## 2 Materials and Methods
### Characterization of La-EPRC Particles for Static Adsorption of Phosphorus Removal  
#### 2.3.1 Adsorption Isotherm Experiments  
Two widely used adsorption isotherm models, Langmuir and Freundlich, were employed to investigate the adsorption mechanism of La-EPRC particles [15]. The experimental design involved adding phosphorus solution concentrations ranging from 5 mgP/L to 100 mgP/L into labeled stoppered conical flasks (labeled 1-9). Phosphorus standard reserve solutions were prepared and added to the flasks, followed by dilution to 250 mL with deionized water. Nine portions of La-EPRC material (2 g each) were weighed and added to the respective conical flasks. The solutions underwent continuous shaking in a shaker (25 \(\pm\) 1 \({}^{\circ}\)C, 150 \(\pm\) 5 r/min), and supernatants were collected after reaching adsorption equilibrium. These supernatants were filtered through a 0.45 \(\upmu\)m pore size filter, and UV-Vis spectrophotometry was used to determine absorbance values to calculate corresponding adsorption concentrations [16], and adsorption isotherms at 25 \({}^{\circ}\)C were constructed. Similar procedures were followed at 15 \({}^{\circ}\)C and 35 \({}^{\circ}\)C to generate corresponding relationship curves. * Langmuir adsorption isotherm modeling The Langmuir model posits that the adsorbent surface offers a finite number of adsorption sites, and the adsorption process exclusively transpires within a monomolecular layer. According to this model, the adsorbent attains maximum adsorption capacity when its surface reaches a state of saturated adsorption. Furthermore, the model assumes uniform attraction among all adsorption sites, implying equal affinities. As the reaction progresses, the surface adsorption sites on the adsorbent gradually become occupied, resulting in a deceleration of the adsorption rate. When nearly all available adsorption sites are engaged, the adsorption rate approaches zero, signifying the attainment of dynamic equilibrium in the adsorption reaction [17]. The specific expression for the Langmuir model is articulated as follows: \[\mathrm{q_{e}}=\frac{\mathrm{q_{m}K_{L}C_{e}}}{1+\mathrm{K_{L}C_{e}}},\] (1) \[\frac{\mathrm{C_{e}}}{\mathrm{q_{e}}}=\frac{1}{\mathrm{q_{m}K_{L}}}+\frac{ \mathrm{C_{e}}}{\mathrm{q_{m}}}.\] (2)  
In the given equation, \(\mathrm{q_{e}}\) represents the adsorption capacity at equilibrium in mg/g; \(\mathrm{q_{m}}\) denotes the saturated adsorption capacity of the adsorbent in mg/g; \(\mathrm{C_{e}}\) represents the equilibrium concentration of the adsorbent in mg/L; and \(\mathrm{K_{L}}\) signifies the Langmuir constant, expressed in L/mg. To determine these parameters, a linear fit was conducted by plotting \(\mathrm{C_{e}}/\mathrm{q_{e}}\) against \(\mathrm{C_{e}}\). The saturated adsorption capacity (\(\mathrm{q_{m}}\)) can be computed from the slope of the resulting straight line, while the intercept yields the Langmuir constant (\(\mathrm{K_{L}}\)). * Freundlich adsorption isotherm model The Freundlich model characterizes the adsorption process of the adsorbent as intricate, multilayered, and non-uniform in adsorption sites. It acknowledges the presence of heterogeneous adsorption sites on the surface of the adsorbent and postulates that the adsorbent is capable of offering an infinite number of adsorption sites. This model falls under physical adsorption and is applicable to describing the adsorption processes of both gases and liquids [18]. The specific expressions for the Freundlich model are articulated as follows: \[\mathrm{q_{e}}=\mathrm{K_{F}C_{e}}^{\frac{1}{2}},\] (3) \[\mathrm{ln}\,\mathrm{q_{e}}=\frac{1}{\mathrm{n}}\,\mathrm{ln}\,\mathrm{C_{e} }+\mathrm{ln}\mathrm{K_{F}}.\] (4)  
In the provided equation, \(\mathrm{q_{e}}\) represents the adsorption capacity at equilibrium in mg/g; n signifies the Freundlich constant; \(\mathrm{C_{e}}\) denotes the equilibrium concentration of the adsorbent in mg/L; and \(\mathrm{K_{F}}\) denotes the Freundlich constant in mg/g.