## 3 Results and Discussion
### Apparent Reaction Activation Energy  
The leaching process of rare-earth metal from DES\({}_{\text{S}}\) is a homogeneous uncatalyzed reaction in a fluid-solid system; various models have been proposed for exploring the kinetics of this kind of process, including the shrinking-core model, the progressive-conversion model, and the grain model. The progressive-conversion and grain models are focused on describing the leaching kinetics of porous solids [27; 28], while both of the inactivated and activated rare-earth carbonates were compact particles with no porous solids, according to the morphology images by the scanning electron microscopy, and the shrinking-core model is adopted for describing the kinetics of rare-earth carbonate decomposition and the leaching process [29]. The leaching behavior of rare-earth carbonates in ChCl-Ur-MA DESs differs from that of conventional alkaline and acid leaching. The yttrium dissolution process can be described in terms of a complex reaction relating to concentrations of surface ligand (Ur and Cl\({}^{-}\)). During the leaching of DESs, free urea and Cl\({}^{-}\) were gradually consumed due to the production of [Y.urea Cl\({}^{-}\)] complexes. A shrinking core model was developed to describe the dissolving kinetics of Y in the absence of a product layer. The reaction of Y complexation with ligands (Ur and Cl\({}^{-}\)) and diffusion of ligands or produced complexes are involved in the Y leaching process. Therefore, the integrated rate equation is provided below, considering that the ligand concentrations are nearly constant [30]:  
\[1-(1-\alpha)^{\frac{1}{3}}=\frac{k^{\prime}[\text{Ur}]^{m}[Cl^{-1}]^{n}}{\rho _{Y}r_{0}}t=kt \tag{1}\]  
where \(\alpha\) is the leaching efficiency of Y at time t; k represents an apparent rate constant; [Ur] and [Cl\({}^{-}\)] denote the Ur and Cl\({}^{-}\) concentrations on the unreacted Y core surface, respectively, which are both unchanged; m and n correspond to the apparent reaction order; \(\rho_{Y}\) represents the Y molar density; r\({}_{0}\) represents the particle's initial radius; k denotes the kinetic constant calculated from Equation (1). Figure 8 depicts the plots of \(1-(1-\alpha)^{1/3}\) versus leaching time (t) at different temperatures. According to the experimental data, it  
Figure 7: Effect of leaching time on the leaching efficiency of metallic yttrium in different activated samples at different reaction temperatures (T = 80 \({}^{\circ}\)C, t\({}_{\text{MA}}\) = 60 min). (a) Mechanically inactivated samples; (b) Mechanically activated samples. was evident that the experimental data fit well with the shrinking core model. The higher viscosity of ChCl\(-\)Ur\(-\)MA DESs results in a slower rate of reactant diffusion (Ur and Cl\({}^{-}\)). Furthermore, using the k value obtained by fitting, we can calculate the apparent activation energy of the leaching reaction, based on the Arrhenius equation:  
\[lnk=-\frac{\text{Ea}}{RT}+lnA \tag{2}\]  
where A represents a frequency factor; Ea describes the apparent activation energy of the leaching reaction; \(R\) represents a gas constant; and T represents the leaching temperature. The logarithm of the kinetic rate constant (\(lnk\)) versus 1/T had a good linear relationship (Figure 9a). The apparent activation energy was determined to be 83.88 kJ\(\cdot\)mol\({}^{-1}\) based on the slope. These findings indicated that the chemical reaction controls the leaching process. Using the same analysis method as described above, the plots of \(1-(1-\alpha)^{1/3}\) and \(1-3(1-\alpha)^{2/3}+2(1-\alpha)\) versus leaching time, respectively, were tested and are presented in Figure 10. As demonstrated in Figure 10, the linear equation showed good fits with the experimental results. For the plots of \(1-(1-\alpha)^{1/3}\) versus the leaching time, the minimum value of the adjusted coefficient of determination (adjusted (Adj.) R\({}^{2}\)) of these plots was 0.9713, while it was 0.9878 for the plots of \(1-3(1-\alpha)^{2/3}+2(1-\alpha)\) versus the leaching time.