The corrosion rate is to be determined for some divalent metal M in a solution containing hydrogen ions. The following corrosion data are known about the metal and solution: \begin{tabular}{rr} \hline \multicolumn{1}{c}{ For Metal \(M\)} & For Hydrogen \\ \hline\(V_{\left(M M^{2}+\right)}=-0.47 \mathrm{~V}\) & \(V_{\left(\mathrm{H}^{+} / H_{2}\right)}=0 \mathrm{~V}\) \\ \(i_{0}=5 \times 10^{-10} \mathrm{~A} / \mathrm{cm}^{2}\) & \(i_{0}=2 \times 0^{-9} \mathrm{~A} / \mathrm{cm}^{2}\) \\ \(\beta=+0.15\) & \(\beta=-0.12\) \\ \hline \end{tabular} (a) Assuming that activation polarization controls both oxidation and reduction reactions, determine the rate of corrosion of metal \(\mathrm{M}\left(\mathrm{in} \mathrm{mol} / \mathrm{cm}^{2} \cdot \mathrm{s}\right)\) (b) Compute the corrosion potential for this reaction.

Electrochemical corrosion is a common problem where metal surfaces deteriorate due to chemical reactions with their environment. It occurs when a metal forms a voltaic cell with its environment, resulting in the flow of electrons from the metal to the electrolyte. This process typically involves an anodic reaction, where the metal is oxidized and loses electrons, and a cathodic reaction, where a reduction occurs, and electrons are gained.

Let’s take a divalent metal M, for instance. In an environment containing hydrogen ions, the metal may corrode via an electrochemical reaction. Oxidation happens at the metal's surface, releasing metal ions into solution and electrons into the metal. These electrons then travel to another spot on the metal, or to another electrode, where they reduce hydrogen ions into hydrogen gas. As this process continues, the metal is gradually consumed.

The Tafel equation is crucial for understanding the kinetics of electrochemical reactions during corrosion. It describes the relationship between the overpotential, which is the potential deviation from the equilibrium potential, and the current density. The equation is given by:
\[\begin{equation}\eta = \beta \log\left(\frac{i}{i_0}\right)\end{equation}\]
where \(\eta\) is the overpotential, \(\beta\) is the Tafel slope (indicative of the rate of the reaction), \(i\) is the current density, and \(i_0\) is the exchange current density (a measure of the reaction rate when there is no net current).

In corrosion science, both the anodic and cathodic branches of the Tafel plot are used to extrapolate the corrosion current density. This allows us to estimate the rate of corrosion without having to measure it directly in an aggressive environment, instead using controlled experimental conditions.

The Nernst equation is another cornerstone of electrochemistry that is employed to calculate the equilibrium potential of an electrode reaction based on concentration. It is an expression for the potential of a chemical reaction to move towards equilibrium. Mathematically, the Nernst equation is written as:\[\begin{equation}E = E^\circ - \frac{RT}{nF} \ln\left(\frac{[\text{products}]}{[\text{reactants}]}\right)\end{equation}\]
where \(E\) is the electrode potential, \(E^\circ\) is the standard electrode potential, \(R\) is the gas constant, \(T\) is the temperature in Kelvin, \(n\) is the number of moles of electrons, and \(F\) is the Faraday’s constant.

In corrosion calculations, we often use a simplified form of this equation to find the corrosion potential. By comparing the corrosion potential to other potentials, one can infer how noble or active a metal is under specific conditions; the more negative the potential, the more active the metal behaves.

Corrosion current density is a critical parameter in electrochemical corrosion, representing the amount of electrical current per unit area that passes through the electrolyte as a result of corrosion. It is denoted by \(i_{\text{corr}}\) and is usually measured in amperes per square centimeter (A/cm²).

The corrosion current density is directly related to the rate of metal dissolution in the case of anodic reactions, or the rate of reduction reaction in cathodic processes. In other words, it tells us how quickly a metal is corroding in a given environment. By using the previously mentioned Tafel equation and experimental data, we can calculate this important value.

Once the corrosion current density is known, it can be converted to the corrosion rate by considering the number of electrons transferred and the Faraday constant. This provides a quantitative measurement of how much metal is lost over time due to corrosion, which is essential for predicting the longevity and durability of metal structures in corrosive environments.