## 3 Results and Discussion
### Sorption Isotherms  
In most of adsorption studies, adsorption models such as Langmuir and Freundlich isotherms have been widely used to evaluate adsorption phenomena [9]. In this work, we use Langmuir and Freundlich model to fit the adsorption data. Sorption isotherms and fitting parameters for the examined metals are shown in Figure 3 and Table 2, respectively. The isotherms of Pb\({}^{2+}\), Cu\({}^{2+}\), and Zn\({}^{2+}\) adsorbed by PPG-nZVI beads are found to better fit with the Langmuir model, judging from the higher correlation coefficients (\(r^{2}>0.98\)) than Freundlich model (Table 2). The well fit by Langmuir model suggested a monolayer adsorption of above heavy metal ions on the surface of PPG-nZVI beads. The Freundlich equation is based on adsorption on a heterogeneous surface [10] and it is considered to be a multi-layer process. This may be the reason why the Freundlich equation cannot better describe the adsorption of Pb\({}^{2+}\), Cu\({}^{2+}\), or Zn\({}^{2+}\) on PPG-nZVI beads. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Pseudo-First-Order Model**} & \multicolumn{2}{c}{**Pseudo-Second-Order Model**} \\ \hline  & \(k_{1}\) (L/min) & \(q_{\text{e}}\) (mg/g) & \(r^{2}\) & \(k_{2}\) (g/mg\(\cdot\)min) & \(q_{\text{e}}\) (mg/g) & \(r^{2}\) \\ \hline Pb\({}^{2+}\) & \(0.54\pm 0.01\) & \(3.10\pm 0.02\) & \(0.97\) & \(0.21\pm 0.01\) & \(3.43\pm 0.06\) & \(0.98\) \\ Cu\({}^{2+}\) & \(1.83\pm 0.14\) & \(2.47\pm 0.04\) & \(0.94\) & \(1.03\pm 0.02\) & \(2.66\pm 0.01\) & \(0.99\) \\ Zn\({}^{2+}\) & \(0.25\pm 0.01\) & \(2.03\pm 0.05\) & \(0.99\) & \(0.01\pm 0.01\) & \(2.30\pm 0.05\) & \(0.98\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Adsorption kinetic constants obtained by different models. Figure 2: Adsorption kinetics of Pb\({}^{2+}\), Cu\({}^{2+}\), or Zn\({}^{2+}\) removal by PPG-nZVI beads. Initial Pb\({}^{2+}\), Cu\({}^{2+}\), or Zn\({}^{2+}\) concentrations, \(20\) mg\(\cdot\)L\({}^{-1}\); adsorbent dose, \(4.0\) g\(\cdot\)L\({}^{-1}\); total solution volumes, \(400\) mL; pH, \(5.0\); temperature, \(25\)\({}^{\circ}\)C, and shaking time, \(48\) h.  
As depicted, there are pronounced differences for sorption capacities of the PPG-nZVI bead to the different metals examined (Figure 3). The maximum adsorption capacities (\(q_{\rm m}\)) for Pb\({}^{2+}\), Cu\({}^{2+}\) and Zn\({}^{2+}\)obtained by Langmuir isotherm were 16.01, 5.23, 2.31 mg/g, respectively (Table 2). In the above section, we observed that the adsorption rates decreased in the order of Cu\({}^{2+}\)\(>\) Pb\({}^{2+}\)\(>\) Zn\({}^{2+}\), in consistent with the order of the standard reduction potential. However, for the adsorption capacity, the \(q_{\rm m}\) decreased following in the order of Pb\({}^{2+}\)\(>\) Cu\({}^{2+}\)\(>\) Zn\({}^{2+}\), not well correlated with the order of standard reduction potential. Besides the influence of reduction potential, other factors such as the electronegativity, paramagnetic, and atomic weight could all make contribute to affect their sorption capacity. The maximum adsorption capacity for Pb\({}^{2+}\) was much higher than the values of Cu\({}^{2+}\) and Zn\({}^{2+}\), showing the following capacity order: Pb\({}^{2+}\)\(>\) Cu\({}^{2+}\)\(>\) Zn\({}^{2+}\). Pb\({}^{2+}\) has the preferential uptake, which may be because of the higher affinity of PPG-nZVI beads towards Pb\({}^{2+}\) than the other two heavy metal ions. A previous study by [PERSON] et al. [11]. showed that the sorption of Pb\({}^{2+}\), Cu\({}^{2+}\) and Cd\({}^{2+}\) by gelation with alginate following the same order of Pb\({}^{2+}\)\(>\) Cu\({}^{2+}\)\(>\) Cd\({}^{2+}\). They explained the difference is due to the strength of covalent bonds between the metal ions and beads that follows the sequence of Pb\({}^{2+}\)\(>\) Cu\({}^{2+}\).