Impedance of Reaction E in Non-steady State with Quiescent Electrolyte
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 studied under semi-infinite diffusion at an electrode in a quiescent electrolyte. The Faradaic impedance used here is the Warburg impedance, which is described in the Details section.
The steady-state current density for a redox reaction () at a plane electrode in a quiescent electrolyte is . The electrode potential is defined by the Nernst equation
where and are the bulk concentrations of the electroactive species and , respectively; is the standard potential of the redox couple ; ; and is the number of exchanged electrons. At steady state, the interfacial concentrations of the electroactive species and . The reduction and oxidation reaction rates are
respectively, with the electrode potential and and the symmetry factors of the reduction and oxidation reactions, respectively.
The Faradaic impedance is
with , where is the frequency. Assuming equal bulk concentrations and diffusion coefficients for both species (), we can write:
After simplification, this gives:
The electrode impedance is
with the double-layer capacitance related to the electrode/electrolyte interface.
In this Demonstration, it is considered that the bulk concentration of the electroactive species changes with time due to an increase of the support electrolyte:
with the initial concentration of electroactive species, the initial volume of the solution, the flow rate of additional electrolyte and the time. Assuming that the change of the concentration occurs instantaneously after the addition of the solution, due to the relatively low concentration of electroactive species, we can express the instantaneous Faradaic impedance:
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 and at the final concentration in orange. The dots, whose sizes increase with time, show the measurement performed during the dilution.
 A. J. Bard and L. R. Faulkner, Electrochemical Methods: Fundamentals and Applications, 2nd ed., New York: Wiley, 2001.
 J.-P. Diard, B. Le Gorrec and C. Montella, Cinétique électrochimique, Hermann éd., 1996.
 "EIS Measurements on a Rotating Disk Electrode (RDE) Part I: Determination of a Diffusion Coefficient Using the New Element ," BioLogic Application Note 66. www.biologic.net/wp-content/uploads/2019/08/rde-diffusionv_electrochemistry-an66.pdf.
 R. Pachimatla and R. Srinivasan, "Non-linear Electrochemical Impedance Spectroscopic Analysis of Instabilities in Electrochemical Systems," ECS Transactions, 85(13), 2018 pp. 1145–1153. iopscience.iop.org/article/10.1149/08513.1145ecst.
 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. iopscience.iop.org/article/10.1149/2.0571908jes.