A rotating disk electrode: Levich and Koutecký–Levich

A platinum disk spinning in 1 mM Fe³⁺ and Fe²⁺ (in 1 M KCl with 0.1 M HCl): the steady current against potential at five rotation rates, and the analysis an electrochemist would run on them. The Levich plot takes the limiting currents; Koutecký–Levich plots take the transport out; and a Tafel plot of what's left gives back the electrode's rate constant k⁰ and transfer coefficient α. Vary k⁰ and the potential the Koutecký–Levich plot is read at. (More about the model below.)

About this simulation

A spinning disk pulls solution up to its face and flings it outward, and the flow thins the layer the ions must diffuse across to the same thickness everywhere on the disk, Levich's δ = 1.61 D^1/3ν^1/6ω^−1/2. So the limiting current grows as √ω: the Levich plot, i_lim against ω^1/2, is a line through the origin whose slope gives D. Where the kinetics are slow, even the plateau at E° − 0.30 V falls short of the limit, more so at the faster rates, and the Levich plot bends (try a small k⁰). At any potential the current is the kinetics and the transport in series, 1/i = 1/i_k + (1/i_l,c + e^Fη/RT/i_l,a)/(1 − e^Fη/RT), with i_l,c and i_l,a the cathodic and anodic limiting currents, both as ω^1/2. So 1/i against ω^−1/2 is a straight line (the Koutecký–Levich plot), its slope Levich's toward the plateau and steeper near E°, where the back reaction counts, and its intercept the current the electrode would pass with no transport limit at all. Those intercepts, one per potential and corrected for the back reaction, i_k/(1 − e^Fη/RT) with equal Fe³⁺ and Fe²⁺ (the Allen–Hickling form of the Tafel plot), fall on a line of slope −αF/RT (in log₁₀, 118 mV per decade for α = ½) that meets η = 0 at the exchange current Fk⁰c. The Levich slope gives D back: here D_Fe³⁺ = 6.04 × 10⁻⁶ cm²/s, the value the model uses (D_Fe²⁺ = 7.19 × 10⁻⁶).

The page runs that analysis on the simulated currents alone, as on measured ones: the plateaus for the Levich plot, a straight-line fit per potential across the five rotation rates, and a line through the corrected intercepts where i_k is at most ten times the fastest disk's plateau. Without noise that cut isn't needed: the simulation's intercepts stay within 0.06% of the true kinetic current however far past it they go (the grey points; its small migration error goes into the slopes, not the intercepts). The cut is for measured data, where an intercept far past it is the small difference of big numbers. Tick "0.5% noise" (and draw again with "new noise"): over many draws an intercept scatters by about 10% at ten times the plateau, and by its own size or more at a hundred, and k⁰ comes back within about 1% for a slow electrode, 4% at 0.01 cm/s and 12% at 0.03 cm/s (medians over draws), where a third of the draws leave fewer than the three potentials a line needs. Potentials within 20 mV of E° are left out too: there the correction divides by nearly zero. So when k⁰ is large the readout calls the electrode too fast for the disk: the rule speaking, for measured data. The readout gives what comes back against what went in.

The model stands one stagnant layer of Levich's thickness for Fe³⁺ in for the flow. The flux to the disk is then Levich's (the disk is uniformly accessible), though the profile across the layer is a straight line where the real one curves, and Levich's formula itself runs about 3% above a real disk at this Schmidt number (Newman's correction). Fe²⁺'s layer would be 6% thicker (δ goes as D^1/3); here they share one, so the anodic plateau is 1.19 times the cathodic, where a real disk's would be 1.12. The lines are the exact steady solution of this model, i = F(k_fc − k_bc)/(1 + k_fδ/D_Fe³⁺ + k_bδ/D_Fe²⁺), less migration, which the supporting salt keeps to 0.1% (it scales with the iron's concentration). The line through the intercepts inside the cut still gives k⁰ and α back within 0.05%. The solution is acidified, as Fe³⁺ must be (above about pH 2 it hydrolyses and precipitates). The potentials are on an ideal-solution scale, from table μ°: E° = 0.77 V, where the formal potential in 1 M HCl is nearer 0.70 V. k⁰ is in cm/s; driftlet's k0 is that times its 1 M reference concentration. ν = 0.89 mm²/s, water's at 25 °C.