## 3 Results and Discussion
### Reactive Azo Dye Oxidation
#### 3.2.5 Kinetics and Thermodynamics  
In this section, the reaction kinetics and thermodynamics of Levafix dye oxidation via the photo-Fenton-modified system are investigated. In the current study, the photo-Fenton oxidation kinetics of Levafix dye using a modified reaction, the [PERSON]'s earth-based photo-Fenton process, was evaluated for various contact times varying from 0 to 20 min under isothermal conditions at the following operating temperatures: 25, 40, 50, and 60 \({}^{\circ}\)C. Moreover, the zero-, first-, and second-order reaction kinetics were assessed for the modified Fenton oxidation reaction according to the following equations: Equation (11) for the zero-, Equation (12) for the first-, and Equation (13) for the second-order reaction kinetics [31]:  
\[\mathrm{C_{t}}=\mathrm{C_{o}}-\mathrm{k_{o}t} \tag{11}\]  
\[\mathrm{C_{t}}=\mathrm{C_{o}}-\mathrm{e^{k_{t}t}} \tag{12}\]  
\[\left(\frac{1}{\mathrm{C_{t}}}\right)=\left(\frac{1}{\mathrm{C_{0}}}\right)- \mathrm{k_{2}t} \tag{13}\]  
where C is the concentration of the Levafix dye; \(\mathrm{C_{t}}\) is the concentration of the Levafix dye at time t; \(\mathrm{C_{o}}\) is the Levafix dye's initial concentration; t is the reaction time; and \(\mathrm{k_{0}}\), \(\mathrm{k_{1}}\), and \(\mathrm{k_{2}}\) represent the kinetic rate constants for the zero-, first-, and second-order reaction kinetics, respectively. The reaction kinetics most appropriate for Levafix removal were assessed by plotting Equations (11)-(13) for the experimental results data. The kinetic parameters, as well as the regression coefficients (\(R^{2}\)), for each reaction order were investigated, and the data are tabulated in Table 3. Examining the data in Table 3 revealed that the reaction is well-fitted to a second-order reaction. Further, the kinetics constant of the second-order reaction constant, \(\mathrm{k_{2}}\), was notably affected by the reaction temperature, increasing with the temperature elevation. This investigation is associated with the generation of the 'OH species since it is a product of the reaction of [PERSON]'s earth with hydrogen peroxide. Moreover, another kinetics value of importance is \(\mathrm{t_{1/2}}\) (the half-life of a reaction), which signifies the essential time needed for the reactant's initial concentration to decrease by half. An examination of Table 3 revealed that the calculated \(\mathrm{t_{1/2}}\) is a function of the reaction temperature; \(\mathrm{t_{1/2}}\) declines with increasing temperature. Various researchers in other studies have confirmed that the Fenton reaction follows second-order reaction kinetics. To fully understand the modified Fenton reaction based on the use of Fullers' earth oxidation to oxidize Levafix dye molecules, thermodynamic parametric data were quantified. Arrhenius formula was used to investigate the activation energy, \(k_{2}=Ae^{\frac{-E_{a}}{R^{2}}}\), with the activation energy (\(\mathrm{E_{a}}\)) of the Levafix dye oxidation being based on the second-order kinetic constant, where R is the gas constant (8.314 J mol\({}^{-1}\)K\({}^{-1}\)), T is the temperature in kelvin, and A is the pre-exponential factor that is considered to be constant with respect to temperature [32]. Taking the natural log of the Arrhenius formula yields the following:  
\[\mathrm{ln}\,\mathrm{k_{2}}=\mathrm{ln}\,\mathrm{A}-\frac{\mathrm{E_{a}}}{ \mathrm{RT}} \tag{14}\]
A plot of \(\ln\) k\({}_{2}\) versus 1/T could be used to investigate the value of E\({}_{\rm a}\) (Figure 11). Figure 11 displays the relationship of the Levafix Dark Blue dye oxidation through the modified [PERSON]'s earth-based photo Fenton system reaction. The value of E\({}_{\rm a}\) of the process was recorded to be 50.8 K J mol\({}^{-1}\).