## 3 Results and Discussion
The rate law written in a differential form was used to describe the progress of the photocatalytic reaction, which follows first-order kinetics and is given by the following:  
\[\ln\frac{c}{c_{0}}=-K_{p}t \tag{6}\]  
where \(c_{0}\) and \(c\) are the concentrations of RB19 at the beginning and at time \(t\) of the photocatalytic reaction, respectively, and \(K_{p}\) is the first-order constant of photocatalytic reaction rate. The photocatalytic reaction rate, defined as the amount of RB19 degraded in an infinitely short time interval, can be found as the slope of the plot showing the dependence of \(\ln\frac{c}{c_{0}}\) on time (Figure 10). For short reaction times, the slope can be considered linear. The results are presented in Table 3.  
It can be seen that modification of TiO\({}_{2}\) increased the photocatalytic reaction rates. The highest value was obtained in the case of Cu and Zr/Cu introduction as a result of the highest decrease in band gap coupled with the longest carrier lifetimes; thus, the photocatalytic reaction rates are related to the slopes of the curves presented in Figure 10.  
The results summarized in Table 3 show clear differences at the 95% confidence level between the slopes of regression lines for all samples (TiO\({}_{2}\)/Cu (0.05%), TiO\({}_{2}\)/Zr (1%) and TiO\({}_{2}\)/Cu (0.05%) Zr (1%)). This confirms that the modification of TiO\({}_{2}\) with Cu (0.05%), Zr (1%) and Zr (1%)Cu (0.05%) accelerates the photocatalytic degradation process of RB19, as indicated by the photocatalytic reaction rate constants presented in Table 3. The results are in line with  
Figure 10: Decay profiles for first-order photocatalytic reactions during simulated solar light irradiation of RB19 aqueous solutions in presence of unmodified TiO\({}_{2}\) and Cu- and/or Zr-modified TiO\({}_{2}\) materials. \begin{table}
\begin{tabular}{c c} \hline \hline
**Sample** & 1**Kp 1, min\({}^{-1}\)** \\ \hline TiO\({}_{2}\) & 0.039 \(\pm\) 0.001 \\ TiO\({}_{2}\)/Cu (0.05\%) & 0.054 \(\pm\) 0.002 \\ TiO\({}_{2}\)/Zr (1%) & 0.044 \(\pm\) 0.001 \\ TiO\({}_{2}\)/Cu (0.05\%) Zr (1%) & 0.063 \(\pm\) 0.002 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Photocatalytic reaction rate constants for different TiO\({}_{2}\) materials. those presented in our previous paper [26]. The photocatalytic reaction rates correspond to a 99.5% degradation efficiency as calculated previously using same experimental data [26]. Moreover, these results indicate an improvement considering that the maximum efficiencies (98%, 100%) were achieved before with 20% longer (120 min) irradiation times [27]. As expected, the decrease in the band gap by addition of 0.05% Cu increased the reaction rate since such materials are able to absorb a larger portion of the simulated solar light. Further improvement of the efficiency of the photocatalyst in degradation of RB19 dye was achieved by modification with 1% Zr, which increased the charge carrier lifetime. It has to be noted that despite the insignificant effect of Zr on the material's band gap, as discussed above, the TiO\({}_{2}\)/Zr (1%) material showed an increased photocatalytic efficiency, reflected in the 10% increase in the reaction rate constant as compared to pure TiO\({}_{2}\). This might be attributed in part also to the change in the specific surface of the material (50.7 m\({}^{2}\)/g), which might also play some role in other investigated materials. The particle size of the pure TiO\({}_{2}\) crystallite material did not change substantially when introducing Cu and Zr particles, as demonstrated in a previous study [28]. Their average size lies around 25 nm for both unmodified and modified samples. The addition of Zr to the TiO\({}_{2}\) lattice results in an increase in the material surface area. This in turn enables better pollutant adsorption on the material and consequently more efficient photodegradation [26; 27; 28; 29] and thus improves the photocatalytic properties.