## Materials and Methods
### Physicochemical Model  
The dissolution reaction of LiCoO\({}_{2}\) particles in acid media, expressed as a redox half-reaction, can be described as [25]:  
\[\text{LiCoO}_{2}(\text{s})+4\text{H}^{+}+e^{-}\rightleftarrows\text{Li}^{+}+ \text{Co}^{2+}+2\text{H}_{2}\text{O}, \tag{4}\]  
where Co\({}^{3+}\) within the LiCoO\({}_{2}\) particle is reduced to soluble Co\({}^{2+}\). Considering that water acts as the reducing agent of the Co\({}^{3+}\) as  
\[\text{O}_{2}(\text{g})+4\text{H}^{+}+4e^{-}\rightleftarrows 2\text{H}_{2}\text{O}, \tag{5}\]  
the redox pair that describes the LiCoO\({}_{2}\) particles' dissolution can be expressed as:  
\[\text{LiCoO}_{2}(\text{s})+3\text{H}^{+}\rightleftarrows\text{Li}^{+}+\text{ Co}^{2+}+\frac{3}{2}\text{H}_{2}\text{O}+\frac{1}{4}\text{O}_{2}(\text{g}). \tag{6}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(m_{\text{LiCoO2}}\) (mg) & \(V_{\text{solution}}\) (mL) & S/L (g L\({}^{-1}\)) & [HCl] (M) \\ \hline
**Experiment A** & 125 & 25 & 5 & 0.1 \\
**Experiment B** & 1250 & 25 & 50 & 2.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Experimental parameters. As mentioned before, it was experimentally observed that the amount of extracted Li was nearly twice as much as that of Co, and that, in the absence of stronger reducing agents, the reaction reaches a maximum near 65-70% of Li extraction. A possible explanation for this phenomenon is the formation of a crust of insoluble Co\({}_{3}\)O\({}_{4}\)(s) according to:  
\[\text{LiCoO}_{2}(\text{s})+\text{H}^{+}\rightleftarrows\text{Li}^{+}+\frac{1} {3}\text{Co}_{3}\text{O}_{4}(\text{s})+\frac{1}{2}\text{H}_{2}\text{O}+\frac{1 }{12}\text{O}_{2}(\text{g}). \tag{7}\]  
As the Co\({}_{3}\)O\({}_{4}\) is insoluble even in concentrated acid, it is likely to form a crust in the particle that will impose resistance to the diffusion of the reactants, H\({}^{+}\) in this case, and could eventually stop the reaction [26]. Accordingly, the model presented here is a variation of the unreacted shrinking core model, and it is depicted in Figure 1.  
The model is based on the following assumptions:  
1. LiCoO\({}_{2}\) particles are assumed to be spherical with a mean initial radius of \(R_{p}=5\) um that follows a normal distribution. During the reaction, the radius of the unreacted core decreases while the number of particles is considered constant. 2. The extraction of lithium and cobalt from LiCoO\({}_{2}\) particles in acidic aqueous solution takes place according to the stoichiometry in Equation (8), which takes into account that the dissolution of LiCoO\({}_{2}\) particles produces Li and Co in a proportion of 2:1, as well as the formation of an outer crust layer of Co\({}_{3}\)O\({}_{4}\)(s) in the external surface of LiCoO\({}_{2}\)(s) core. \[\text{LiCoO}_{2}(\text{s})+2\text{H}^{+}\,\rightarrow\text{Li}^{+}+\frac{1} {2}\text{Co}^{2+}+\frac{1}{6}\text{Co}_{3}\text{O}_{4}(\text{s})+\text{H}_{2} \text{O}+\frac{1}{6}\text{O}_{2}(\text{g})\] (8)
3. The kinetic rate for the reaction in Equation (8) is given by: \[\frac{dN_{\text{LiCoO}_{2}}}{dt}=\rho_{A}\ n_{p}\left(4\pi r_{c}^{2}\right) \frac{dr_{c}}{dt}=-\eta_{p}k_{r}\left(4\pi r_{c}^{2}\right)\left(C_{\text{H}^ {+}}^{\text{corr}}\right)^{\alpha}=-r_{Li^{+}}\] (9) where \(\rho_{A}\) (mol m\({}^{-3}\)) is the molar density of LiCoO\({}_{2}\), \(n_{p}\) (-) is the number of particles, \(r_{c}\) (m) is the radius of the unreacted core, \(k_{r}\) (mol\({}^{1-\alpha}\)m\({}^{3\alpha-2}\)s\({}^{-1}\)) is the kinetic constant and \(C_{\text{H}^{+}}^{\text{corr}}\) (mol m\({}^{-3}\)) is the concentration of protons in the position \(r_{c}\), and \(\alpha\) (-) is the reaction order with respect to the protons. Figure 1: Time transition of the radius core LiCoO\({}_{2}\) particles. (**a**) Initial particles. (**b**) Process rate controlled by the chemical kinetics. (**c**) Process rate controlled by the diffusion through the outer layer. 4.