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How does a reversible electrode respond in a.c. voltammetry? Part 2: solutions for the periodic current amplitudes

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Abstract

In a.c. voltammetry, a programmed electrical potential—a linear ramp modulated by a sine wave of frequency ω and modest amplitude—is applied to the working electrode of a voltammetric cell. If the electrolyte solution contains a solute that undergoes a reversible electron exchange at the electrode, then the ensuing faradaic current incorporates two aperiodic components, together with a pseudosinusoidal component of every frequency that is an integer multiple of ω. An exact mathematical model is presented that predicts how the time-dependent amplitudes of each of these latter harmonic currents evolve during the experiment. This analytical model, which does not invoke simulation or Fourier transformation, is useful only at early voltammetric times, but a numerical scheme extends the solution to encompass the entire scan. Amazingly, the time-dependent amplitudes of the periodic currents match their semiintegrals in shape.

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Notes

  1. The acronym FTACV is in use, but is inappropriate in this work because Fourier transformation is not employed.

  2. The name “a.c./d.c. voltammetry” would be more descriptive, helping to distinguish the method from “a.c. only” techniques [14].

  3. Diverse expansions are represented as summations in this document. In each instance, the Σ symbol is embellished with the first, second, and final members of the arithmetic sequence of values adopted by the summation index.

  4. or otherwise convolve [17]

  5. Moreover, correction for interfering resistance and capacitance is more easily applied to the current than to its semiintegral.

  6. In “A.c. voltammetry: the current at early times”, the initial potential cited in Table 1 corresponds to a lower limit that is early enough that an earlier choice would have no significant numerical effect.

  7. See [16, p89] or [25] for details. When Euler transformation is used to reformulate a power series it replaces \( \kern0.36em {\sum}_{j=1}^{\infty }{a}_j{x}^j\ \mathrm{by}\ \frac{1}{1+x}{\sum}_{k=0}^{\infty }{\left(\frac{x}{1+x}\right)}^k{\sum}_{j=1}^k\left({}_j^k\right)\kern0.1em {a}_j. \)

  8. Forty if the data in Table 1 are adopted.

  9. Otherwise replace ωt by ωt + sin {φ1/2}, and μt by μt − ρ sin {φ1/2} throughout the ensuing algebra.

  10. The symbolism in Part 1 differs somewhat from that used here. In particular, ρ ≡ 2σ, μt ≡ 2λ(t) and ΔM ≡ m

  11. Please avoid confusing the Roman symbol I, standard for these Bessel functions [16, ch 49], with the italic I signifying current.

  12. Thus (−1)H = (−1)Int{h/2} =  + 1  for h = 1, 4, 5, 8, ⋯, but =  − 1  for h = 2,3,6,7, ⋯

  13. One such alternative is Eq. 9:4 below.

  14. The stacking in 7:7 and 7:8 is merely for typographic economy; unlike its role in 7:9, stacking does not reflect the parity of h.

  15. Sgn{x} extracts the sign of x.

  16. Formula 9:4 encounters a 00 indeterminacy at t = 0 when h is odd. The correct interpretation is 00 = 1.

  17. For the data in Table 1, this choice corresponds to a sampling interval δ=125 milliseconds for h = 1 and proportionately less for other h values. Correspondingly the length of the ΔMh file is 1 + 8 ωh[E1/2 − Einitial]/2πv, equal to 641 for h = 1.

  18. Any larger G would merely add more summands of a magnitude that would not significantly affect the sum.

  19. This equality applies to all h’s, not just the four values that we use illustratively. The pattern of less height and more wiggles continues indefinitely as h increases.

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Authors and Affiliations

  1. Department of Chemistry, Trent University, 1600 West Bank Drive, Peterborough, Ontario, K9L 0G2, Canada

    Jan C. Myland & Keith B. Oldham

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  1. Jan C. Myland
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Correspondence to Keith B. Oldham.

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Myland, J.C., Oldham, K.B. How does a reversible electrode respond in a.c. voltammetry? Part 2: solutions for the periodic current amplitudes. J Solid State Electrochem 23, 2061–2071 (2019). https://doi.org/10.1007/s10008-019-04238-0

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