Simplifying Mechanistic Impedances
Authors: Harrington David A. 
Published: May 2020

The methodology to derive the impedance corresponding to a reaction mechanism involves linearizing the kinetic equations. It has been used for a long time for quite complicated mechanisms, and has been systematized in a matrix form so that it is now straightforward to compute the impedance spectrum for any set of reactions and kinetic parameters. There has also been some progress on the inverse problem; deducing which types of mechanisms can correspond to which shapes of impedance spectra.
The complexity of the spectrum, in terms of the number of semicircles (loops) in a Nyquist plot or bends in a Bode plot is known to be related to the number of reaction steps with linearly independent stoichiometry. According to these rules, a complicated reaction mechanism is expected to give a complicated impedance spectrum at least at some potentials. However, impedance spectra are often deceptively simple, and mask the complexity of the underlying mechanism. I discuss some of the reasons for this and how to simplify mechanistic impedances in order to better understand the relationship between mechanism and impedance shapes. Aside from the increased understanding of what is going on, this also has the practical advantage that data can be fitted to expressions with a smaller number of (combined) parameters, which leads to more reliable location of the global minimum.
A simple reason for simplification is that an adsorbed intermediate has very low coverage (or coverage close to one), so oscillations in the coverage are too small to show up. For the hydrogen evolution reaction, for example, the two semicircle spectrum reduces to a single semicircle under these conditions. This is a special case of a mechanism having a rate-determining step (RDS). When a reaction mechanism with multiple adsorbed species has a single RDS, the impedance collapses to a single semicircle1.
This result comes from applying the preequilibrium approximation widely used in conventional chemical kinetics. Reaction steps before the RDS are fast and approximately in equilibrium, so the coverages associated with them oscillate in phase with the potential and so are not “seen”; reaction steps after the RDS are fast and any adsorbed species have very low coverages and again are not “seen”. The case of collapse to a single semicircle is an extreme case of simplification and I show here how less drastic simplifications are possible using the similar principles of classifying reaction steps. (A matrix formulation for simplifying mechanisms based on fast and slow species has been developed by Sengoku et al2, but reaction steps do not play a key role in that theory.)
Consider the case where an electron-transfer step 1 of a mechanism involving a coverage θ is treated as an equilibrium. Then:
(1) the rate and rate constants for that step cannot appear in the rate equations, though the equilibrium constant K
1 = k
1/k
-1 may.
(2) the coverage θ must be eliminated from the kinetics by noting that it will oscillate in phase with the potential, so dθ/dt can be rewritten as (dθ/dE)(dE/dt) = f(K
1, dE/dt) before the equations are linearized.
I will show how consistent use of these principles can lead to simplified impedance spectra. Cases related to the production of hydrogen and its migration through an electrode in a Devanathan-Stachurski cell will be used as examples. The effect on the derived equivalent circuits will be discussed.
I thank the Natural Sciences and Engineering Research Council of Canada for financial support of this research.
1. D.A. Harrington, J. Electroanal. Chem., 30, 737 (2015).
2. J. Sengoku, M. Naito, H. Okamoto, T. Ogita, and S. Ichikawa, Chem. Phys. Lett., 525, 125 (2012).