Fundamentals of faradaic electrochemistry

The boundary conditions for the potential step experiments described above are  

Equation (6.1.4.1) can be solved using Laplace transform techniques to give the time evolution of the current, i(t), subject to the boundary conditions described. Equation (6.1.4.2) is then obtained  

Equation (6.1.4.2) shows that the current response following a potential step contains both time-independent and time-dependent terms. The differences in the electrochemical responses observed at macroscopic and microscopic electrodes arise because of the relative importance of these terms at conventional electrochemical timescales. It is possible to distinguish two limiting regimes depending on whether the experimental timescale is short or long. 

At sufficiently short times, the thickness of the diffusion layer that is depleted of reactant is much smaller than the electrode radius and the spherical electrode appears to be planar to a molecule at the edge of this diffusion layer. Under these conditions, the electrode behaves like a macroelectrode and mass transport is dominated by linear diffusion to the electrode surface as illustrated in Eigure 6.1.4.1 A. 

At these short times, the dependence of the second term in equation (6.1.4.2) makes it significantly larger than the first and the current response induced by the potential step initially decays in time according to the Cottrell equation (Chapter 11)  

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