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Lesson 6 · 30 min

Overpotential and electrode kinetics

Why does evolving hydrogen on mercury cost more than a volt beyond platinum? Draw a Tafel plot from Tafel’s original 1905 data, extract the transfer coefficient and exchange current density, and use the Butler–Volmer equation to see how potential controls the rate of an electrode reaction.

In this lesson

  • Explain overpotential: how far an electrode must be pushed from its equilibrium potential to make a reaction run at a given rate
  • Describe current against overpotential with the Butler–Volmer and Tafel equations, η = a + b lg j, and extract α and j₀ from a Tafel plot
  • Use the differences in hydrogen overpotential between metals to explain what happens in electrolysis, electroplating and lead–acid batteries

The Nernst equation gives an electrode’s potential at equilibrium — but at equilibrium the net current is zero and nothing is made. To make electrolysis actually run, the electrode has to be pushed away from equilibrium, and the further it is pushed, the faster the reaction goes. That extra potential is the overpotential η.

In 1905 Julius Tafel in Würzburg systematically measured the potentials at which hydrogen is evolved on cathodes of different metals. He found two things. First, metals differ enormously: evolving hydrogen on mercury costs more than a volt beyond what platinised platinum needs. Second, on any one metal, each tenfold increase in current raises the overpotential by an almost fixed amount. The second finding is today’s Tafel equation, the starting point of electrode kinetics.

Predict first

Electrolysing 2 N sulfuric acid at a mercury cathode, the current density is raised from 0.001 to 0.01 A/cm² (tenfold). Roughly how does the cathode overpotential change?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

How potential changes the activation energy

Reducing H⁺ to an H atom adsorbed on the metal, the electron must cross an energy barrier. Every Δφ by which the electrode potential is made more negative raises the electron’s energy by FΔφ per mole; a fraction α of this — the transfer coefficient — lowers the activation energy for reduction, and the rest raises it for oxidation. With Arrhenius-type rates, the currents in the two directions each vary exponentially with overpotential, and their difference is the net current: the Butler–Volmer equation.

Two parameters set how “fast” an electrode reaction is. The exchange current density j₀ is the current flowing in each direction at equilibrium — equilibrium is not stillness but two opposing currents of j₀ that cancel. α sets how steeply current grows with potential and is close to 0.5 for most one-electron steps.

j=j0[eαaFη/RT−e−αcFη/RT],∣η∣=a+b lg⁡∣j∣,b=2.303 RTαFj = j_0\left[e^{\alpha_\mathrm{a} F\eta/RT} - e^{-\alpha_\mathrm{c} F\eta/RT}\right],\qquad |\eta| = a + b\,\lg|j|,\quad b = \frac{2.303\,RT}{\alpha F}
When |η| is well above RT/F (about 25 mV) only one term survives and the Tafel equation follows; extrapolating to η = 0 gives j₀. For small η, j ≈ j₀Fη/RT and the electrode behaves like a resistance RT/(Fj₀).
j0j_0
exchange current density
α\alpha
transfer coefficient
η\eta
overpotential E − E_eq
bb
Tafel slope (per decade of current)
j = j₀[exp(α_a fη) − exp(−α_c fη)] · one electron, 12 °C

The Butler–Volmer equation: the rate of an electrode reaction

Presets
Near equilibrium: current against overpotential
−60−3003060η / mV−3−2−10123j / (mA/cm²)
Large overpotential: lg|j| against overpotential (Tafel region)
0.1 A/cm² −1.5−1−0.500.511.5η / V−14−10−6−22lg(|j| / (A/cm²))

Cathodic Tafel slope b = 0.113 V per decade; charge-transfer resistance RT/(Fj₀) = 25 Ω cm²; evolving hydrogen at 0.1 A/cm² needs η = −0.23 V.

