    # Theoretical Determination of Collection Efficiency (N)

Last Updated: 1/5/23 by Alex Peroff ### 1General Overview

The theoretical collection efficiency can be computed from the three principle diameters describing the RRDE geometry:  the disk outer diameter (d1), the ring inner diameter (d2), and the ring outer diameter (d3).  This somewhat tedious computation is made easier by normalizing the ring diameters with respect to the disk diameter as follows: $\displaystyle{\sigma_{OD} = d_3/d_1}$
and $\displaystyle{\sigma_{ID} = d_2/d_1}$

Three additional quantities are defined in terms of the normalized diameters as follows: $\displaystyle{\sigma_{A} = \sigma_{ID}^{3} - 1}$ $\displaystyle{\sigma_{B} = \sigma_{OD}^{3} - \sigma_{ID}^{3}}$ $\displaystyle{\sigma_{C} = \frac{\sigma_A}{\sigma_B}}$

If a complex function, G(x), is defined as follows, $\displaystyle{G(x) = \frac{1}{4} + \left(\frac{\sqrt{3}}{4\pi}\right)ln\left[\frac{(x^{(1/3)}+1)^3}{x+1}\right] + \left(\frac{3}{2\pi}\right)arctan\left[\frac{2x^{1/3}-1}{\sqrt{3}}\right]}$

then the theoretical collection efficiency (Ntheoretical) for a rotating ring-disk electrode is given by the following equation first reported by Albery and Bruckenstein:  $\displaystyle{N_{theoretical} = 1 - \sigma_{OD}^{2} + \sigma_{B}^{2/3} - G(\sigma_{C}) - \sigma_{B}^{2/3}G(\sigma_{A}) + \sigma_{OD}^{2}G(\sigma_{C}\sigma_{OD}^{3})}$

### 2References

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