Reversibility and the Randles-Sevcik Equation

Battery and supercapacitor research relies on Randles–Sevcik-type scan-rate analysis to separate diffusion-controlled (battery-like, bulk intercalation) charge storage from capacitive (surface-confined, pseudocapacitive) charge storage in the same electrode material — a $v^{1/2}$-versus-$v^1$ decomposition applied at the electrode-material level, directly extending the diffusion-versus-adsorption distinction worked out in Example 2. Fitting $i(V) = k_1 v + k_2 v^{1/2}$ across a family of voltammograms, and extracting the two coefficients $k_1$ and $k_2$, is now a routine step in every paper reporting a new lithium-ion, sodium-ion, or supercapacitor electrode material. In pharmaceutical and biomedical electroanalysis, cyclic voltammetry with Randles–Sevcik analysis quantifies diffusion coefficients of drug molecules and metabolites in physiological buffers, supports the design of amperometric glucose and neurotransmitter (dopamine, serotonin) sensors, and — because $\Delta E_p$ is exquisitely sensitive to surface fouling — is a standard quality-control check for whether an electrode surface has been properly cleaned or modified before deployment as a biosensor. Materials and catalysis research uses the quasi-reversible framework directly: measuring $k^0$ for a molecular electrocatalyst attached to an electrode surface, or embedded in a metal–organic framework film, is exactly the kind of $\Delta E_p(v)$-versus-Nicholson's-$\Lambda$ analysis carried out in Example 3, and it is the standard first kinetic screen before more elaborate rotating-disk or spectroelectrochemical rate measurements are attempted. Analytical and environmental chemistry extends this same framework toward trace metal and trace organic analysis, where deliberately exploiting adsorption (rather than avoiding it) and controlled irreversibility during a pre-concentration step is the operating principle of Stripping Voltammetry — a technique that inverts the usual "diffusion-controlled, reversible, well-behaved" ideal of this article into a deliberately concentrated, surface-confined signal for detecting metal ions at parts-per-billion levels.

Ask the tutor: How would the Randles–Sevcik analysis change if the diffusion coefficients of the oxidized and reduced forms are significantly different, rather than equal?

Picture a planar electrode of area $A$ immersed in a quiescent solution containing only the oxidized form O at bulk concentration $C^*$ (reduced form R absent initially). A potentiostat imposes a potential $E(t)$ that ramps linearly from a starting value, through the formal potential $E^{0\prime}$ of the O/R couple, and back again — the classic triangular waveform of cyclic voltammetry. Near $E^{0\prime}$, O is reduced to R at the electrode surface, consuming O locally and creating a concentration gradient that drives more O toward the electrode by diffusion. As the potential continues to sweep negative (for a reduction), the surface concentration of O is driven essentially to zero, and the current — no longer limited by kinetics but by how fast diffusion can resupply the depleted layer — begins to fall even as the driving force keeps increasing. This competition between an ever-growing thermodynamic pull and an ever-thinning, ever-slower-refilling diffusion layer is exactly what produces a peaked, not sigmoidal, current-potential curve. The key physical quantity controlling the peak's height is the diffusion layer thickness at the moment of the peak, which itself depends on how long diffusion has had to act — set by the scan rate $v = dE/dt$. A fast scan reaches the peak potential quickly, so the diffusion layer is thin, gradients are steep, and current is large. A slow scan gives diffusion more time to spread the depletion zone, gradients relax, and the peak current is smaller. Because the diffusion layer thickness scales as $\sqrt{Dt}$, and the "time available" scales inversely with $v$, the peak current must scale as $\sqrt{v}$ — the central scaling law of the Randles–Sevcik equation, derivable rigorously from Fick's second law with a moving (Nernstian) boundary condition. Whether this diffusion-controlled picture holds at all depends on the standard heterogeneous rate constant $k^0$ for electron transfer, measured in cm/s. If $k^0$ is large compared to the mass-transfer velocity implied by the scan rate, electron transfer is so fast that the O and R surface concentrations remain in constant Nernstian equilibrium with the instantaneous electrode potential — this is the reversible limit. If $k^0$ is small, the surface concentrations lag behind what the Nernst equation would predict, an extra activation overpotential is needed to force the reaction at the required rate, and the peaks separate and shift — the quasi-reversible and, eventually, totally irreversible regimes.

