## 3 Results and Discussion
### Parameters Estimation

Table 4: Statistical analysis of cathodic Tafel slope \(\text{b}_{\text{H}_{2}}\) [\(\text{V}/\text{dec}\)] for valeric acid and acetic acid at 20 \({}^{\circ}\)C, 40 \({}^{\circ}\)C, and 60 \({}^{\circ}\)C.  
Figure 3: Cathodic Tafel slope as function of pH and temperature for valeric acid solutions (**a**,**c**) and acetic acid solutions (**b**,**d**). The experimental cathodic exchange current density \(\mathrm{i_{0,H_{2}}}\) was derived and its dependance on pH and temperature was evaluated and compared to the predictions of the model. In particular, the complete statistical analysis of the experimental and predicted data for valeric and acetic acids is reported in Table 6, where the mean values (\(\upmu\)), the interquartile range values (IQR), the mean variation in the logarithm of \(\mathrm{i_{0,H_{2}}}\) with pH (\(\partial\mathrm{log_{10}}(\mathrm{i_{0,H_{2}}})/\partial\mathrm{pH}\)), the 95% confidence interval of the results (CI), and the error metrics MAE and RMSE are specified. A graphical representation of the results is shown in Figure 5 for the case of valeric acid, and Figure 6 for the case of acetic acid. Figure 4: Anodic Tafel slope as function of pH and temperature for valeric acid solutions (**a,c**) and acetic acid solutions (**b,d**). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Acid**} & **T** & \multirow{2}{*}{**Theoretical \(\mathrm{b_{Fe}}\)**} & \multirow{2}{*}{\(\mu_{\mathrm{b_{Fe}}}\)} & \multirow{2}{*}{\(\mathrm{IQR_{b_{Fe}}}\)} & \multirow{2}{*}{**MAE**} & \multirow{2}{*}{**RMSE**} \\  & [\({}^{\circ}\)C] & & & & & \\ \hline \multirow{2}{*}{Valeric (\(\mathrm{C_{5}}\mathrm{H_{10}}\mathrm{O_{2}}\))} & 20 & +0.039 & +0.045 & [+0.044, +0.047] & +0.006 & +0.007 \\  & 40 & +0.041 & +0.044 & [+0.044, +0.046] & +0.003 & +0.004 \\  & 60 & +0.044 & +0.047 & [+0.045, +0.048] & +0.003 & +0.003 \\ \hline \multirow{2}{*}{Acetic} & 20 & +0.039 & +0.046 & [+0.045, +0.046] & +0.007 & +0.007 \\  & 40 & +0.041 & +0.046 & [+0.045, +0.047] & +0.005 & +0.005 \\ \cline{1-1}  & 60 & +0.044 & +0.046 & [+0.045, +0.047] & +0.002 & +0.002 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Statistical analysis of anodic Tafel slope \(\mathrm{b_{Fe}}\) [V/dec] for valeric and acetic acids at 20 \({}^{\circ}\)C, 40 \({}^{\circ}\)C, and 60 \({}^{\circ}\)C.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multirow{3}{*}{**Acid**} & \multirow{3}{*}{**T [’C]**} & \multicolumn{3}{c}{**Experimental**} & \multicolumn{6}{c}{**T-P Model**} \\ \cline{2-13}  & & \multicolumn{3}{c}{**Temperature**} & \multicolumn{3}{c}{**pH**} & \multicolumn{3}{c}{**Temperature**} & \multicolumn{3}{c}{**pH**} \\  & & \multicolumn{3}{c}{**Dependence**} & \multicolumn{3}{c}{**Dependence**} & \multicolumn{3}{c}{**Dependence**} & \multicolumn{3}{c}{**Dependence**} \\ \cline{3-13}  & & \(P_{\text{high}_{\text{high}_{\text{high}}}}\) & \(\text{IQR  
Starting with valeric acid, Figure 5 reports the regression lines of the experimental log\({}_{10}\)(i\({}_{0,\text{H}_{2}}\)) with respect to pH as defined in (19) at 20, 40, and 60 \({}^{\circ}\)C and compares them to the estimations of the Tafel-Piontelli model. As can be seen, the experimental cathodic exchange current density shows an exponential decay with pH. This behavior is expected, since a reduction in pH corresponds to an increase in the solution acidity and therefore in the concentration of hydrogen evolution reagents, moving the equilibrium of the reaction towards the dissolution of iron and pushing forward the corrosion process. The Tafel-Piontelli model is able to account for this dynamic behavior, as can be seen from the almost parallel regression lines between the experimental and predicted results, that tends to get closer and closer with the increase in temperature.