## 3 Results and Discussion
### Sorption Properties
#### 3.2.2 Uptake Kinetics  
The uptake kinetics are remarkably reproducible (note fine superposition of the triplicated curves, Figure 3). The equilibrium is reached within 30-60 min. The micron size of the sorbent particles may explain these fast equilibria. However, the t\({}_{50}\) (time required to reach 50% of the total sorption) is close to 15 min. The sorption kinetics may be controlled by the proper reaction rate (which is frequently associated with a pseudo-first- or pseudo-second-order rate equation, PFORE or PSORE, Table S4) and/or resistance to diffusion (external and intraparticle diffusions). The lines in Figure 3 show the PFORE fits of the kinetic profiles. The model roughly fits the experimental data. This is consistent with the comparison of the statistical criteria (i.e., R\({}^{2}\) and AIC) for PFORE, PSORE, and RIDE (resistance to intraparticle diffusion, according to the Crank equation, Table S12a). The quality of the fitting is demonstrated by the higher values of R\({}^{2}\) and lower values of AIC (the difference is significant when \(\mid\Delta\)AIC \(\mid\) > 2): PFORE \(>\) PSORE \(>\) RIDE.  
The PFORE slightly overestimates the q\({}_{\rm eq}\) values (by 5-10%), which increase with the successive functionalizations: 0.287 \(\pm\) 0.012, 0.844 \(\pm\) 0.010, and 1.233 \(\pm\) 0.021 mmol Ce g\({}^{-1}\) (for the experimental values; 0.310 \(\pm\) 0.017, 0.898 \(\pm\) 0.008, and 1.310 \(\pm\) 0.024 mmol  
Figure 3: Uptake kinetics for Ce(III) sorption using the MC, MC-UR, and MC-UR/S sorbents (pH\({}_{0}\) 5; C\({}_{0}\): 0.732 mmol Ce L\({}^{-1}\); SD: 400 mg L\({}^{-1}\); v: 210 rpm; T: 22 \(\pm\) 1 \({}^{\circ}\)C).  
Ce g\({}^{-1}\) for the calculated values). The apparent rate coefficients (k\({}_{1}\)\(\times\) 10\({}^{2}\) min\({}^{-1}\)) are of the same order for the three sorbents (though, in the case of MC, a greater dispersion of the data is observed): 3.11 \(\pm\) 0.73, 3.26 \(\pm\) 0.05, and 3.81 \(\pm\) 0.08 min\({}^{-1}\), for MC, MC-UR, and MC-UR/S, respectively. These values are of the same order of magnitude as the apparent rate coefficients reported by [PERSON] et al. (2011) (0.8-1.4 10\({}^{-2}\) min\({}^{-1}\)) for Nd(III), Ce(III), and La(III) using composite biochar.  
Sometimes, the fit of the kinetic profiles with the PFORE and/or PSORE is associated with the prevalence of physi- or chemisorption. However, [PERSON] (2010) and [PERSON] et al. (2011) demonstrated the difficulty of correlating the controlling mechanism with the mathematical fit because of the inappropriate selection of the experimental conditions. Herein, the apparent rate coefficients of PFORE are simply discussed as a tool for objectively comparing the systems. The RIDE gives less accurate fits of the kinetic profiles, but the effective diffusivity coefficients deduced from these curves can be compared with the self-diffusivity of Ce(III) in water (i.e., D\({}_{0}\)(Ce\({}^{3+}\)): 3.72 \(\times\) 10\({}^{-8}\) m\({}^{2}\) min\({}^{-1}\)) ([PERSON] et al., 2011)). The values are several orders of magnitude lower than D\({}_{0}\)(Ce\({}^{3+}\)), meaning that the resistance to intraparticle diffusion contributes to the overall control of the uptake kinetics.