For a reversible (Nernstian) redox couple, planar diffusion, and a linear potential sweep, the peak current at 25 °C is given by the Randles–Sevcik equation:

where: $i_p$ is peak current (A); $n$ is the number of electrons transferred; $A$ is electrode area (cm²); $D$ is the diffusion coefficient of the electroactive species (cm²/s); $C^*$ is bulk concentration (mol/cm³ — note the cm³, not liters); $v$ is scan rate (V/s); the constant $2.69\times10^5$ has units C·mol⁻¹·V⁻¹/² absorbing $F$, $R$, $T$ at 298 K. The fully explicit, temperature-general form, from which that constant is derived, is

where: $F$ is the Faraday constant (96 485 C/mol); $R$ is the gas constant (8.314 J·mol⁻¹·K⁻¹); $T$ is absolute temperature (K); $0.4463$ is the dimensionless peak current function $\chi(bt)$ evaluated at its maximum, a pure number arising from the solution of the diffusion equation with a Nernstian boundary condition.

Reading: peak current grows with the square root of scan rate, not linearly — doubling the scan rate increases $i_p$ by only $\sqrt2 \approx 1.41\times$, not $2\times$. This sublinear scaling is the fingerprint of semi-infinite linear diffusion control, and any strong deviation from it (linear-in-$v$, or scan-rate independent) signals a different rate-limiting process (adsorption, or kinetic control). The companion equation, diagnostic of reversibility itself, is the peak separation:

where: $E_{pa}$ and $E_{pc}$ are the anodic and cathodic peak potentials; the numeric constant 0.0592 V (often quoted as "59 mV/n") is exact only in the reversible limit and at 298.15 K.

Reading: a reversible one-electron couple gives peaks separated by almost exactly 59 mV regardless of scan rate; if you measure 57–63 mV (allowing for uncompensated resistance and instrumental limits) across a range of scan rates, reversibility is well supported. If $\Delta E_p$ grows as $v$ increases, the system is quasi-reversible or irreversible. For a totally irreversible one-step, one-electron (or rate-determining single-electron) reduction, the peak current becomes

where: $\alpha$ is the transfer coefficient (typically 0.3–0.7); $n_a$ is the number of electrons transferred up to and including the rate-determining step; the prefactor $2.99\times10^5$ replaces Randles–Sevcik's $2.69\times10^5$, reflecting the different (Gaussian rather than Nernstian) surface boundary condition. The single parameter that unifies all three regimes is Nicholson's dimensionless kinetic parameter:

where: $k^0$ is the standard heterogeneous electron-transfer rate constant (cm/s); $a$ is the transfer coefficient (often taken as $\alpha$); the denominator has units of velocity, so $\Lambda$ is dimensionless.

Reading: $\Lambda \gtrsim 15$ corresponds to reversible behavior, $\Lambda$ between roughly $10^{-3}$ and 15 is quasi-reversible with $\Delta E_p$ a specific tabulated function of $\Lambda$, and $\Lambda \lesssim 10^{-3}$ is totally irreversible. Because $\Lambda$ depends on $v^{-1/2}$, every system becomes "more irreversible" at sufficiently fast scan rates — reversibility is not a fixed property of a molecule but a statement about the time scale of the experiment relative to $k^0$.

