## 3 Results and Discussion
### Photocatalytic Performance  
The initial concentration (\(C_{0}\)) and final concentration (\(C\)) of colorant values were obtained from the absorbance spectra of intensity versus wavelength. Afterwards, the Nb\({}_{2}\)O\({}_{5}\) and Mn-doped Nb\({}_{2}\)O\({}_{5}\) photocatalytic performance was evaluated through kinetics degradation; colorant concentration, as a function of exposure time under UV-visible radiation, was analyzed. The analyzed reaction has a first order kinetics [23], as shown in Equation (2). \[v=\frac{dC}{dt}kC \tag{2}\]  
\(k\) represents the exchange rate constant, obtained by integration (2)  
\[ln\frac{C}{C_{0}}=-kt \tag{3}\]  
Thus, the ln\(C/C_{0}\) graph versus \(t\) has a lineal tendency with slope at \(k\). The \(C/C_{0}\) values were calculated and their tendency is represented in Figure 10. The lineal fitting was performed and the \(k\) values are presented in Table 4. \begin{table}
\begin{tabular}{l c c c c} \hline
**Temperature** & **1\%** & **2.5\%** & **5\%** & **10\%** \\ \hline \multicolumn{5}{c}{Hydrodynamic dimension (nm)} \\
500 \({}^{\circ}\)C & 404.2 & 349.9 & 410.1 & 396.8 \\
600 \({}^{\circ}\)C & 419.2 & 739.4 & 473.2 & 338.0 \\
700 \({}^{\circ}\)C & 554.6 & 375.4 & 516.4 & 456.1 \\ \multicolumn{5}{c}{\(Z\) potential (mV)} \\
500 \({}^{\circ}\)C & \(-\)25.7 & \(-\)24.9 & \(-\)23.6 & \(-\)22.6 \\
600 \({}^{\circ}\)C & \(-\)34.8 & \(-\)31.4 & \(-\)29.0 & \(-\)27.9 \\
700 \({}^{\circ}\)C & \(-\)41.8 & \(-\)40.9 & \(-\)32.5 & \(-\)30.6 \\ \hline \end{tabular}
\end{table} 
Table 3: Hydrodynamic stability parameters of samples subjected to 700 \({}^{\circ}\)C for all samples with the addition of Mn. Figure 9: Photoluminescence (PL) spectra of Mn-doped Nb\({}_{2}\)O\({}_{5}\) powders at different Mn concentration annealed 700 \({}^{\circ}\)C.  
The linear fitting had correlation parameters greater than 0.95, which confirms the first order kinetics behavior. The Nb\({}_{2}\)O\({}_{5}\):Mn (5.0 wt.%) sample is the most photoactive of the doped samples; it showed significant photoactivity in Rhodamine B degradation. However, its response in the Rhodamine B photodegradation is comparable to the response of Nb\({}_{2}\)O\({}_{5}\) un-doped. The Nb\({}_{2}\)O\({}_{5}\):Mn (5.0 wt.%) sample has the Nb\({}_{2}\)O\({}_{5}\) orthorhombic phase, which makes it a good photocatalyst. This latter occurs even though the sample shows the presence of secondary phase MnNb\({}_{2}\)O\({}_{6}\). The MnNb\({}_{2}\)O\({}_{6}\) compound had previously evidenced good photocatalytic performance in combination with other systems [24], so that it can act as co-agent in the photocatalytic process. The Nb\({}_{2}\)O\({}_{5}\):Mn (5.0 wt.%) micrography (Figure 5) distinguishes particles of size minor as compared to the un-doped, 1.0% and 2.5% doped samples. Therefore, the Mn-doped Nb\({}_{2}\)O\({}_{5}\) photocatalytic performance strongly depends on reaching small particle size. In addition to the surface level, the sample modified at 5.0% presents specific surface area and pore size values suitable for the process of Rhodamine B photodegradation. Finally, the band gap energy value of this sample is lesser than the same Nb\({}_{2}\)O\({}_{5}\) value. This indicates that the reduction of this parameter together with the morphological and surface properties of the Nb\({}_{2}\)O\({}_{5}\):Mn (5.0 wt.%) sample was a good photocatalyst to complete the Rhodamine B degradation. On the other hand, the Nb\({}_{2}\)O\({}_{5}\):Mn (1.0 wt.%) sample evidenced better photocatalytic response due to its structural properties; however, its surface characteristics were not enough to exhibit good degradation of pollutant analyzed. The Nb\({}_{2}\)O\({}_{5}\):Mn (10 wt.%) sample exhibited similar behavior. It showed good morphological, surface and optical properties, with the minor energy band gap value, but since its crystallinity was poor, its photocatalytic response was not good. The Nb\({}_{2}\)O\({}_{5}\):Mn (10 wt.%) sample has a lower crystallinity degree and a greater number of defects.