Semi-infinite External Spherical Diffusion

In this case, the boundary and initial conditions are as follows  

4 Impedance of the Faradaic Reactions in the Presence of Mass Transfer 

X 00 Co 0 and Cr 0, where ro is the sphere radius. Keeping in mind that 

A comparison of Eqs. (4.97) and (4.30) indicates that an additional real term, Dj/ro, appeared in the denominator under the square root. The Warburg impedance, using Eq. (4.69), becomes 

Impedance plots are modified at spherical electrodes. Examples of such plots, using an equivalent circuit in Eig. 4.1, are shown in Eig. 4.14. The high-frequency semicircle is related to the coupling of Ret and C ji, while the low-frequency, depressed semicircle is related to spherical diffusion. 

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Fig. 4.14 Complex plane and Bode plots for semi-infinite external spherical diffusion Tq. (a) qo semi-infinite linear diffusion, (b) 0.005, (c) 0.01, (d) 0.02, (e) 0.05 cm other parameters as in Fig. 4.12...

These equations can be solved for semi-infinite external diffusion, where both Red and Ox forms are in the solution outside the sphere (diffusion to a spherical or hemispherical hanging mercury electrode, metallic solid spherical electrode), or they may diffuse inside the sphere (amalgam formation at mercury electrode, intercalation of Li into particles, hydrogen absorption into spherical hydrogenabsorbing particles). 

Exercise 4.4 Write a program in Maple or Mathematica to simulate semi-infinite spherical fusion (external) diffusion and create the corresponding complex plane and Bode plots. The parameters are as in Exercise 4.3, except Kq = 0.01 cm. 

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