Randles-Sevcik Equation: Peak Current vs Scan Rate in Voltammetry

What the Randles-Sevcik equation is and where it applies

The Randles-Sevcik equation quantifies the height of the current peak seen when you sweep an electrode's potential linearly through a redox couple's formal potential. As the potential passes E°′, the electroactive species is oxidized or reduced at the surface; the current rises, peaks, then falls as the near-electrode region becomes depleted and diffusion can no longer keep pace with the sweep.

It applies specifically to a reversible (Nernstian), diffusion-controlled electron transfer at a planar electrode, with the redox species freely dissolved in a well-supported electrolyte. Key assumptions:

• Fast, reversible electron transfer (kinetics never limit the current).
• Semi-infinite linear diffusion — no convection, negligible migration (excess supporting electrolyte), no ohmic drop.
• A flat electrode large compared with the diffusion layer, so edge/spherical effects are ignored.

Where it matters: measuring diffusion coefficients, determining the number of electrons n, quantifying analyte concentration, characterizing new redox mediators, battery and fuel-cell materials, and screening electrocatalysts. It is arguably the single most-cited equation in practical electrochemistry.

Deriving it step by step from Fick's second law

The derivation starts with mass transport. The flux of the electroactive species obeys Fick's second law: ∂C/∂t = D · ∂²C/∂x², with C the concentration, x the distance from the electrode, and D the diffusion coefficient.

• Boundary condition at the surface: because the couple is reversible, the surface concentrations of oxidized and reduced forms track the applied potential through the Nernst equation, E = E°′ + (RT/nF) ln([O]s/[R]s). As E is ramped, the potential-dependent factor is θ = exp[(nF/RT)(E − E°′)].
• Sweep: the potential is E(t) = Ei − v·t, so time and potential are interchangeable through the scan rate v.
• Solution: substituting the moving boundary into the diffusion equation gives an integral (a convolution / Abel-type integral) that has no closed form. Randles and Ševčík solved it numerically.

The result: the current is i = nFAC(πDσ)^(1/2)·χ(σt), where σ = nFv/RT and χ is a tabulated dimensionless function. Its maximum value is χmax = 0.4463. Collecting terms gives the general form ip = 0.4463 · nFAC · (nFvD/RT)^(1/2). Plugging in F, R, and T = 298 K bundles the constants into the familiar 2.69 × 10⁵.

Key quantities and a worked example

Track the units carefully — the equation is written for CGS-flavored electrochemistry, not pure SI:

• ip in amperes; n dimensionless; A in cm²; D in cm²/s; C in mol/cm³ (so 1 mM = 1×10⁻⁶ mol/cm³); v in V/s.
• The constant 2.69 × 10⁵ carries units of C·mol⁻¹·V⁻¹ᐟ² and is temperature-specific (25 °C). The general 0.4463 form is temperature-independent.

Worked example: the classic ferricyanide couple, [Fe(CN)6]³⁻/⁴⁻, with n = 1, D ≈ 7.6 × 10⁻⁶ cm²/s, C = 1.0 mM = 1×10⁻⁶ mol/cm³, at a 3 mm-diameter disk (A ≈ 0.071 cm²), swept at v = 0.1 V/s.

• ip = 2.69×10⁵ × 1^(3/2) × 0.071 × (7.6×10⁻⁶)^(1/2) × 1×10⁻⁶ × (0.1)^(1/2)
• = 2.69×10⁵ × 0.071 × 2.76×10⁻³ × 1×10⁻⁶ × 0.316 ≈ 1.7 × 10⁻⁵ A ≈ 17 µA.

Raising v to 0.4 V/s (4×) doubles ip to ~34 µA — the square-root law made concrete.

How it is measured and used in the lab

In practice you run cyclic voltammetry with a three-electrode cell (working, reference, counter) driven by a potentiostat, and record ip at a series of scan rates.

