Table 2: A list of maximum adsorption capacities for different types of adsorbents.  
Figure 7: Effect of pH on the adsorption of methyl orange and sunset dye by Fe\({}_{3}\)O\({}_{4}\)@MSN-PDMAEMA and Fe\({}_{3}\)O\({}_{4}\)@MSN-QPDMAEMA at equilibrium.