# Impedance of Reaction E in Non-steady State at a Rotating Disk Electrode

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This Demonstration considers the effect of decreasing electroactive species concentration on the Nyquist diagram of the impedance. The electrochemical process is a single-electron transfer redox reaction (E) studied under steady-state bounded diffusion conditions at a rotating disk electrode (RDE). The Faradaic impedance used here is an analytical approximation, described in more detail in the Details section.

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Contributed by: Jean-Paul Diard and Nicolas Murer (August 25)

Open content licensed under CC BY-NC-SA

## Details

The current density versus potential ( vs.* **)* curve for a redox reaction () under steady-state conditions at a rotating disk electrode is given by

, , , , , ,

with the electrolyte cinematic viscosity, the diffusion coefficient of the species , both in and the rotation speed of the RDE in rad .

The Faradaic () and electrode () impedance for the redox electrochemical reaction () at an RDE, taking into account the series ohmic resistance

, are given by

, , , ,

with the frequency in Hz,

, , = , ,, .

For ,

, ,

with , the dimensionless Schmidt number.

This expression of the impedance is an improvement of the Nernst approximation previously described. It assumes that the concentration changes continuously away from the interface and that both diffusion and convection contribute to the mass transport of the electroactive species.

Rearranging and simplifying the terms and considering that the concentration of the electroactive species changes with time,

.

The instantaneous concentration , where is the initial concentration of electroactive species, is the initial volume of the solution, is the flow rate of additional electrolyte, and is time. At each frequency of the time-varying measurement, the Faradaic impedance is calculated using the value of the concentration at the beginning of the measurement.

The top graph shows the change of the bulk concentration during the measurement, with the final volume of electrolyte. The bottom graph shows the impedance graph at the initial electrolyte concentration in blue; at the final concentration in orange. The dots, whose size increases with time, show the measurement performed during the dilution.

References

[1] C. Deslouis, C. Gabrielli and B. Tribollet, "An Analytical Solution of the Nonsteady Convective Diffusion Equation for Rotating Electrodes," *Journal of the Electrochemical Society*, 130(10), 1983 pp. 2044–2046. https://iopscience.iop.org/article/10.1149/1.2119518.

[2] B. Tribollet and J. Newman, "Analytic Expression of the Warburg Impedance for a Rotating Disk Electrode," *Journal of the Electrochemical Society*, 130(4), 1983 pp. 822–824. https://iopscience.iop.org/article/10.1149/1.2119828.

[3] M. T. T. Tran, B. Tribollet, V. Vivier and M. E. Orazem, "On the Impedance Response of Reactions Influenced by Mass Transfer," *Russian Journal of Electrochemistry*, 53(9), 2017 pp. 932–940. https://doi.org/10.1134/S1023193517090142.

[4] R. Michel and C. Montella, "Diffusion–Convection Impedance Using an Efficient Analytical Approximation of the Mass Transfer Function for a Rotating Disk," *Journal of Electroanalytical Chemistry*, 736, 2015 pp. 139–146. http://dx.doi.org/10.1016/j.jelechem.2014.11.009.

[5] J.-P. Diard and C. Montella, "Re-examination of the Diffusion–Convection Impedance for a Uniformly Accessible Rotating Disk. Computation and Accuracy," *Journal of Electroanalytical Chemistry*, 742, 2015 pp. 37–46. https://doi.org/10.1016/j.jelechem.2015.01.017.

[6] "EIS Measurements on a Rotating Disk Electrode (RDE) Part I: Determination of a Diffusion Coefficient Using the New Element Winf," BioLogic Application Note 66. https://www.biologic.net/wp-content/uploads/2019/08/rde-diffusionv_electrochemistry-an66.pdf.

[7] R. Pachimatla and R. Srinivasan, "Non-linear Electrochemical Impedance Spectroscopic Analysis of Instabilities in Electrochemical Systems," *ECS Transactions*, 85(13), 2018 pp. 1145–1153. https://iopscience.iop.org/article/10.1149/08513.1145ecst.

[8] R. Pachimatla, M. Thomas, S. Rahman OC and R. Srinivasan, "Analysis of Instabilities in Electrochemical Systems Using Nonlinear Electrochemical Impedance Spectroscopy," *Journal of the Electrochemical Society*, 166(8) 2019 pp. H304–H312. https://iopscience.iop.org/article/10.1149/2.0571908jes.

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