## 3 Results and Discussion
### Characterization of La-EPRC Particles for Static Adsorption of Phosphorus Removal
#### 3.2.2 Adsorption Kinetics  
Corresponding to phosphorus concentration and adsorption amount, a relationship curve between reaction time (t) and adsorption amount (Q) was plotted, as illustrated in Figure 9. Notably, the adsorption rate during the initial period is faster for each reaction with different initial concentrations of phosphorus solution. This phenomenon is attributed to the abundance of material adsorption sites in the initial phase, leading to less competition and facilitating a rapid reaction. As the reaction progresses, the concentration difference in the solution decreases, resulting in a gradual reduction in adsorption rate, indicating the transition to the slow reaction stage. With diminishing phosphorus solution concentration, the competitiveness of adsorption sites intensifies, further slowing down the adsorption rate. Over time, the material's adsorption eventually reaches equilibrium, entering the adsorption equilibrium stage, represented by a flat curve. Additionally, higher initial concentrations of the reaction correspond to greater adsorption capacities in the early stage and higher capacities upon reaching equilibrium. Furthermore, lower initial concentrations of the phosphorus solution lead to faster equilibrium times, whereas higher concentrations result in slower equilibrium times. Pseudo-first-order (PFO) and pseudo-second-order (PSO) kinetic models were applied to the data obtained from the aforementioned experiments. The fitting results are depicted in Figure 9, and detailed outcomes are summarized in Table 5. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multicolumn{6}{c}{**PFO Adsorption Kinetics**} & \multicolumn{6}{c}{**PSO Adsorption Kinetics**} \\ \hline
**Initial Phosphorus** & **q\({}_{e}\) (mg/g)** & **K\({}_{1}\)** & **R\({}^{2}\)** & **Initial Phosphorus** & **q\({}_{e}\) (mg/g)** & **K\({}_{2}\)** & **R\({}^{2}\)** \\
**Concentration (mg P/L)** & **q\({}_{e}\) (mg/g)** & **K\({}_{1}\)** & **R\({}^{2}\)** & **Concentration (mg P/L)** & **q\({}_{e}\) (mg/g)** & **K\({}_{2}\)** & **R\({}^{2}\)** \\ \hline
5 & 0.56935 & 0.1028 & 0.9706 & 5 & 0.67577 & 0.17179 & 0.99366 \\
10 & 1.07083 & 0.12135 & 0.96394 & 10 & 1.26282 & 0.11048 & 0.99021 \\
15 & 1.52834 & 0.12648 & 0.98207 & 15 & 1.77373 & 0.08435 & 0.99186 \\
20 & 2.02956 & 0.13264 & 0.97696 & 20 & 2.3399 & 0.06823 & 0.99038 \\ \hline \end{tabular}
\end{table} 
Table 5: Kinetics fitting results. Figure 9: Kinetic fitting results. (**a**) Results of fitting PFO adsorption kinetics. (**b**) Results of fitting PSO adsorption kinetics. As indicated in Table 5, the correlation coefficients for PFO kinetics (R\({}^{2}\) > 0.96) and PSO kinetics (R\({}^{2}\) > 0.99) are highly regressive. However, the results from PSO kinetics surpass those of the PFO kinetics model across different initial concentrations of the phosphorus solution. This suggests that the adsorption kinetics of La-EPRC particles align more closely with the PSO kinetics model. The q\({}_{\rm e}\) values calculated from the PSO kinetic fitting also exhibit closer agreement with the experimentally obtained values. Consequently, it can be concluded that the adsorption of phosphorus by the modified particles primarily involves chemisorption, with the adsorption process being predominantly rate-limited by chemisorption [22]. The adsorption kinetics curves at various temperatures are illustrated in Figure 10. The results of the PFO and PSO kinetic fits can be observed in Figures 11 and 12, as well as in Table 6.  
From the fitting results depicted in Figures 11 and 12 and Table 6, it is evident that the elevation in temperature enhances the correlation of both kinetic models, with PFO and PSO kinetic fits at 35 \({}^{\circ}\)C closely resembling each other. This suggests a complex adsorption process involving both physical and chemical mechanisms for La-EPRC particles. Overall, the PFO adsorption kinetic model appears more suitable for elucidating the phosphate removal adsorption process by La-EPRC particles.