Tafel Approximation

The Tafel approximation simplifies the Butler-Volmer equation for large overpotentials, where one reaction direction (anodic or cathodic) dominates. It is used to analyze kinetic parameters in electrochemistry, such as in cyclic voltammetry (e.g., Nicholson-Shain theory). The Tafel equation for the cathodic reaction is:

\[ \eta = -a - b \log_{10}(|j|) \]

where \( \eta = E - E^0 \) is the overpotential (V), \( j \) is the current density (A/cm²), \( a = \frac{2.303 R T}{\alpha n F} \log_{10}(j_0) \), \( b = \frac{2.303 R T}{\alpha n F} \), \( j_0 \) is the exchange current density, \( \alpha \) is the transfer coefficient, \( n \) is the number of electrons, \( F = 96485 \, \text{C/mol} \), \( R = 8.314 \, \text{J/(mol·K)} \), and \( T \approx 298 \, \text{K} \).

Step 1: Start with the Butler-Volmer Equation

The Butler-Volmer equation describes the net current density for a redox reaction (\( \text{O} + n e^- \leftrightarrow \text{R} \)):

\[ j = j_0 \left[ \exp\left( \frac{(1-\alpha) n F \eta}{R T} \right) - \exp\left( -\frac{\alpha n F \eta}{R T} \right) \right] \]

The first term represents the anodic current (oxidation), and the second term represents the cathodic current (reduction).

Step 2: Large Overpotential Assumption

For the cathodic reaction (large negative \( \eta \)), the anodic term (\( \exp\left( \frac{(1-\alpha) n F \eta}{R T} \right) \)) becomes negligible because \( \eta < 0 \) makes the exponent large and negative. Thus, the Butler-Volmer equation simplifies to:

\[ j \approx -j_0 \exp\left( -\frac{\alpha n F \eta}{R T} \right) \]

For the anodic reaction (large positive \( \eta \)), the cathodic term becomes negligible, yielding:

\[ j \approx j_0 \exp\left( \frac{(1-\alpha) n F \eta}{R T} \right) \]

The negative sign for the cathodic current reflects the convention that reduction currents are negative.

Step 3: Deriving the Tafel Equation

Focus on the cathodic case. Take the natural logarithm of the simplified equation:

\[ \ln(|j|) = \ln(j_0) - \frac{\alpha n F \eta}{R T} \]

Convert to base-10 logarithm (\( \ln(x) = 2.303 \log_{10}(x) \)):

\[ 2.303 \log_{10}(|j|) = 2.303 \log_{10}(j_0) - \frac{\alpha n F \eta}{R T} \]

Rearrange to solve for \( \eta \):

\[ \eta = -\frac{2.303 R T}{\alpha n F} \log_{10}(j_0) - \frac{2.303 R T}{\alpha n F} \log_{10}(|j|) \]

This is the Tafel equation:

\[ \eta = -a - b \log_{10}(|j|) \]

where:

\( a = \frac{2.303 R T}{\alpha n F} \log_{10}(j_0) \) (Tafel constant),
\( b = \frac{2.303 R T}{\alpha n F} \) (Tafel slope).

Step 4: Tafel Slope and Parameters

The Tafel slope \( b \) depends on \( \alpha \), \( n \), and \( T \). At \( T = 298 \, \text{K} \):

\[ b = \frac{2.303 \cdot 8.314 \cdot 298}{\alpha n \cdot 96485}\] \[ b \approx \frac{0.059}{\alpha n} \, \text{V/decade} \]

For example, if \( \alpha = 0.5 \), \( n = 1 \), then \( b \approx 0.118 \, \text{V/decade} \). The exchange current density \( j_0 \) is found from the intercept \( a \).

Step 5: Applications

The Tafel approximation is used to analyze electrochemical kinetics in:

Cyclic Voltammetry: Provides the boundary condition for irreversible reactions (e.g., in Nicholson-Shain theory), where the cathodic term dominates at large negative overpotentials.
Corrosion: Tafel plots (\( \eta \) vs. \( \log_{10}(|j|) \)) determine corrosion rates and \( j_0 \).
Electrocatalysis: Evaluates catalyst performance by measuring Tafel slopes.

The linear relationship in a Tafel plot allows extraction of \( \alpha \) and \( j_0 \), key parameters for understanding reaction kinetics.

Tafel Approximation