## 2 Materials and Methods
### Mathematical Model
Table 3 shows the values of the estimated effects of each factor and some interactions, according to a linear statistical model for a wavelength of 1060 and 1456 cm\({}^{-1}\)[20], as well as the associated standard errors, which measure the sampling error. These are based on the total error with eight degrees of freedom. The effects of each of the factors are shown, as well as of certain contrasts (linear combinations) of their interactions. Because the design was fractionated, the effects of the variable interactions are confused in pairwise sums. In this case, all the estimated effects of the factors interacted. They represent the deviations from the mean between the upper and lower levels for each of them. For example, A is the mixed interaction between the Factor FN plus BD and CE, whereas B is the mixed interaction between the Factor FC plus the interaction between A and D. According to these results, this system has complex levels of interaction between the factors. Phenomenologically speaking, these results show the interactions between the primary factors and the combinations between them, which means that the variables interact in the bioprocess. From a physical point of view, they have interaction nodes, which is common in bioprocesses, since metabolic, physicochemical, and thermodynamic aspects converge. Consequently, the traditional polynomial mathematical models provided in statistical software do not fit the actual behavior of the system. The signs of the estimated effects indicate the level of the control variable. These also apply to interactions because the analysis of these is not direct. It is about contrasts instead of separate interactions. #### 3.1.2 \(t\)-test and Analysis

After estimating the effects of the factors, the significant factors affecting the efficiency of the bioleaching process were determined by Student's \(t\)-test, which allows assessing whether the calculated effects are significantly different from zero. If the absolute value of the standardized effect is greater than the value of t, with a probability of \(1-\alpha/2\) and with the same degrees of freedom of the error, then this effect is statistically significant. The Pareto charts are shown in Figures 2 and 3, where the factors and their interactions are ordered according to the absolute values of their standardized effects. \begin{table}
\begin{tabular}{l c c} \hline \hline \multirow{2}{*}{**Factor**} & **Factor Bioleaching of Gold Using** & **Factor Bioleaching of Gold Using** \\  & **the Infrared Spectrum at 1060 cm\({}^{-1}\)** & **the Infrared Spectrum at 1456 cm\({}^{-1}\)** \\ \hline \hline  & Estimated \(\pm\) standard error & Estimated \(\pm\) standard error \\ \hline Average & 0.0749 \(\pm\) 0.00061 & 0.0647 \(\pm\) 0.000441 \\ A: FN + BD + CE & 0.0133 \(\pm\) 0.00122 & 0.0020 \(\pm\) 0.000883 \\ B: FC + AD & 0.0255 \(\pm\) 0.00122 & 0.0064 \(\pm\) 0.000883 \\ C: FK + AE & \(-\)0.0093 \(\pm\) 0.00122 & \(-\)0.0016 \(\pm\) 0.000883 \\ D: Trace + AB & 0.0043 \(\pm\) 0.00122 & \(-\)0.0009 \(\pm\) 0.000883 \\ E: Mixing + AC & 0.0118 \(\pm\) 0.00122 & 0.0018 \(\pm\) 0.00883 \\ BC + DE & \(-\)0.0046 \(\pm\) 0.00122 & \(-\)0.0006 \(\pm\) 0.000883 \\ BE + CD & 0.0047 \(\pm\) 0.00122 & 0.0003 \(\pm\) 0.000883 \\ \hline \hline \end{tabular} The first order factors were: FN: N Source; FC: C Source; FK: K Source; Trace; Mixing and other combinations.

\end{table} 
Table 3: Estimated Effects for the \(2^{5-2}\) Fractional Factorial. For the bioleaching of gold using the infrared spectrum at a wavenumber of 1060 cm\({}^{-1}\), all the contrasts are statistically significant, while, for the bioleaching of gold using the infrared spectrum at a wavenumber 1456 cm\({}^{-1}\), they were not significant. The contrasts B: FC + AD are statistically representative. First-order factors are not presented, demonstrating the nonlinearity behavior and the influence of strong interactions between the bioprocess variables.