• Randles-Sevcik plot: plot ip against v^(1/2). A straight line through the origin confirms diffusion control; the slope equals 2.69×10⁵·n^(3/2)·A·D^(1/2)·C.
• Extracting D: if n, A, and C are known, solve the slope for the diffusion coefficient — the most common analytical use.
• Counting electrons: with D and C known, the slope gives n^(3/2), pinning down how many electrons the step transfers.
• Quantifying concentration: ip is linear in C, so calibrated CV becomes a sensor readout.

Practical cautions: subtract capacitive (double-layer) charging current, which itself scales linearly with v and can swamp small faradaic peaks at high scan rates; add ample supporting electrolyte to kill migration; compensate for uncompensated resistance (iR drop), which artificially widens ΔEp and depresses ip at fast sweeps. Always verify reversibility first: ΔEp ≈ 59/n mV and a return-to-forward peak ratio near 1.

How it relates to neighboring concepts

The Randles-Sevcik equation sits inside a family of transport-and-kinetics relations, and knowing its cousins prevents misuse:

• Cottrell equation: governs a potential step (chronoamperometry), giving i = nFAC·D^(1/2)/(π^(1/2) t^(1/2)) — same D, same planar diffusion, but time-decay rather than a scanned peak.
• Nernst equation: supplies the surface boundary condition; the 0.4463 peak factor only exists because the couple is Nernstian.
• Butler-Volmer / Nicholson analysis: take over when electron transfer is slow (quasi-reversible or irreversible), replacing 0.4463 with 0.496·(αnα)^(1/2) and letting peak potential drift with v.
• Levich equation: the rotating-disk analog where forced convection, not diffusion alone, sets a steady limiting current proportional to ω^(1/2).
• Surface (Laviron) behavior: for adsorbed species ip ∝ v, not v^(1/2) — the litmus test that distinguishes solution-phase from surface-confined redox.

Exceptions, limits, and lasting significance

Because the equation encodes so many assumptions, its deviations are as diagnostically useful as the law itself.

• Non-planar diffusion: at ultramicroelectrodes (radius ≤ ~25 µm) radial diffusion dominates and the CV becomes a sigmoidal steady-state wave with no peak — Randles-Sevcik simply does not apply.
• Slow kinetics: quasi-reversible/irreversible couples show ΔEp > 59/n mV that grows with v, and ip falls below the reversible prediction; use the irreversible form ip = 2.99×10⁵·n(αnα)^(1/2)·A·D^(1/2)·C·v^(1/2).
• Coupled chemistry (EC, ECE): following reactions distort ip/v^(1/2), which is exactly how mechanisms are unraveled.
• Adsorption: a peak scaling linearly with v flags surface confinement or thin-layer behavior.

Its lasting significance: introduced in 1948 by John E. B. Randles in England and, independently, Augustin Ševčík in Czechoslovakia, it turned voltammetry from a qualitative curiosity into a quantitative analytical method. Nearly eight decades on, it remains the first equation any electrochemist reaches for to interrogate a new redox system.

Randles-Sevcik forms and diagnostic behavior for reversible vs irreversible vs surface-confined systems

System	Peak current expression (25 °C)	Scan-rate dependence	Peak-to-peak signature
Reversible (Nernstian), diffusion	ip = 2.69×10⁵ · n^(3/2) A D^(1/2) C v^(1/2)	ip ∝ v^(1/2)	ΔEp = 59/n mV, ip,a/ip,c = 1, Ep independent of v
Irreversible, diffusion	ip = 2.99×10⁵ · n(αnα)^(1/2) A D^(1/2) C v^(1/2)	ip ∝ v^(1/2)	No reverse peak; Ep shifts ~30/(αnα) mV per decade of v
Quasi-reversible, diffusion	Between the two limits (Nicholson analysis)	≈ v^(1/2), curvature at high v	ΔEp grows with v; use ψ vs v to extract k0
Surface-confined / adsorbed	ip = (n²F²/4RT) A Γ v	ip ∝ v (linear)	ΔEp ≈ 0, symmetric peak, thin-layer shape