## 2 Materials and Methods
### Electrolyte Speciation  
The activity of iron ions in a solution is a fundamental element in the estimation of the corrosion rate. In order to define it, a mechanistic model for the general case of an n-protic acid was developed by coupling the charge balance condition with the chemical equilibrium condition between the iron, iron salts, and acid.  
The acid dissociation, iron salt dissolution, and water self-ionization reactions are reported in Table 3, assuming that the infinite dilution hypothesis is valid and therefore all the activity coefficients are unitary.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \(\text{z}_{\text{H}^{+}}\) & \(\text{z}_{\text{Fe}}\) & \(\alpha_{\text{H}_{2}}\) & \(\alpha_{\text{Fe}}\) & \(m_{\text{H}_{2}}\) & \(m_{\text{Fe}}\) & \(k_{\text{H}_{2}}^{\circ}\) & \(k_{\text{Fe}}^{\circ}\) & \(\Delta H_{\text{H}_{2}}\) & \(\Delta H_{\text{Fe}}\) & \(T_{\text{0}}\) \\ \hline Value & 1 & 2 & 0.5 & 0.5 & 0.5 & 0.0 & \(7.58\times 10^{-8}\) & \(2.05\times 10^{-7}\) & 30.0 & 29.5 & 20 \\ \hline Units & - & - & - & - & - & - & \(\text{mol/m}^{2}\cdot\text{s}\cdot\) & \(\text{mol/m}^{2}\cdot\text{s}\cdot\) & \(\text{mol/m}^{2}\cdot\text{s}\cdot\) & kJ/mol & kJ/mol & \({}^{\circ}\text{C}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: The coefficients used in the kinetic parameters of the Tafel–Piontelli model.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Reaction** & **Equilibrium Equation** \\ \hline \(\text{H}_{\text{n}}\text{A}_{\text{(aq)}}+\text{H}_{\text{2}}\text{O}_{\text{(l)}} \rightleftharpoons\text{H}_{\text{3}}\text{O}_{\text{(aq)}}^{+}+\text{H}_{ \text{n-1}}\text{A}_{\text{(aq)}}^{-}\) & \(\text{K}_{\text{a,1}}=\frac{\text{k}_{\text{a,l}}}{\text{k}_{\text{a,l}}}=\frac{ \left[\text{H}_{\text{n-1}}\text{A}^{-}\right]\left[\text{H}^{+}\right]}{\left[ \text{H}_{\text{a}}\text{A}\right]}\) & (9) \\ \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) \\ \(\text{H}_{\text{n+1-i}}\text{AFrom the electroneutrality condition  
\[2\left[\mathrm{Fe^{2+}}\right]+\left[\mathrm{H^{+}}\right]=\left[\mathrm{OH^{-}} \right]+\sum_{i=1}^{n}i\cdot\left[\mathrm{H_{n-i}A^{i-}}\right] \tag{14}\]  
it is possible to derive the activity of iron ions in solution for unsaturated solutions.  
\[\left[\mathrm{Fe^{2+}}\right]=\frac{\left(\sum_{i=1}^{n}\left[\mathrm{H^{+}} \right]^{n-i}\cdot i\cdot\prod_{j=1}^{i}K_{j}\right)\left[\mathrm{H_{n}A} \right]+K_{w}\left[\mathrm{H^{+}}\right]^{n-1}-\left[\mathrm{H^{+}}\right]^{n +1}}{2\left[\mathrm{H^{+}}\right]^{n}}. \tag{15}\]  
In the case of the saturated solution, considering the iron salt dissolution reaction constant \(\mathrm{K_{sp}}\), the activity of iron ions becomes the following:  
\[\left[\mathrm{Fe^{2+}}\right]=\left(\frac{\mathrm{K_{sp}}}{\left[\mathrm{A^{ n-}}\right]^{2}}\right)^{\frac{1}{n}}=K_{sp}^{\frac{3+(-1)n}{2n}}\left(\frac{ \left[\mathrm{H^{+}}\right]^{2n}}{\prod_{i=1}^{n}K_{a,i}^{2}\cdot\left[ \mathrm{H_{n}A}\right]^{2}}\right)^{\frac{1}{n}}. \tag{16}\]  
In both cases, \(\left[\mathrm{Fe^{2+}}\right]\) is a function of the concentration of hydrogen ions \(\left[\mathrm{H^{+}}\right]\) and therefore pH, the concentration of acid \(\left[\mathrm{H_{n}A}\right]\), the acid dissociation constants \(\mathrm{K_{a,j}}\), and the water self-dissociation constant \(\mathrm{K_{w}}\).  
The water self-dissociation constant is equal to \(\mathrm{K_{w}}=1.0\times 10^{-14}\frac{\mathrm{mol}^{2}}{\mathrm{L^{2}}}\), while the dissociation constant of valeric acid is equal to \(\mathrm{K_{a,v}}=1.44\times 10^{-5}\frac{\mathrm{mol}}{\mathrm{L}}\) and the dissociation constant of acetic acid is equal to \(\mathrm{K_{a,ac}}=1.75\times 10^{-5}\frac{\mathrm{mol}}{\mathrm{L}}\).