## 2 Materials and Methods
### Characterization of La-EPRC Particles for Static Adsorption of Phosphorus Removal
#### 2.3.2 Adsorption Kinetics  
The following amounts of the prepared phosphorus standard reserve solution were added to a 250 mL stoppered conical flask: 1.25 mL, 2.5 mL, 3.75 mL, and 5 mL. They were then diluted with deionized water to achieve concentrations of 5, 10, 15, and 20 mg/L. A total of 2 g of La-EPRC particles was weighed, added to the conical flask with varying phosphorus concentrations, and mixed thoroughly. The flasks were placed on a shaker for the adsorption experiment (25 \(\pm\) 1 \({}^{\circ}\)C, 150 \(\pm\) 5 r/min). Sampling occurred at 2 h, 4 h, 8 h, 12 h, 18 h, 24 h, 30 h, 36 h, 48 h, and 72 h. The upper layer of the supernatant was extracted during sampling and filtered through 0.45 \(\mathrm{\SIUnitSymbolMicro m}\) filters. Absorbance was determined using the ammonium molybate spectrophotometric method.  
The experimental procedure for La-EPRC particle adsorption kinetics at different temperatures mirrored the aforementioned method, with shaker temperatures set at 15 \({}^{\circ}\)C, 25 \({}^{\circ}\)C, and 35 \({}^{\circ}\)C. The data were analyzed using pseudo-first-order (PFO) and pseudo-second-order (PSO) adsorption kinetic models.  
* PFO adsorption kinetic model  
[PERSON]'s research introduced the PFO adsorption kinetic model to delineate monolayer adsorption primarily governed by boundary diffusion. This model is particularly adept at capturing the swift reaction phase within the adsorption process, offering a more nuanced representation of this specific stage rather than the entirety of the adsorption process [19]. The expression for this PFO adsorption kinetic model is as follows:  
\[\ln(\mathrm{Q_{e}}-\mathrm{Q_{t}})=\ln\mathrm{Q_{e}}-\mathrm{k_{1}t}. \tag{5}\]  
In the formula, \(\mathrm{Q_{e}}\) represents the adsorption capacity at equilibrium (mg/g), while \(\mathrm{Q_{t}}\) signifies the adsorption capacity of the adsorbent at time t (h) in milligrams per gram. The PFO kinetic equilibrium constant is denoted as \(\mathrm{k_{1}}\), expressed in units of 1/h. When plotting t against \(\ln(\mathrm{Q_{e}}-\mathrm{Q_{t}})\), a well-fitted straight line with a correlation coefficient (R2) approaching 1 indicates that the material's adsorption process aligns with the PFO kinetic model. The adsorption equilibrium constant is determined by the slope of this line, while the intercept allows for the calculation of the adsorbent's equilibrium adsorption amount.
* PSO adsorption kinetic model  
The PSO adsorption kinetic model, elucidated and synthesized by [PERSON] and [PERSON], primarily addresses adsorption processes constrained by chemisorption [20]. This model aptly characterizes the entirety of the adsorption process and is represented by the following expression:  
\[\frac{\mathrm{t}}{\mathrm{Q_{t}}}=\frac{1}{\mathrm{k_{2}Q_{e}^{2}}}+\frac{ \mathrm{t}}{\mathrm{Q_{e}}}. \tag{6}\]  
In this equation, \(\mathrm{k_{2}}\) denotes the PFO kinetic equilibrium constant, expressed in units of g/(mg \(\times\) h). By plotting t/\(\mathrm{Q_{t}}\) against t, the slope of the resulting straight line, derived from the regression fit, provides the value of \(\mathrm{Q_{e}}\), while the intercept corresponds to the value of \(\mathrm{k_{2}}\).