Tafel Equation: Overpotential and Exchange Current
The Tafel equation is a core concept in electrochemical kinetics tested in advanced examinations including CSIR NET, GATE Chemistry, and IIT JAM. It describes the logarithmic relation between the overpotential applied to an electrode and the resulting net current density during charge-transfer controlled reactions. Mastering exchange current density, transfer coefficients, and Tafel slopes is essential for analysing electrode processes, corrosion mechanisms, and fuel-cell kinetics.
Exam details
- Source: practice problem written for this article (not a past-paper question)
- Relevant exams: CSIR NET Chemical Sciences, GATE Chemistry, IIT JAM, BARC OCES, CUET-PG
- Topic: Electrode Kinetics and Tafel Overpotential
- Difficulty: Advanced
Overpotential and the Butler-Volmer Origin
At equilibrium, forward anodic and backward cathodic reaction rates are balanced, yielding zero net current. This dynamic equilibrium rate is defined by the exchange current density (j0). When the electrode potential (E) departs from the reversible equilibrium potential (Eeq), an overpotential (η = E − Eeq) develops. Current density (j) follows the Butler-Volmer equation:
j = j0 × [exp((1 − α) × F × η / (R × T)) − exp(−α × F × η / (R × T))]
Here α is the cathodic charge transfer coefficient, (1 − α) is the anodic transfer coefficient, F is the Faraday constant (96485 C/mol), R is the gas constant (8.3145 J/(mol·K)), and T is absolute temperature. At high overpotential (|η| > 52 mV at 298.15 K), the opposing reaction becomes negligible, yielding the linear Tafel relationship.
The Tafel Equation and Tafel Slope
The Tafel equation relates overpotential linearly to the base-10 logarithm of current density.
Tafel Equations:
General expression: η = a + b × log10(j)
Anodic form (η > 0): ηa = [2.3026 × R × T / ((1 − α) × F)] × log10(ja / j0)
Cathodic form (η < 0): ηc = −[2.3026 × R × T / (α × F)] × log10(|jc| / j0)
Where:
• η = Overpotential in volts (V) or millivolts (mV).
• j = Net current density in A/cm2 or mA/cm2.
• j0 = Exchange current density in A/cm2 or mA/cm2.
• α = Charge transfer coefficient (dimensionless, typically 0.5).
• b = Tafel slope in V/decade or mV/decade. At 298.15 K, 2.3026 × R × T / F = 0.05916 V (59.16 mV). For α = 0.5, b = 118.3 mV/decade.
Worked Numerical Examples
Question: An anodic electrochemical reaction proceeds at 298.15 K with exchange current density j0 = 2.00 × 10−5 A/cm2 and anodic transfer coefficient (1 − α) = 0.50. Calculate the anodic overpotential ηa required to produce an operating current density j = 8.00 × 10−2 A/cm2.
Step 1: Compute the anodic Tafel slope (ba).
ba = 2.3026 × R × T / ((1 − α) × F)
ba = (2.3026 × 8.3145 × 298.15) / (0.50 × 96485) = 5708.2 / 48242.5 = 0.11832 V/decade = 118.32 mV/decade
Step 2: Compute current ratio and logarithm.
j / j0 = (8.00 × 10−2) / (2.00 × 10−5) = 4000.0
log10(4000.0) = log10(4) + 3 = 0.60206 + 3.00000 = 3.60206
Step 3: Calculate overpotential ηa.
ηa = 0.11832 V × 3.60206 = 0.4262 V = 426.2 mV
Double-check by natural logarithm:
ηa = [R × T / ((1 − α) × F)] × ln(j / j0) = (2478.98 / 48242.5) × ln(4000.0) = 0.051386 V × 8.29405 = 0.4262 V = 426.2 mV. Consistent.
Answer: 0.426 V (426.2 mV)
Question: During cathodic reduction at 298.15 K, two steady-state measurements give overpotential η1 = −0.150 V at j1 = 2.50 mA/cm2, and η2 = −0.270 V at j2 = 25.0 mA/cm2. Determine the cathodic Tafel slope (bc), transfer coefficient (α), and exchange current density (j0).
Step 1: Calculate the cathodic Tafel slope (bc).
bc = (η2 − η1) / [log10(j2) − log10(j1)] = [−0.270 − (−0.150)] / log10(25.0 / 2.50) = −0.120 V / 1.000 decade = −0.120 V/decade
Step 2: Calculate the transfer coefficient (α).
|bc| = 2.3026 × R × T / (α × F) = 0.05916 V / α
α = 0.05916 V / 0.120 V = 0.493
Step 3: Calculate exchange current density (j0).
η1 = −0.120 × log10(j1 / j0) ⇒ −0.150 = −0.120 × log10(2.50 / j0)
log10(2.50 / j0) = 1.250 ⇒ 2.50 / j0 = 101.250 = 17.783
j0 = 2.50 mA/cm2 / 17.783 = 0.1406 mA/cm2 = 1.41 × 10−4 A/cm2
Double-check using second point:
−0.270 = −0.120 × log10(25.0 / j0) ⇒ log10(25.0 / j0) = 2.250 ⇒ j0 = 25.0 / 177.83 = 0.1406 mA/cm2. Consistent.
Answer: α = 0.493; j0 = 1.41 × 10−4 A/cm2 (0.141 mA/cm2)
Common Mistakes to Avoid:
• Mixing natural and base-10 logarithms: With log10, use the 2.3026 pre-factor (2.3026 RT / αF = 0.05916 V / α at 298.15 K). With ln, use RT / αF.
• Confusing transfer coefficients: The anodic slope uses (1 − α), while the cathodic slope uses α.
• Inconsistent units: Always ensure j and j0 use identical units (A/cm2 or mA/cm2) before taking ratios.
• Applying Tafel equation at low overpotential: When |η| < 20 mV, use the linear Butler-Volmer approximation (j ∝ η), not the logarithmic form.
| Competitive Examination | Core Concepts Tested | Typical Question Format |
|---|---|---|
| CSIR NET Chemical Sciences | Calculation of exchange current density, transfer coefficient, and high overpotential approximations. | Multi-step numerical and graphical slope interpretation problems. |
| GATE Chemistry | Determination of Tafel slopes, overpotentials for gas evolution, and corrosion rates. | Numerical answer type (NAT) questions requiring exact decimal precision. |
| IIT JAM Chemistry | Definitions of overpotential, activation energy of charge transfer, and exchange current density. | Multiple-choice and numerical questions on electrode kinetics fundamentals. |
| CUET-PG Chemistry | Limiting forms of Butler-Volmer equation, definition of Tafel slope, and temperature dependence. | Direct multiple-choice questions testing formulas and definitions. |
Previous-year CSIR-NET questions and timed quizzes to check what you just read.
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