## 3 Results
### Phosphate Adsorption in Batch Experiments
#### 3.3.2 Adsorption Kinetics  
An essential feature of adsorbents is the adsorption kinetics, which allows the measurement of the adsorption rate. Figure 8a shows the phosphate adsorption capacity of the synthesized adsorbents at different contact times. The phosphate adsorption capacities of BS-GMA-EDA-Q and BS-GMA-PEHA-Q increased rapidly with increasing contact time and reached equilibrium at 180 min.  
Figure 7: Effect of pH on phosphate adsorption on the BS based adsorbents (**a**) and Zeta potential of the adsorbents (**b**).  
Pseudo-first-order, pseudo-second-order, and Weber-Morris models were used to predict the adsorption mechanism and describe the phosphate adsorption rate of BS-GMA-EDA-Q and BS-GMA-PEHA-Q, expressed by Equations (4)-(6), respectively [37].  
\[\ln(q_{e}-q_{t})=\ln q_{e}-k_{1}t \tag{4}\]  
\[\frac{t}{q_{t}}=\frac{1}{k_{2}q_{e}^{2}}+\frac{t}{q_{e}} \tag{5}\]  
\[q_{t}=k_{\text{in}}t^{1/2}+I \tag{6}\]  
where \(q_{t}\) and \(q_{e}\) (mg/g) are the amounts of phosphate adsorbed on 1 g adsorbent at time t (min) and equilibrium, respectively. \(k_{1}\) and \(k_{2}\) are the rate constant for the respective model [36]. \(K_{\text{in}}\) is the reaction rate constant (mg/g:min\({}^{1/2}\)), and \(I\) is the intercept. The fitting parameters and the correlation coefficients (R2) are listed in Table 1.  
The correlation coefficients (R2) of the pseudo-second-order kinetic model for BS-GMA-EDA-Q and BS-GMA-PEHA-Q were 0.9995 and 0.9995, respectively. The high R2 revealed that the pseudo-second-order adsorption mechanism was predominant in the adsorption process. The pseudo-second-order kinetic model shows that there was a chemical adsorption mechanism between the phosphate and the two adsorbents.  
The Weber-Morris model was used to investigate whether intra-particle diffusion was the only rate-controlling process in the adsorption process. The Weber-Morris model describe the ions' transport from aqueous solutions to the adsorbent. In this adsorption process, the plots present three distinct linear regions. The initial region represents the  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Parameters** & **BS-GMA-EDA-Q** & **BS-GMA-PEHA-Q** \\ \hline pseudo-first-order kinetics & \(k_{1}\) (n\({}^{-1}\)) & 0.0458 & 0.0421 \\  & \(q_{e}\) (mg/g) & 22.266 & 21.177 \\  & R2 & 0.9843 & 0.9616 \\ pseudo-second-order kinetics & \(k_{2}\) (g/(mg:min)) & 0.0025 & 0.0021 \\  & \(q_{e}\) (mg/g) & 24.504 & 23.981 \\  & R2 & 0.9995 & 0.9995 \\  & \(K_{\text{id}1}\) & 3.401 & 2.8049 \\  & \(I_{1}\) & \(-\)2.094 & \(-\)0.6285 \\  & R2 & 0.9557 & 0.9860 \\  & \(K_{\text{id}2}\) & 1.1673 & 1.1649 \\  & \(I_{2}\) & 9.8019 & 8.3497 \\  & R2 & 0.9278 & 0.9775 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Kinetic parameters obtained from pseudo-first-order, pseudo-second-order kinetic and intra-particle diffusion model.  
Figure 8: Adsorption kinetics of phosphate removal onto BS in 50 mL, 25 mg/L, 0.05 g, pH 7, (**a**) effect of adsorption time; (**b**) pseudo-second-order; (**c**) intra-particle diffusion model.  
phosphate transport from the aqueous solution to the BS-GMA-EDA-Q and BS-GMA-PEHA-Q surfaces. The second region presents the gradual adsorption corresponding to intra-particle diffusion. The third plateau region indicates equilibrium adsorption. The first linear part did not pass through the origin, indicating that intra-particle diffusion was not the sole rate-determining step [38].