5.2 Butler-Volmer Equation and Tafel Analysis

The Butler-Volmer equation is a cornerstone of electrochemical kinetics. It describes how current density relates to electrode potential, considering both oxidation and reduction reactions. This equation is crucial for understanding the behavior of electrochemical systems and predicting reaction rates.

Tafel analysis, derived from the Butler-Volmer equation, is a practical tool for extracting kinetic parameters from experimental data. By examining the linear regions of Tafel plots, researchers can determine exchange current densities and transfer coefficients, providing insights into reaction mechanisms and electrode performance.

Butler-Volmer Equation

Derivation of Butler-Volmer equation

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• Describes relationship between current density and electrode potential in electrochemical systems considers both oxidation and reduction reactions
• Assumes rate of electron transfer is the rate-determining step
• Derivation steps:
  1. Start with Arrhenius equation for forward and backward reaction rates
  2. Introduce activation energy concept and its dependence on electrode potential
  3. Apply Nernst equation to relate electrode potential to standard electrode potential and concentrations of reactants and products
  4. Combine forward and backward reaction rates to obtain net current density
• Resulting Butler-Volmer equation: $i = i_0 [\exp(\frac{\alpha_a nF\eta}{RT}) - \exp(\frac{-\alpha_c nF\eta}{RT})]$
  • $i$: current density
  • $i_0$: exchange current density
  • $\alpha_a$ and $\alpha_c$: anodic and cathodic transfer coefficients
  • $n$: number of electrons transferred
  • $F$: Faraday's constant
  • $\eta$: overpotential
  • $R$: universal gas constant
  • $T$: absolute temperature
• Equation implies current density depends exponentially on overpotential higher overpotentials lead to larger current densities (non-linear relationship due to exponential terms)

Branches of Butler-Volmer equation

• Consists of two exponential terms representing anodic and cathodic branches
• Anodic branch: $i_a = i_0 \exp(\frac{\alpha_a nF\eta}{RT})$
  • Describes oxidation reaction
  • Current density increases exponentially with increasing overpotential
• Cathodic branch: $i_c = -i_0 \exp(\frac{-\alpha_c nF\eta}{RT})$
  • Describes reduction reaction
  • Current density decreases exponentially with increasing overpotential
• At equilibrium ($\eta = 0$), anodic and cathodic current densities are equal in magnitude but opposite in sign, resulting in zero net current
• Relative contributions of anodic and cathodic branches depend on overpotential and transfer coefficients

Tafel Analysis

Tafel analysis for kinetic parameters

• Method to extract kinetic parameters from experimental current-potential data
• At high overpotentials, one exponential term in Butler-Volmer equation becomes negligible
  • Anodic branch dominates at high positive overpotentials: $i \approx i_0 \exp(\frac{\alpha_a nF\eta}{RT})$
  • Cathodic branch dominates at high negative overpotentials: $i \approx -i_0 \exp(\frac{-\alpha_c nF\eta}{RT})$
• Taking logarithm of simplified equations yields Tafel equations:
  • Anodic Tafel equation: $\log i = \log i_0 + \frac{\alpha_a nF}{2.303RT}\eta$
  • Cathodic Tafel equation: $\log |i| = \log i_0 - \frac{\alpha_c nF}{2.303RT}\eta$
• Plotting $\log i$ vs. $\eta$ (Tafel plot) results in linear regions at high overpotentials
  • Slope of linear region is Tafel slope, $b = \frac{2.303RT}{\alpha nF}$
  • y-intercept of linear region is $\log i_0$
• From Tafel slope and exchange current density, transfer coefficient and number of electrons transferred can be determined

Overpotential and Butler-Volmer relationship

• Overpotential ($\eta$): difference between applied potential and equilibrium potential
  • $\eta = E - E_{eq}$
  • Represents additional potential required to drive reaction at certain rate
• Types of overpotential:
  1. Activation overpotential: related to energy barrier for electron transfer
  2. Concentration overpotential: arises from mass transport limitations
  3. Ohmic overpotential: caused by resistance of electrolyte and electrode materials
• Butler-Volmer equation relates current density to overpotential
  • Higher overpotentials lead to larger current densities (exponential relationship described by anodic and cathodic branches)
• Exchange current density ($i_0$): current density at zero overpotential
  • Measure of intrinsic rate of electron transfer
  • Higher $i_0$ values indicate faster kinetics and lower activation barriers
• Transfer coefficients ($\alpha_a$ and $\alpha_c$): describe symmetry of energy barrier
  • Determine relative contributions of anodic and cathodic branches to overall current density
  • Typically, $\alpha_a + \alpha_c = 1$, often assumed to be 0.5 for symmetric barriers