The model covers only charge transfer at the electrode surface — no mass-transport limit and no ohmic drop in the solution; for a one-electron step α_a + α_c = 1. The mercury preset is the straight-line fit to Tafel’s runs 5–8 between 0.0004 and 0.04 A/cm². The platinum preset assumes α_c = 0.5 and back-calculates j₀ from the 0.07 V overpotential at 0.1 A/cm² in Tafel’s Table 12; it is only an order-of-magnitude estimate, because at such large j₀ real currents are usually limited by hydrogen diffusion and solution resistance.
Press “mercury”, then “platinised platinum”: their j₀ differ by more than eleven orders of magnitude. Evolving hydrogen at 0.1 A/cm² needs about 1.3 V on mercury and under 0.1 V on platinum. Lower α and the Tafel line flattens while the overpotential needed grows.

The metal as catalyst

Such different values of j₀ show that the electrode is not just a wire but a catalyst: hydrogen evolution first adsorbs H atoms on the metal, which then combine to H₂. Binding H too weakly (mercury, lead, cadmium) or too strongly is unfavourable; the platinum metals are just right. Platinised platinum also has a huge surface area, so its overpotential is almost zero — which is why it serves as the standard hydrogen electrode.

A high overpotential is useful too. When a lead–acid battery is charged, the large hydrogen overpotential on lead lets PbSO₄ be reduced to lead first; on a mercury cathode hydrogen evolution is so difficult that the chlor-alkali industry once used it to reduce Na⁺ to sodium amalgam.

Tafel’s 1905 Table 12 · 0.1 A/cm², 2 N sulfuric acid, about 12 °C

Same hydrogen, different metals: how much extra voltage

Hydrogen overpotential (cathode)

Mercury: to evolve hydrogen at 0.1 A/cm², the cathode must sit 1.30 V more negative than a reversible hydrogen electrode (ε = 1.97 V against mercury–mercurous sulfate) — 1.23 V more than platinised platinum.

Each mole of H₂ takes 2 mol of electrons, so this extra voltage costs 2Fη = 251 kJ, all of it turned into heat.

Values are Tafel’s time-maximum readings at a constant 0.1 A/cm²; π = ε − 0.674 V. On copper, nickel, gold and bright platinum the potential kept drifting for hours, so those values are less reliable than mercury, lead and cadmium; the value for silver is queried in the original. Overpotential is not a material constant: it changes with current density, surface condition, temperature and solution.
Tafel compared 11 cathodes at the same current density. Click each metal to see how much extra energy each mole of H₂ costs.

Worked example

Reading a Tafel plot

On a mercury cathode at 12 °C, η = 1.078 V at 0.001 A/cm² and 1.185 V at 0.01 A/cm². Find the Tafel slope, the transfer coefficient and the order of magnitude of the exchange current density.

Concept check

Why does electrolysing dilute sulfuric acid with a lead cathode need a much higher voltage than with a platinum cathode?

Try the calculation

A one-electron reaction at 25 °C has α = 0.5. What is its Tafel slope in mV per decade of current?

mV

Key takeaways

  • The overpotential η = E − E_eq is the extra potential needed to drive an electrode reaction at a finite rate; it is lost as heat.
  • Butler–Volmer: j = j₀[exp(α_a fη) − exp(−α_c fη)]; for |η| ≫ RT/F it gives the Tafel equation |η| = a + b lg|j| with b = 2.303RT/(αF).
  • The exchange current density j₀ measures how catalytic an electrode is: about 10⁻¹³ A/cm² for Tafel’s mercury, which needs 1.30 V to evolve hydrogen at 0.1 A/cm², against 0.07 V on platinised platinum.
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Data and references

  1. J. Tafel, “Über die Polarisation bei kathodischer Wasserstoffentwicklung”, Z. Phys. Chem. 50, 641–712 (1905): Tables 2, 3 and 12.
  2. J. A. V. Butler, “Studies in heterogeneous equilibria. Part II. The kinetic interpretation of the Nernst theory of electromotive force”, Trans. Faraday Soc. 19, 729–733 (1924); T. Erdey-Grúz, M. Volmer, Z. Phys. Chem. 150A, 203–213 (1930).
  3. S. Trasatti, “Work function, electronegativity, and electrochemical behaviour of metals: III. Electrolytic hydrogen evolution in acid solutions”, J. Electroanal. Chem. 39, 163–184 (1972).
  4. A. J. Bard, L. R. Faulkner, Electrochemical Methods, 2nd ed., Wiley (2001), ch. 3.