Example 1 (Foundational) — computing $i_p$ and checking $D$ from a single reversible voltammogram. A 1.00 mM solution of ferrocene in acetonitrile/0.10 M TBAPF₆ is scanned at a glassy carbon disk electrode of radius 1.50 mm ($A = \pi r^2$) at $v = 0.100$ V/s, 298 K. The measured cathodic peak current is $i_{pc} = 8.90\ \mu\text{A}$, and $\Delta E_p = 61$ mV, essentially scan-rate independent over 0.02–0.50 V/s — consistent with reversible one-electron transfer ($n=1$). Electrode area:

Concentration in the required units:

Rearranging the Randles–Sevcik equation for $D$:

Substituting $i_p = 8.90\times10^{-6}$ A, $n=1$, $A = 7.069\times10^{-2}$ cm², $C^*=1.00\times10^{-6}$ mol/cm³, $v^{1/2} = (0.100)^{1/2}=0.3162$:

This value, $D_{\text{ferrocene}} \approx 2.2\times10^{-6}\ \text{cm}^2/\text{s}$, matches well-established literature values for ferrocene diffusion in acetonitrile (typically $1.7$–$2.4\times10^{-6}$ cm²/s), a good consistency check that the electrode area, concentration, and current were all correctly measured. If the experimenter instead ran the scan at $v=0.400$ V/s and measured $i_{pc}=17.6\ \mu\text{A}$, the ratio test confirms diffusion control: $17.6/8.90 = 1.978$, compared with the predicted $\sqrt{0.400/0.100}=\sqrt{4}=2.000$ — a 1.1% deviation well within experimental scatter, supporting the $v^{1/2}$ law.

Example 2 (Applied) — distinguishing adsorption from diffusion control in a biosensor redox probe. A researcher immobilizes an azurin-modified electrode (a blue copper protein with a surface-confined redox center) and records cyclic voltammograms at scan rates 0.020, 0.050, 0.100, 0.200, and 0.500 V/s, obtaining peak currents (µA): 0.84, 2.08, 4.15, 8.30, 20.7. A log–log plot of $i_p$ against $v$ is the correct diagnostic here, because surface-confined (adsorbed) redox couples obey

where $\Gamma^*$ is the surface coverage (mol/cm²) — note the first power of $v$, not $v^{1/2}$, because there is no diffusion layer to build up; every immobilized molecule is already at the electrode. Testing linearity: $i_p/v$ should be constant if adsorption-controlled. Computing:

The ratio is constant to within 1.4% across a 25-fold range of scan rate — decisive confirmation of surface-confined, adsorption-controlled behavior rather than diffusion control (for which $i_p/\sqrt{v}$, not $i_p/v$, would be the constant quantity). Using the slope of $i_p$ vs. $v$ (here $\approx 41.6\ \mu\text{A}\,\cdot\,\text{s/V} = 4.16\times10^{-5}$ A·s/V) to extract $\Gamma^*$ for a one-electron protein couple ($n=1$) on an electrode of $A = 0.196\ \text{cm}^2$ (0.500 cm diameter disk):

Numerator: $4\times8.314\times298\times4.16\times10^{-5} = 4.126\times10^{-1}$. Denominator: $(96485)^2 \times 0.196 = 9.310\times10^9 \times0.196 = 1.825\times10^9$.

This corresponds to $2.26\times10^{-10}\times6.022\times10^{23} = 1.36\times10^{14}$ molecules/cm², or an average footprint per protein of $1/(1.36\times10^{14}) = 7.4\times10^{-15}\,\text{cm}^2 = 74\ \text{nm}^2$ — physically sensible for a globular protein of a few nanometers diameter, giving confidence that a genuine monolayer (rather than multilayer aggregation) is present, information that a Randles–Sevcik ($v^{1/2}$) analysis would have entirely misdiagnosed.

Example 3 (Exam/competition level) — extracting $k^0$ for a quasi-reversible couple from the growth of $\Delta E_p$ with scan rate. A quasi-reversible one-electron couple (an FeIII/FeII complex, $D = 6.50\times10^{-6}\ \text{cm}^2/\text{s}$, assumed equal for both oxidation states, $\alpha = 0.50$) is studied at a $A = 0.0314\ \text{cm}^2$ electrode. The measured peak separations at three scan rates are:

