Derivation of the Cottrell Equation

The Cottrell equation was developed in 1903 by solving a simple model system for chronoamperometry at a large electrode and at high overpotential [F.G. Cottrell, Z. Physik. Chem. 42 (1903) 385]. 

The equation can be solved by the method of Boltzmann transformation, as in Understanding Voltammetry, or alternatively by a mathematical method known as Laplace transformation, in which an integral transform is used to convert a partial differential equation into an ordinary differential equation. The transformation is  

We can Laplace transform the boundary conditions as well, so c = 0 at X — oo and so A(s) = 0. Then from the electrode boundary condition  

Therefore in finding B s), recalling that the Laplace transform of a constant 

because both differentiation and inverse Laplace transformation are Knear operations, we can note without further effort that 

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The area to be used in the Cottrell equation, or in other similar equations describing current flow in electrochemical experiments, depends on the time scale of the measurements. In the derivation of the Cottrell equation (Section 5.2.1), the current is defined by the flux of species diffusing across the plane at x = 0. The total rate of reaction in moles per second, giving the total current in amperes, is the product of that flux and the cross sectional area of the diffusion field, which is the area needed for the final result. 

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