Diffusion-Limited Current Calculator
Find the diffusion-limited current at a stationary electrode at any time after a potential step, using the Cottrell equation for chronoamperometry.
🧪 What is Diffusion-Limited Current?
Diffusion-limited current is the current measured at an electrode when the electrochemical reaction happens instantly at the surface, so the reaction rate is controlled entirely by how fast fresh reactant diffuses in from the bulk solution. The Cottrell equation describes exactly how this current decays over time after a large potential step.
Electrochemists use this constantly, to design and interpret chronoamperometry experiments, to extract diffusion coefficients from measured current-time transients, to model the response of amperometric sensors like glucose biosensors, and to distinguish diffusion-controlled behavior from kinetically or convectively limited behavior.
A common misconception is that this current stays constant over time. It doesn't, unlike a rotating or microelectrode's steady-state limiting current, a stationary planar electrode's diffusion-limited current keeps falling as 1/sqrt(t), because the depleted diffusion layer at the surface keeps growing thicker.
This calculator computes the instantaneous current at any chosen time using the Cottrell equation from your inputs for electrons transferred, electrode area, diffusion coefficient, and bulk concentration, and plots the full decay curve so you can see the characteristic 1/sqrt(t) shape.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — One-Electron Reaction at 1 Second
n=1, A=0.07 cm², D=7.6×10⁻⁶ cm²/s, C=5 mM, t=1 s
Example 2 — Two-Electron Reaction at 2 Seconds
n=2, A=0.05 cm², D=6.5×10⁻⁶ cm²/s, C=10 mM, t=2 s
Example 3 — Fast Diffuser, Short Time
n=1, A=0.1 cm², D=10×10⁻⁶ cm²/s, C=2 mM, t=0.5 s
❓ Frequently Asked Questions
🔗 Related Calculators
What is the Cottrell equation?
The Cottrell equation describes how current decays with time at a stationary planar electrode after a potential step that instantly reduces the surface concentration of analyte to zero. It is i(t) = nFAD^0.5*C / (pi^0.5 * t^0.5), where the current falls off as 1 over the square root of time.
Why does diffusion-limited current decrease over time?
Immediately after the potential step, analyte right at the electrode surface is consumed instantly, and the current is limited by how fast more analyte diffuses in from the bulk solution. As time passes, the depleted region (diffusion layer) grows thicker, so each new molecule has farther to travel, slowing the current.
What does each variable in the Cottrell equation represent?
n is the number of electrons transferred per molecule reacted, F is the Faraday constant (96,485 C/mol), A is electrode area, D is the analyte's diffusion coefficient, C is its bulk concentration, and t is elapsed time since the potential step. Current comes out largest for large n, A, D, and C, and smallest at long times.
How is chronoamperometry different from cyclic voltammetry?
Chronoamperometry holds the electrode at a fixed, large potential step and records how current changes over time, producing the Cottrell decay curve. Cyclic voltammetry instead sweeps the potential linearly and records current as a function of that changing potential, producing a peak-shaped voltammogram.
What is a diffusion coefficient and typical values?
The diffusion coefficient D measures how quickly a species spreads through solution due to random thermal motion, in units of cm2/s. Small ions and molecules in water typically have D values around 0.5 to 2 x 10^-5 cm2/s, larger molecules like proteins diffuse more slowly.
How can the Cottrell equation be used to find a diffusion coefficient experimentally?
Since current is proportional to D^0.5, plotting measured current against 1/sqrt(t) gives a straight line whose slope is proportional to D^0.5 (given known n, F, A, and C). Solving that slope for D is a standard way to measure diffusion coefficients experimentally.
Does the Cottrell equation apply to all electrode geometries?
No, it strictly applies to a planar electrode with semi-infinite linear diffusion. Microelectrodes and rotating disk electrodes reach a steady (non-decaying) limiting current instead, governed by different equations like the Levich equation, because their geometry supports a different diffusion regime.
What units does this calculator use?
Number of electrons is dimensionless, electrode area is in square centimeters (cm2), diffusion coefficient is entered as a value times 10^-6 cm2/s, concentration is in millimolar (mM), and time is in seconds. Output current is in microamps (µA).
Why does the current diverge (become infinite) as t approaches zero?
Mathematically, i(t) contains 1/sqrt(t), which grows without bound as t approaches zero. Physically this reflects the idealized assumption of an instantaneous potential step, real electrodes and instruments have finite response times, so the current is always finite and simply very large at very short times.
How is diffusion-limited current used in practice?
Diffusion-limited current measurements underlie amperometric sensors (like glucose sensors), electroanalytical concentration determinations, and diffusion coefficient measurements. Because the current at a fixed time is directly proportional to bulk concentration, chronoamperometry is a common quantitative analytical technique.