I'm working through the derivation of current in a spherical electrode, and so far I've been able to get it into the following, starting from Fick's 2nd Law:
Any help would be appreciated.
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$\Delta c$ is a function of (among other things) $r$, so you can't just swap $v/r$ into the differentiation operator. Instead you have to use the quotient rule for differentiating to get:
$$\frac{\text{d}^2vr^{-1}}{\text{d} r^2}+\frac{2\text{d}vr^{-1}}{r\text{d} r}-\frac{svr^{-1}}{D}=0$$ $$\frac{\text{d}}{\text{d}r}\left(\frac{v'}{r}-\frac{v}{r^2}\right)+\frac{2}{r}\left(\frac{v'}{r}-\frac{v}{r^2}\right)-\frac{sv}{rD}=0$$ $$\left(\frac{v''}{r}-\frac{v'}{r^2}\right)-\left(\frac{v'}{r^2}-\frac{2v}{r^3}\right)+\frac{2}{r}\left(\frac{v'}{r}-\frac{v}{r^2}\right)-\frac{sv}{rD}=0$$ Then a bunch of stuff cancels, and we're let with
$$\frac{v''}{r}-\frac{sv}{rD}=\frac{1}{r}\left(v''-\frac{sv}{D}\right)=0$$
And, since we know that $r\neq0$, we're done.
Trold
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