# Kinetics of Corrosion - the Tafel Equation

## Tafel equation

Armed with the new Arrhenius expression and the generalised reaction:

M ⇌ M^{z+} + ze,

where M is a metal that forms M^{z+} ions in solution, we can now derive an equation describing corrosion kinetics.

Consider the rate of the anodic (oxidation, corrosion) reaction, k_{a}

\[{k_{\rm{a}}} = k_{\rm{a}}^{'}\;\exp \left( {\frac{{ - \Delta G^0}}{{RT}}} \right)\]

Since the reaction involves the release of electrons, its progress can be expressed as a current density, i (current per unit area).

The **exchange** current density, i_{0} is defined as the current flowing in both directions per unit area when an electrode reaction is at equilibrium (and, hence, at its equilibrium potential).

If i_{0} is small, then little current flows and the reactions at dynamic equilibrium are generally slow. Likewise, a high i_{0} gives a fast reaction. The metal itself affects the value of i_{0}, even if the reaction does not involve the metal directly.

\[{i_0} = z{\rm{F}}{k_{\rm{a}}} = z{\rm{F}}k_{\rm{a}}^{'}\;\exp \left( {\frac{{ - \Delta G^0}}{{RT}}} \right)\]

If overpotential is applied, the activation energy is changed, as described on the previous page:

\[{i_{\rm{a}}} = i_{0}\;\exp \left( {\frac{{\alpha z{\rm{F}}\eta }}{{RT}}} \right)\]

This is one form of the **Tafel equation**.

The Tafel equation can also be written in several equivalent ways, as shown here.

The quantity \(\frac{{2.303\,RT}}{{\alpha z{\rm{F}}}}\) is given the symbol *b*_{a} and is known as the anodic Tafel slope. It has units of volts per decade of current.

Similarly, if the cathodic reaction were to be considered, the quantity would be \(\frac{{ - 2.303\,RT}}{{(1 - \alpha )z{\rm{F}}}}\) since (1 − α) is applicable instead of α and E - E_{e} is negative. This quantity is the cathodic Tafel slope, b_{c}. .

The usual form of Tafel’s equation is η = *a* + *b*_{a} log *i*_{a} where \(a = \frac{{ - 2.303\,RT}}{{\alpha z{\rm{F}}}}\;\log {i_0}\)

Through consideration of the reaction as both a chemical and electrical process and manipulation of algebra, we have found that the applied potential is proportional to the log of the resulting corrosion current. This is certainly different to Ohmic behaviour where applied potential is directly proportional to the resulting current.