At $v=0.050$ V/s the couple looks nearly reversible; by $v=5.00$ V/s it clearly is not. We use Nicholson's tabulated $\Delta E_p$–$\Lambda$ correspondence (for $\alpha=0.5$): $\Delta E_p = 65$ mV corresponds to $\Lambda \approx 3.5$; $\Delta E_p=92$ mV to $\Lambda\approx1.0$; $\Delta E_p = 210$ mV to $\Lambda\approx0.16$ (values read from Nicholson's original working curve, standard in electroanalytical reference tables). Recall

Solve for $k^0$ at each scan rate as an internal consistency check. At $v=0.500$ V/s, $n=1$, $T=298$ K:

Repeat at $v = 0.050$ V/s: $nFv/RT = 1.948\ \text{s}^{-1}$, so $\sqrt{\pi\times1.948\times6.50\times10^{-6}} = \sqrt{3.977\times10^{-5}} = 6.306\times10^{-3}$ cm/s, giving

And at $v=5.00$ V/s: $nFv/RT = 194.8\ \text{s}^{-1}$, $\sqrt{\pi\times194.8\times6.50\times10^{-6}}=\sqrt{3.977\times10^{-3}}=6.306\times10^{-2}$ cm/s, giving

The three estimates — $2.21\times10^{-2}$, $1.99\times10^{-2}$, $1.01\times10^{-2}$ cm/s — agree only to within a factor of two, a common real-world outcome reflecting both the coarseness of reading $\Lambda$ off a working curve by eye and the fact that uncompensated resistance inflates $\Delta E_p$ at the fastest scan rates more than the theory (which assumes zero cell resistance) accounts for. A careful analysis would correct $\Delta E_p$ for the measured solution resistance $R_u$ (via $iR_u$ drop, itself measurable by impedance spectroscopy) before reading $\Lambda$; doing so typically brings such estimates into agreement to within 10–20%, and the resistance-corrected consensus value here would be reported as $k^0 = (2.0\pm0.3)\times10^{-2}$ cm/s — comfortably in the "quasi-reversible" regime (roughly $10^{-2}$ to $10^{-5}$ cm/s spans quasi-reversible behavior for typical scan rates in the 0.02–5 V/s range used).

The limiting-case row deserves emphasis: reversibility is not an intrinsic, scan-rate-independent property of a molecule. Any redox couple with finite $k^0$ will appear reversible if $v$ is slow enough (giving diffusion, and hence the experimental clock, plenty of time relative to electron transfer) and will appear irreversible if $v$ is fast enough. The correct statement is always "reversible at this scan rate," and a full kinetic characterization requires scanning across a wide range of $v$ and fitting the resulting $\Delta E_p(v)$ or $i_p(v)$ curve, not a single measurement at one arbitrary scan rate.

The theory bears two names because it was worked out independently and almost simultaneously in 1948: John E. B. Randles in Birmingham, England, and Antonín Ševčík in Czechoslovakia, both tackling the same diffusion-controlled boundary-value problem that arises when a linearly-swept potential is applied to a planar electrode. Randles, a physical chemist working on polarography and fast electrode processes, was interested in extending the Ilkovič-type steady-state treatments of dropping-mercury electrodes to the newer technique of linear-sweep and cyclic voltammetry at solid or hanging-drop electrodes, where the potential (rather than simply time) is the controlled variable. He derived the integral equation governing the surface concentration under a Nernstian boundary condition and obtained, by careful approximate and then numerical solution, the peak current function whose maximum value — 0.4463 — still appears unchanged in every modern textbook. Ševčík arrived at essentially the same integral equation from a similar physical starting point and pursued the numerical evaluation of the current function independently; the near-simultaneous, unconnected derivation of an identical, non-trivial result is itself a small illustration of how physical boundary conditions, once correctly identified, admit only one correct solution. The technique's real explosion in importance came a generation later, in the 1960s, with Richard S. Nicholson's rigorous digital-simulation treatment of quasi-reversible electron transfer — precisely the problem this article's second derivation walks through — which for the first time gave electrochemists a systematic, tabulated way to extract a real heterogeneous rate constant $k^0$ from the shape and scan-rate dependence of a single family of cyclic voltammograms, rather than requiring a separate, slower steady-state technique.