Butler-Volmer Equation describes the relationship between the current density at an electrode and the electrode potential (specifically the overpotential, which is the deviation from the equilibrium potential). It considers both the forward (oxidation or reduction) and reverse reactions and incorporates parameters like the exchange current density (the rate of reaction at equilibrium) and the charge transfer coefficient (which reflects the symmetry of the activation energy barrier).

The simplified form of the Butler-Volmer equation is:

$$ i=i_0\left(\exp \left(\frac{\alpha_n n F_\eta}{R T}\right)-\exp \left(\frac{-\alpha_c n F \eta}{R T}\right)\right) $$

where:

• $i$ is the current density
• $i_0$ is the exchange current density
• $\alpha_a$ and $\alpha_c$ are the anodic and cathodic transfer coefficients
• $n$ is the number of electrons transferred
• $F$ is the Faraday constant
• $\eta$ is the overpotential $\left( E - E _{\text {eq }}\right)$
• $R$ is the ideal gas constant
• $T$ is the absolute temperature

Tafel Equation: At sufficiently high overpotentials, one of the exponential terms in the Butler-Volmer equation dominates, leading to the Tafel equation. This equation shows a linear relationship between the logarithm of the current density and the overpotential. The slope of this linear region (the Tafel slope) provides information about the reaction mechanism.

The Tafel equation can be expressed as:

$$ \eta=a+b \log |i| $$

where:

• $\eta$ is the overpotential
• $a$ is the Tafel intercept
• $b$ is the Tafel slope

🧠Example: For the Butler-Volmer Equation (finding $E_{\text {corr }}$ ):