## 3 Results and Discussion
### Sorption Properties
#### 3.2.3 Sorption Isotherms  
The sorption isotherms (i.e., plots of the sorption capacity vs. equilibrium concentration at fixed values of the pH and temperature) are represented in Figure 4 (herein, pH\({}_{0}\) 5; T: 22 \(\pm\) 1 \({}^{\circ}\)C). The superposition of the triplicated curves confirm the good reproducibility of the sorption experiments. The sorption isotherms are characterized by a pseudo-asymptotic shape. The asymptote is correlated with the maximum sorption capacity (or sorption capacity at the saturation of the monolayer in the Langmuir model). This maximum sorption capacity (i.e., q\({}_{\rm m,exp}\)), consistent with previous observations, increases with the successive functionalizations of the support: 0.549, 1.328, and 2.176 mmol Ce g\({}^{-1}\) (for MC, MC-UR, and MC-UR/S, respectively). The sorption capacity increases almost linearly with the number of potential reactive groups on the sorbent (i.e., \(\Sigma\) = n(N) + n(S), in molar units g\({}^{-1}\)): q\({}_{\rm m}\) = 0.647 \(\Sigma\) \(-\) 1.558. The sorption isotherms were fitted using the Langmuir, Freundlich, Sips, Temkin and Dubinin-Radushkevich (D-R) equations (Table S12b). The Freundlich equation is an empirical model based on the concept of heterogeneous sorption, with possible interactions  
Figure 4: Ce(III) sorption isotherms for MC, MC-UR, and MC-UR/S sorbents at pH\({}_{0}\) 5—modeled with the Langmuir equation (sorbent dose, SD: 400 mg L\({}^{-1}\); C\({}_{0}\): 0.007-3.57 mmol Ce L\({}^{-1}\); time: 48 h; v: 210 rpm; T: 22 \(\pm\) 1 \({}^{\circ}\)C). of the sorbent molecules. Its mathematical expression (i.e., power-like function) is not consistent with the asymptotic trend of the experimental profiles. Logically, this model gives the poorest correlation between the experimental and fitted curves. The Langmuir equation accounts for homogeneous sorption without interactions with the sorbed molecules and with a similar distribution and binding energies. The Sips equation is a combination of the Langmuir and Freundlich equations. Introducing a third adjustable parameter facilitates the fitting of the experimental data (with a weaker physical significance). The adaptations of the Temkin and the Dubinin-Raduskevich equations to liquid/solid systems are strongly debated in terms of both the equations and interpretations [73; 74]. Therefore, a case where these equations closely fit the experimental profiles must be considered very carefully. Tables S5-S7 summarize the parameters and the statistical criteria for the triplicates (and for the composite isotherm obtained by combining the triplicates) for the five models and the three sorbents. In the case of the MC sorbent, the Langmuir and the Sips equations gave comparable qualitative fittings of the experimental curves. However, the fitting advantage of the Langmuir equation consists of the closer values for \(\mathrm{q_{m,L}}\) with \(\mathrm{q_{m,exp}}\) (compared with \(\mathrm{q_{m,S}}\)). The Temkin equation fits the isotherms for MC-UR slightly better (preferentially to Langmuir \(\approx\) Sips). It is noteworthy that the energetic term (\(\mathrm{b_{T}}\)) decreases with the functionalization of the material (according \(\approx\)21 \(>\)\(\approx\)11 \(>\)\(\approx\)5 J kg mmol \({}^{-2}\)). In the case of MC-UR/S, the Langmuir equation is again the most appropriate for describing the Ce(III) isotherm (Langmuir \(>\) Sips \(\approx\) Temkin). Despite the less accurate fit of the isotherm with the D-R equation, the characteristic free energy of the adsorption (\(\mathrm{E_{DR}}\)) can be used to analyze and compare the sorption behavior of the three sorbents. The EDR for MC is a little lower (i.e., around 3.6 kJ mol\({}^{-1}\)) than the values obtained for the functionalized sorbents (i.e., around 6 kJ mol\({}^{-1}\)). The Langmuir equation shows a better fit for all of the three systems, making the comparison of the model parameters easier (in addition to the more accurate determination of the sorption capacity at saturation).