## Materials and Methods
### Modeling  
Uptake kinetics have been modeled using the pseudo-first order rate equation (PFORE, Equation (1)) [20], the pseudo-second order rate equation (PSORE, Equation (2)) [21], and the resistance to the intraparticle diffusion equation (IPDE, Equation (3)) [22].  
\[q_{t}=q_{eq}\Big{(}1-e^{-k_{1}t}\Big{)} \tag{1}\]  
\[q_{t}=\frac{q_{eq}^{2}k_{2}t}{1+q_{eq}k_{2}t} \tag{2}\]  
\[q_{t}=k_{id}\sqrt{t}+C \tag{3}\]  
where q\({}_{t}\) and q\({}_{eq}\) (mmol g\({}^{-1}\)) are the sorption capacities sorbed at t and at equilibrium, respectively. The parameters k\({}_{1}\), k\({}_{2}\), and k\({}_{id}\) are the apparent rate constants of PFORE (h\({}^{-1}\)), PSORE (g mmol\({}^{-1}\) h\({}^{-1}\)), and IPDE (mmol g\({}^{-1}\) h\({}^{-1/2}\)), respectively. C is a constant of IPDE associated with the thickness of the boundary layer; a higher C value indicates a greater effect of the resistance to film diffusion (larger boundary layer). The parameters were also obtained by non-linear regression analysis using Origin 9.0 (Origin software Inc., San Clemente, CA, USA).  
Sorption isotherms describe the distribution of metal ions between the liquid and the solid phases (when varying the metal concentration in the system). They plot the sorption capacity (i.e., q\({}_{eq}\)) vs.  
residual metal concentration (i.e., \(\mathrm{C_{eq}}\)). [PERSON] (Equation (4)) [23], [PERSON] (Equation (5)) [24], and [PERSON] (Equation (6)) [25] equations were used to fit the experimental data.  
\[q_{eq}=\frac{q_{mL}bC_{eq}}{1+bC_{eq}} \tag{4}\]  
\[q_{eq}=K_{F}C_{eq}^{n} \tag{5}\]  
\[q_{eq}=\frac{q_{ms}K_{s}C_{eq}^{1/n}}{1+K_{s}C_{eq}^{1/n}} \tag{6}\]  
where b (L mmol\({}^{-1}\)) is the affinity coefficient and qmL (mmol g\({}^{-1}\)) is the sorption capacity at saturation of the monolayer in the Langmuir equation; \(\mathrm{K_{F}}\) [(mmol g\({}^{-1}\))/(mmol L\({}^{-1}\))\({}^{n}\)] is the Freundlich constant and n is the intensity parameter; \(1/n\) is the heterogeneity factor, qmL (mmol g\({}^{-1}\)) is the maximum sorption capacity in the Sips equation, and \(\mathrm{K_{s}}\) (L mmol\({}^{-1}\)) is the affinity coefficient in the Sips equation. qeq (mmol g\({}^{-1}\)) is the amount of palladium uptake at equilibrium and \(\mathrm{C_{eq}}\) (mmol L\({}^{-1}\)) is the palladium concentration at equilibrium. The parameters were also obtained by non-linear regression analysis using Origin 9.0. The objectives of the modeling of sorption isotherms may consist of:  
1. the determination of the sorption capacities in function of the residual metal concentration (principle of the sorption isotherms); this may be helpful for predicting the sorption capacity under selected experimental conditions, where a good fit is of great importance;
2. an approach of binding mechanism (the hypotheses attached to the model are indicative of mechanisms possibly involved in metal binding; however, the mathematical fit does not necessarily mean that the hypothesized mechanisms are fulfilled; this should be confirmed by physico-chemical analyses);
3. the determination of maximum sorption capacities (saturation level) and the affinity (correlated to the initial slope of the curve) between the sorbent and the solute under selected experimental conditions (for the comparison of sorbents).