Engineering & Science
Cottrell Equation: Chronoamperometry Diffusion Current i(t) & Nernst Layer Thickness
Calculate t^(-1/2) planar diffusion current i(t) = (n * F * A * C0 * SQRT(D)) / SQRT(PI() * t) in microamps and Nernst diffusion-layer thickness delta = SQRT(PI() * D * t) in electrochemical sensors.
Model glucose strips and potential-step chronoamperometry with the Cottrell t^-1/2 decay!
Stepping a planar working electrode instantaneously to a mass-transport-limited overpotential depletes electroactive analyte at the surface to zero and drives a growing concentration gradient into the unstirred bulk solution! Frederick Cottrell (1903) solved Fick's second law of planar diffusion to show that Faradaic current decays inversely with the square root of time: i(t) = (n * F * A * C0 * SQRT(D)) / SQRT(PI() * t), where n is electrons transferred, F = 96,485 C/mol is Faraday's constant, A is electrode area (cm^2), C0 is bulk concentration IN MOL/CM^3 (note: 5 mM = 5.0e-6 mol/cm^3!), D is diffusion coefficient (cm^2/s), and t is step time (s)!
Watch the #1 electrochemistry CGS unit trap: if A is in cm^2 and D is in cm^2/s, C0 MUST be in mol/cm^3 (= Molarity / 1,000), NOT mol/L! Multiplying amperes by 1,000,000 gives microamperes (uA). For 5 mM ferricyanide (n = 1, C0 = 5e-6 mol/cm^3, D = 7.6e-6 cm^2/s) on a 3 mm glassy-carbon disk (A = 0.07 cm^2), current at t = 1.0 s is 52.52 uA across a Nernst diffusion layer delta = SQRT(PI() * D * t) = 48.86 um—and quadrupling time to t = 4.0 s halves current to 26.26 uA!
| Step Time t (5 mM Ferricyanide, A = 0.07 cm^2) | Cottrell Current i(t) (uA) | Nernst Diffusion Layer delta (um) | Scaling Law |
|---|---|---|---|
| t = 1.0 second | 52.52 uA | 48.86 um | Baseline t^-1/2 point |
| t = 4.0 seconds (4x longer) | 26.26 uA (Exact 0.5x!) | 97.72 um (Exact 2.0x!) | i * SQRT(t) = const = 52.52 uA*s^1/2 |
| Invalid t <= 0 (Capacitive Spike) | Guarded Input Error | Undefined at t = 0 | "Invalid Cottrell Inputs" |
Cottrell Diffusion-Limited Current i(t) in Microamperes (uA)
Evaluates 1e6 * (n * 96485 * A_cm2 * C0_mol_cm3 * SQRT(D_cm2_s)) / SQRT(PI() * t_s).
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0, B5 <= 0, B6 <= 0), "Invalid Cottrell Inputs", ROUND(1000000 * (B2 * 96485 * B3 * B4 * SQRT(B5)) / SQRT(PI() * B6), 2))Nernst Planar Diffusion Layer Thickness delta = SQRT(PI() * D * t) in Micrometers (um)
Converts SQRT(PI() * D_cm2_s * t_s) from centimeters to micrometers (* 10,000).
=IF(OR(B5 <= 0, B6 <= 0), "Invalid Cottrell Inputs", ROUND(10000 * SQRT(PI() * B5 * B6), 2))Try the example
Use the original values or change them to test your understanding.