In electrochemistry, the observed current is generally determined by ion diffusion in the electrolyte. It was found that diffusion-limited currents can be avoided by using very small electrodes, known as ‘ultramicroelectrodes’ [192, 193, 194]. In order to capture how conductive current densities (also called diffusion current densities) vary with the size of the electrode, consider a spherical electrode and a conductive matter current density with spherical symmetry, j_A = j_Ar ˆr ≡ j_r ˆr. Show that when the system reaches a stationary state, the conductive matter current density is nonzero. The analysis of the transient behaviour would show that the stationary state is reached faster when the electrode is smaller. [195] In spherical coordinates (r, θ, φ), taking into account the spherical symmetry of the conductive matter current density, i.e. ∂/∂θ = 0 and ∂/∂φ = 0, the matter diffusion equation (11.54) for a solute of concentration c (r, t) reads,

The boundary conditions are,where c∗ is the concentration very far away from the electrode and r₀ is the radius of the electrode. According to relation (11.51), the conductive matter current density scalar jr that characterises this electrode is,

Establish the following results:a) The diffusion equation recast in terms of the function w(r, t) = r c (r, t) has the structure of a diffusion equation where the spherical coordinate r plays an analogous role to a Cartesian coordinate.b) The diffusion equation,admits the solution,First, write w(r, t) = f (η) where the variable η is a dimensionless function of r and t given by,c) In the limit where the radius of the electrode is negligible, i.e. r = 0, the scalar conductive matter current density is given by,d) After a transient behaviour, the scalar conductive matter current density reaches a stationary value,

Transcribed Image Text:

a₁n₁ = DV²n₁ (11.54)