## 3 Results and Discussion
### Parameters Estimation
The analysis of the cathodic exchange current density in the case of acetic acid provides similar results, with an almost parallel trend of the experimental and predicted regression lines of the log\({}_{10}\)(i\({}_{0,\text{H}_{2}}\)) with respect to the pH of the electrolyte. In particular, like in the case of valeric acid, the model tends to slightly underestimate the exchange current density at lower temperatures, and it improves its performance at higher temperatures with experimental and predicted data that almost overlap at 60 \({}^{\circ}\)C, as can be seen in Figure 6.  
Overall, the Tafel-Piontelli model is able to capture the exponential dependance of the cathodic exchange current density on pH across the different acids and temperatures considered. Finally, the analysis of the anodic exchange current density i\({}_{0,\text{Fe}}\) was performed via comparison between the experimental and predicted data, with a complete statistical analysis of the temperature and pH dependance for valeric and acetic acids reported in Table 7.  
Figure 5: Comparison between experimental and predicted cathodic exchange current density exponential decay with respect to pH in valeric acid solutions at 20 \({}^{\circ}\)C (a), 40 \({}^{\circ}\)C (b), and 60 \({}^{\circ}\)C (c). Figure 6: Comparison between experimental and predicted cathodic exchange current density exponential decay with respect to pH in acetic acid solutions at 20 \({}^{\circ}\)C (a), 40 \({}^{\circ}\)C (b), and 60 \({}^{\circ}\)C (c). \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{3}{*}{**Acid**} & \multirow{3}{*}{**T [’C]**} & \multicolumn{4}{c}{**Experimental**} & \multicolumn{4}{c}{**T-P Model**} \\ \cline{2-11}  & & \multicolumn{2}{c}{**Temperature**} & \multicolumn{2}{c}{**pH**} & \multicolumn{2}{c}{**Temperature**} & \multicolumn{2}{c}{**pH**} \\  & & \multicolumn{2}{c}{**Dependence**} & \multicolumn{2}{c}{**Dependence**} & \multicolumn{2}{c}{**Dependence**} & \multicolumn{2}{c}{**Dependence**} \\ \cline{3-11}  & & \(\mathrm{H}_{\mathrm{H}}\mathrm{e}_{\mathrm{H}}\  
The experimental results suggest that i0,Fe depends on temperature, with a behavior that is partially captured by the Tafel-Piontelli model, with MAE and RMSE relatively high across the different temperatures for both valeric and acetic acids. For what concerns the dependance on the pH of the electrolyte, the model estimates of the anodic exchange current density do not take into account possible variations of i0,Fe with pH. Nonetheless, the empirical evidence suggests a more complex picture, with distinct patterns of pH dependence for the different acids and temperatures considered. This behavior is illustrated by the trends of the log\({}_{10}\)(10,Fe) with respect to pH at 20, 40, and 60 \({}^{\circ}\)C for valeric acid in Figure 7 and for acetic acid in Figure 8, that also include the trends predicted by the Tafel-Piontelli model. While valeric acid displays an increasing trend with pH, acetic acid displays a decreasing trend. The complex interaction between pH, temperature, and i0,Fe highlights the necessity for models that can effectively account for both pH and temperature influences to accurately predict anodic exchange current densities, emphasizing the crucial role of comprehensive statistical analysis in understanding and modeling electrochemical systems.