# Cottrell equation

In electrochemistry, the Cottrell equation describes the change in electric current with respect to time in a controlled potential experiment, such as chronoamperometry. Specifically it describes the current response when the potential is a step function. It was derived by Frederick Gardner Cottrell in 1903.[1] For a simple redox event, such as the ferrocene/ferrocenium couple, the current measured depends on the rate at which the analyte diffuses to the electrode. That is, the current is said to be "diffusion controlled." The Cottrell equation describes the case for an electrode that is planar but can also be derived for spherical, cylindrical, and rectangular geometries by using the corresponding laplace operator and boundary conditions in conjunction with Fick's second law of diffusion.[2]

${\displaystyle i={\frac {nFAc_{j}^{0}{\sqrt {D_{j}}}}{\sqrt {\pi t}}}}$

where,

i = current, in unit A
n = number of electrons (to reduce/oxidize one molecule of analyte j, for example)
F = Faraday constant, 96485 C/mol
A = area of the (planar) electrode in cm2
cj0 = initial concentration of the reducible analyte j in mol/cm3;
Dj = diffusion coefficient for species j in cm2/s
t = time in s.

Deviations from linearity in the plot of i vs t−1/2 sometimes indicate that the redox event is associated with other processes, such as association of a ligand, dissociation of a ligand, or a change in geometry.

In practice, the Cottrell equation simplifies to

i = kt−1/2, where k is the collection of constants for a given system (n, F, A, cj0, Dj).

## References

1. F.G. Cottrell, Z. Phys. Chem. 42 (1903), p. 385.
2. Bard, A. J.; Faulkner, L. R. “Electrochemical Methods. Fundamentals and Applications” 2nd Ed. Wiley, New York. 2001. ISBN 0-471-04372-9