## 3 Results
### Kinetics and Comparison of Energy Requirements for Indigo Carmine  
The degradation of Indigo Carmine in an aqueous solution through advanced oxidation processes followed first-order reaction kinetics (Figure 9, Table 2). The exception was the ozonation process after introducing the catalyst at natural pH, where the reaction order was second (Figure 9d). The reaction rate constant \(k\) varied significantly depending on the technology applied. The highest value for the process using ultraviolet radiation (\(k=0.1018\,\mathrm{min}^{-1}\)) was achieved using the UV + TiO\({}_{2}\) process with a catalyst dosage of \(1\,\mathrm{g/L}\). Reducing the catalyst dosage five-fold resulted in a two-fold reduction in the reaction rate constant, while completely eliminating the catalyst decreased \(k\) by over thirty times. In the studies by [PERSON] and [PERSON] (2022) ([PERSON] and [PERSON], 2022), a five-fold reduction in catalyst dosage from \(1\,\mathrm{g/L}\), under UV irradiation with an intensity of \(0.5\,\mathrm{mW/cm^{2}}\), resulted in a nearly 1.5-fold decrease in the \(k\) value. The ozonation process was effective in removing Indigo Carmine (\(k=0.08\,\mathrm{min}^{-1}\)). This efficiency is attributed to the generation of hydroxyl radicals, which attack the aromatic rings and sulfonic groups in the dye's structure, leading to its breakdown. However, this process can result in the formation of refractory intermediates that require further oxidation stages ([PERSON] and [PERSON], 2022). After introducing the catalyst and oxidant into the system, the process reached the highest rate constant at the level of \(k=0.7691\,\mathrm{min}^{-1}\) (pH = 3). When the process was carried out in a natural pH, the reaction rate constant was more than four times lower. Despite the lower value of the reaction rate constant, the process was characterized by high efficiency, taking second place in terms of efficiency among all the analyzed methods. Indigo Carmine exhibits moderate susceptibility to photodegradation under UV radiation, mainly due to the stability of the aromatic rings in its structure. Degradation becomes more effective in the presence of oxidants such as hydrogen peroxide (H\({}_{2}\)O\({}_{2}\)) or ozone (O\({}_{3}\)), which generate highly reactive \(\cdot\)OH radicals. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**AO** & **UV** & \(\mathbf{UV+TiO_{2}}\) & \(\mathbf{UV+TiO_{2}}\) & \(\mathbf{O_{3}}\) & \(\mathbf{O_{3}+5-mgS}\) & \(\mathbf{O  
Processes utilizing ultrasound (sonication) were characterized by a low efficiency in removing Indigo Carmine. This is attributed to the ionic form of the dye present in the solution, which limits interactions with the generated radicals. The correlation coefficient (R\({}^{2}\)), indicating the fit of the function to the experimental data, is above 0.92 in all cases, confirming the validity of the assumed kinetic equations for the analyzed decolorization processes of Indigo Carmine. The dye removal process is fast in the initial phase and then slows down. In the initial phase, the high concentration of Indigo Carmine causes \(\cdot\)OH radicals to react quickly with its molecules. As the concentration of Indigo Carmine decreases, the reaction rate slows down in accordance with the first-order reaction kinetics. As the reaction progresses, intermediate products may compete with Indigo Carmine for \(\cdot\)OH radicals, further slowing the process. The generated intermediate products may be more stable and more difficult to degrade further. In energy-intensive water treatment processes, such as ozonation, ultrasound (US), or photochemical reactions induced by UV radiation, key parameters include exposure to ozone, UV, and ultrasound as well as the degree of reduction in the contaminant. These parameters can be expressed through the indicator of electrical energy consumption per order of pollutant removal (_EE/O_).