## 4 Discussion
\[Ti_{s}\to Ti^{2+}+2e^{-} \tag{5}\]0  
\[Ti_{s}\to Ti^{2+}+2e^{-} \tag{5}\]1  
\[Ti_{s}\to Ti^{2+}+2e^{-} \tag{5}\]2  
\[Ti_{s}\to Ti^{2+}+2e^{-} \tag{5}\]3  
By substituting Equations (14) and (16) into Equation (12), it is possible to obtain the velocity equation for the photoelectrochemical reaction, as shown in Equation (17). This can be further simplified (Equation (18)) by considering two generalized constants: the electrochemical constant \(K_{el}\) and the photoelectrochemical constant \(K_{fel}\). \[Ti_{s}\to Ti^{2+}+2e^{-} \tag{5}\]4  
\[Ti_{s}\to Ti^{2+}+2e^{-} \tag{5}\]5  
Finally, using the integration method, Equation (19) is obtained, which represents the concentration of the degraded dye as a function of time (Equation (19)). Equations (20) and (21) describe key parameters in the kinetic modeling of dye degradation in the photoelectrochemical reactor. Equation (20) relates the global efficiency constant (\(k_{el}\)) to the product of the first-order and second-order oxidation constants (\(k_{ox1}\) and \(k_{ox2}\)) and the adjusted efficiency constant (\(K_{d}\)'). This highlights the role of oxidation processes in determining efficiency. Equation (21) defines the global degradation constant (\(k_{deg}\)) as the product of the first-order oxidation constant (\(k_{ox1}\)) and the second-order reduction constant (\(k_{rad2}\)), scaled by the fast electron transfer constant (\(k_{fel}\)). It emphasizes the interplay between oxidation and reduction reactions in the degradation process. These equations provide insights into the reaction mechanisms and contribute to optimizing operational parameters. \[Ti_{s}\to Ti^{2+}+2e^{-} \tag{5}\]6  
\[Ti_{s}\to Ti^{2+}+2e^{-} \tag{5}\]7  
\[Ti_{s}\to Ti^{2+}+2e^{-} \tag{5}\]8  
The experimental data were fitted using Equations (8) and (19) for processes without and with ultraviolet light imposition, respectively. In Figure 3, the fraction of degraded dye over time is shown, obtaining values for the electrochemical process (without UV light)of the \(K_{el}\)' constant as 22.39 \(\pm\) 0.49 mL \(\times\) L\({}^{-1}\) of dye, with an adjustment of experimental data to the mathematical model of 0.9893. Meanwhile, for the process with UV light, \(K_{el}\)' presents values of 27.2 \(\pm\) 3.3 mL \(\times\) L\({}^{-1}\), and \(K_{fel}\) presents values of 25.78 \(\pm\) 3.7 mL \(\times\) L\({}^{-1}\), with an adjustment of experimental data to the mathematical model of 0.9360. As observed, the value of K\({}_{el}\)' is like that obtained with the electrochemical process, and the value of \(K_{fel}\) is of a similar magnitude. Further analysis reveals significant differences in the oxidation and reduction constants for both processes. For the process without UV light, the first-order oxidation constant k\({}_{ox1}\) was 0.035 \(\pm\) 0.004 mL \(\times\) L\({}^{-1}\), while the second-order oxidation constant \(k_{ox2}\) reached 22.36 \(\pm\) 0.49 mL \(\times\) L\({}^{-1}\). Similarly, the reduction constant \(K_{rel2}\) was determined to be 23.5 \(\pm\) 6.2 s\({}^{-1}\). The overall degradation rate constant \(k_{deg}\) was 21.2 \(\pm\) 4.0 s\({}^{-1}\), highlighting the efficiency of the process under these conditions. The incorporation of UV light enhanced the degradation process, as evidenced by the increased values of \(k_{el}\) and \(k_{fel}\), suggesting that UV light facilitates additional photochemical pathways or promotes faster electron transfer at the surface. These results underscore the importance of ultraviolet light in improving the efficiency of photoelectrochemical reactors for dye degradation. Overall, the combination of kinetic and efficiency parameters confirms the effectiveness of the proposed reactor design and provides a robust foundation for optimizing operational conditions. Future studies could explore additional variations in operational parameters, such as current density and light intensity, to further improve the reactor's performance. The photoelectrochemical reactor offers the advantage of rapid degradation kinetics at the start of the process.