## 3 Results and Discussion
### Pd(II) Sorption in Mono Metal System
#### 3.2.2 Uptake Kinetics  
The kinetics of Pd(II) sorption onto LD/PEI and CC/PEI beads at two different stirring speeds are shown in Figure 4a. In a solid-liquid sorption process, adsorbate transfer can be controlled by resistances to film diffusion (so called external diffusion), surface diffusion, and pore diffusion, or combined surface and pore diffusion. A higher agitation speed is supposed to decrease the boundary layer (resistance to film diffusion) and thus increase the speed at which the sorption system reaches equilibrium [46; 47]. In this study, the kinetic behaviors obtained from 60 rpm and 150 rpm are almost overlapped, even during the first few hours, suggesting that the external film diffusion should not be the main controlling factor. In general, LD/PEI beads present slightly faster kinetics than CC/PEI beads. The plots of q\({}_{\text{f}}\)/q\({}_{\text{eq}}\) against t in Supplementary Materials in Figure S6 confirm this phenomenon more clearly: LD/PEI beads reach the equilibrium at 24 h, while more than 48 h are required for CC/PEI beads. As discussed in the SEM-EDX analysis section, the PEI-GA particles are mostly distributed in the outer layer of LD/PEI beads and over the whole mass of CC/PEI beads. Hence, for LD/PEI beads, the reaction rate was mainly controlled by surface diffusion, pore diffusion, and sorption itself, whereas for CC/PEI beads, the internal diffusion proceeded in parallel with the sorption reaction and this is expected to require more time for reaching equilibrium. Another reason could be the smaller size of LD/PEI beads (2.48 \(\pm\) 0.18 mm) compared to CC/PEI beads (3.33 \(\pm\) 0.17 mm). Three models including the Pseudo-second order rate equation (PSORE), the Pseudo-first order rate equation (PFORE), and Intraparticle diffusion model equation (IPDE) were used to fit the experimental data. The parameters for the PFORE and PSORE were obtained by non-linear regression analysis and the data before equilibrium time (24 h for LD/PEI and 48 h for CC/PEI beads) were used for modeling. The PSORE fits the experimental data better compared to the PFORE, regardless of the  
Figure 3: Effect of pH on Pd(II) sorption onto LD/PEI and CC/PEI (C\({}_{0}\): 0.8 mmol\({}^{-1}\); contact time: 72 h; temperature: 20 \({}^{\circ}\)C; SD: 0.25 g L\({}^{-1}\); \(\star\): the “negative values of pH correspond to HCl concentrations ranging between 0.1 M and 1 M”). sorbent. The solid lines in Figure 4a represent the fit of the experimental data with the PSORE and the modeling parameters are reported in Table 1. The predicted equilibrium sorption capacities (q\({}_{\text{e,cal}}\)) from PSORE for both beads are in agreement with those obtained from the experiment (q\({}_{\text{e,exp}}\)). However, although the PSORE can adequately predict the sorption kinetics, it does not show the sorption mechanisms, while the intraparticle diffusion model helps in identifying the reaction pathways and sorption mechanisms and reveals the rate-controlling step. As shown in Figure 4b, the plot of q\({}_{\text{t}}\) against t\({}^{0.5}\) presents multiple linear regions, indicating that the sorption process is controlled by a multistep mechanism [47]. Two main stages before achieving equilibrium can be identified in the plots of Pd(II) sorption onto both LD/PEI and CC/PEI sorbents. The first step represents the mass transfer controlled by the resistance to surface diffusion. During this step, a large amount of Pd(II) is bound to the sorption sites that are readily accessible. The second stage refers to the step controlled by the resistance to intraparticle diffusion including pore diffusion and along pore-wall surfaces [22; 47]. At this stage, the sorption rate is far slower than that for the first stage (k\({}_{\text{id,2}}\)\(<\)k\({}_{\text{id,1}}\), shown in Table 1). The same trend was observed in previous studies, for the sorption of dye onto ZnO nanoparticles [48] and Pb(II) binding on the Cu/ZnO composite [49].