## 2 Results and Discussion
### Adsorption
#### 2.5.4 Adsorption Isotherm  
Adsorption isotherm is one of the most important considerations in the application of adsorbents. Langmuir and Freundlich equations were fitted to the experiment data to investigate the isotherm type. The assumption of the Langmuir equation describes that the adsorption is a homogeneous surface by monolayer adsorption without interaction between adsorbed sites. The Freundlich isotherm equation is based on the assumption that the adsorption of EDA-MPMs occurs on a heterogeneous surface by multilayer adsorption.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{\(\mathbf{C_{0}}\) (mg/g)} & \multirow{2}{*}{\(q_{e,exp}\)\({}^{a}\)} & \multicolumn{3}{c}{**Pseudo-First Order Model**} & \multicolumn{3}{c}{**Pseudo-Second Order Model**} \\ \cline{3-8}  & & & \(\mathbf{q_{e,cal}}\) & & \(\mathbf{k_{1}}\) & \(\mathbf{R^{2}}\) & \(\mathbf{q_{e,cal}}\) & \(\mathbf{k_{2}}\) (g\(\cdot\)mg\({}^{-1}\cdot\)min\({}^{-1}\)) \(\times\) 10\({}^{-4}\) & \(\mathbf{R^{2}}\) \\ \hline
100 & 50.5 & 14.206 & 0.0319 & 0.803 & 53.333 & 25.539 & 0.998 \\ \hline
200 & 73.9 & 56.179 & 0.0321 & 0.968 & 80.451 & 8.718 & 0.999 \\ \hline
300 & 145.2 & 141.206 & 0.0303 & 0.985 & 169.205 & 2.295 & 0.997 \\ \hline \end{tabular} \({}^{a}\)\(q_{e,exp}\): the experimental \(q_{e}\) values; \({}^{b}\)\(q_{e,cal}\): the calculated \(q_{e}\) values.  
\end{table}
Table 1: Parameters of kinetic models for Cr(VI) adsorption onto the ethylenediamine-functionalized magnetic microspheres (EDA-MPMs).  
Figure 7: Linear fit of experimental data using pseudo-second-order kinetic model.  
Langmuir Equation [42]:  
\[\frac{c_{e}}{q_{e}}=\frac{c_{e}}{q_{m}}+\frac{1}{q_{m}b} \tag{6}\]  
Freundlich Equation [43]:  
\[\text{ln}q_{e}=\text{ln}K_{F}+\frac{1}{n}\text{ln}c_{e} \tag{7}\]  
where \(\underline{c_{e}}\) (mg/L) is the remaining Cr(VI) concentration in solution; \(q_{e}\) (mg/g) is the equilibrium amount of magnetic microbeads absorption; While \(q_{m}\) (mg/g) represent the maximum adsorption amount of Cr(VI) and \(b\) (L/mg) is the Langmuir adsorption equilibrium constant. \(K_{F}\) (L/g) and n are the Freundlich constants, which indicate the adsorption capacity and intensity, respectively.  
The isotherm data has been linearized using Langmuir and Freundlich isotherm and the constants involved with the models and the correlation coefficients are listed in Table 2 and Table S2. In addition, the isotherm data is plotted between \(c_{e}/q_{e}\)_versus_\(c_{e}\), which is shown in Figure 8 and Figure S2. Compared with two isotherm models, the Langmuir model describes the experimental data better than the Freundlich isotherm due to the higher correlation coefficients (\(R^{2}>0.999\)). Based on the assumptions of the Langmuir model, the adsorption process belongs to monolayer adsorption. The Langmuir constant \(q_{m}\) for Cr(VI) by EDA-MPMs is close to the experimental data. The maximum adsorption capacities at 25-45 \({}^{\circ}\)C are 236-253 mg/g. Most importantly, the maximum adsorption capacity of EDA-MPMs is higher than those of other adsorbents. The values of some of these Cr(VI) adsorbents were given in Table 3. The high capacity indicated that the EDA-MPMs have a potential application in the removal of Cr(VI) from wastewater.