## 2 Materials and Methods
### Mathematical Modeling  
In order to evaluate the corrosion behavior of the system, a theoretical model, called the Tafel-Piontelli model, was proposed and developed, as extensively described in [51; 52; 53; 54]. The model leverages the fundamental kinetic laws of corrosion to define an estimate of the corrosion rate of active metals exposed to stagnant acidic solutions. Considering the overpotential \(\eta\), the Tafel slope of the reaction b, the current density i, and the exchange current density i0, the Tafel law (\(\eta=b\cdot\log\left(\frac{\mathrm{i}}{\mathrm{i}_{0}}\right)\)) is applied to the anodic process of iron dissolution (\(\mathrm{Fe_{(s)}}\to 2\mathrm{e^{-}}+\mathrm{Fe_{(aq)}^{2+}}\)) and the cathodic process of hydrogen evolution (\(2\mathrm{H_{(aq)}^{+}}+2\mathrm{e^{-}}\to\mathrm{H_{2(g)}}\)). This allows us to determine the corrosion current density of the whole process, defined as the meeting point between the anodic curve (\(\mathrm{E}=\mathrm{E_{eq,Fe}}+\mathrm{b_{Fe}}\cdot\log\mathrm{i}/\mathrm{i_{ 0,Fe}}\)) and cathodic curve (\(\mathrm{E}=\mathrm{E_{eq,H_{2}}}-\mathrm{b_{H_{2}}}\cdot\log\mathrm{i}/\mathrm{ i_{0,H_{2}}}\)), where the equilibrium potentials of the reactions (\(\mathrm{E_{eq,Fe}},\mathrm{E_{eq,H_{2}}}\)) are defined via the Nernst equation as follows:  
\[\mathrm{E_{eq,Fe}}=\mathrm{E_{0,Fe}}+\frac{\mathrm{RT}}{\mathrm{z_{Fe^{2+}}Flog \,e}}\cdot\log\left[\mathrm{Fe^{2+}}\right] \tag{1}\]  
\[\mathrm{E_{eq,H_{2}}}=-\frac{\mathrm{RT}_{0}}{\mathrm{z_{H^{+}}Flog\,e}}\cdot \mathrm{pH} \tag{2}\]  
A graphical representation of this approach is reported in the Evans diagram in Figure 2.  
The final application of the Faraday law (\(\text{CR}=\frac{\text{M-k}_{\text{form}}}{2\text{F}\rho}\)) allows us to estimate the uniform corrosion rate of the metal, reported here in the case of iron. \[\text{CR}=\frac{\text{M}_{\text{Fe}}}{\text{z}_{\text{Fe}^{2+}}\cdot\text{F} \cdot\text{P}_{\text{Fe}}}\text{i}_{0,\text{H}_{2}}^{\frac{\text{b}_{\text{H}_ {2}}}{\text{H}_{2}}\cdot\text{k}_{\text{Fe}}}\cdot\text{i}_{0,\text{Fe}}^{ \frac{\text{b}_{\text{Fe}}}{\text{H}_{2}}\cdot\text{k}_{\text{Fe}}}\cdot 10^{ \frac{\text{b}_{\text{Fe}}\text{H}_{2}}{\text{H}_{2}}-\text{k}_{\text{Fe}}\text{ P}_{\text{Fe}}}}, \tag{3}\]  
where \(\text{M}_{\text{Fe}}\) is the iron molecular weight, \(\text{i}_{\text{corr}}\) is the corrosion current density, \(\text{z}_{\text{Fe}^{2+}}\) is the iron valence, F is the Faraday constant, and \(\text{\rho}_{\text{Fe}}\) is the iron mass density. The kinetic parameters of the corrosion process (i.e., the exchange current densities \(\text{i}_{0,\text{H}_{2}}\) and \(\text{i}_{0,\text{Fe}}\) and the Tafel slopes \(\text{b}_{\text{H}_{2}}\) and \(\text{b}_{\text{Fe}}\) of the cathodic and anodic reactions) have been defined through a mechanistic approach, as extensively described in [54]. In particular, the cathodic exchange current density is defined as follows:  
\[\text{i}_{0,\text{H}_{2}}=\text{z}_{\text{H}^{+}}\cdot\text{F}\cdot\left[\text{ H}^{+}\right]^{\text{m}_{\text{H}_{2}}}\cdot\text{k}_{\text{H}_{2}}^{\text{c}} \cdot\text{e}^{-\frac{\Delta\text{H}_{\text{H}_{2}}}{\text{K}}(\frac{1}{1}- \frac{1}{\tau_{0}})}, \tag{4}\]  
where \(\text{z}_{\text{H}^{+}}\) is the valence of the hydrogen evolution reaction, F is the Faraday constant, \(\left[\text{H}^{+}\right]\) is the surface concentration of hydrogen ions in solution, \(\text{m}_{\text{H}_{2}}\) is the cathodic reaction order with respect to pH, \(\text{k}_{\text{H}_{2}}^{\text{c}}\) is the cathodic reaction rate constant at standard temperature, and \(\Delta\text{H}_{\text{H}_{2}}\) is the activation enthalpy of the cathodic reaction. The cathodic Tafel slope is defined as follows:  
\[\text{b}_{\text{H}_{2}}=\frac{\text{RT}\cdot\text{ln}\,10}{\alpha_{\text{H}_{ 2}}\text{F}}, \tag{5}\]  
where \(\alpha_{\text{H}_{2}}\) is the cathodic charge transfer coefficient.