Separating the effects of experimental noise from inherent system variability in voltammetry: the $[$Fe(CN)$_6]^{3-/ 4-}$ process

Recently, we have introduced the use of techniques drawn from Bayesian statistics to recover kinetic and thermodynamic parameters from voltammetric data, and were able to show that the technique of large amplitude ac voltammetry yielded significantly more accurate parameter values than the equivalent dc approach. In this paper we build on this work to show that this approach allows us, for the first time, to separate the effects of random experimental noise and inherent system variability in voltammetric experiments. We analyse ten repeated experimental data sets for the $[$Fe(CN)$_6]^{3-/ 4-}$ process, again using large-amplitude ac cyclic voltammetry. In each of the ten cases we are able to obtain an extremely good fit to the experimental data and obtain very narrow distributions of the recovered parameters governing both the faradaic (the reversible formal faradaic potential, $E_0$, the standard heterogeneous charge transfer rate constant $k_0$, and the charge transfer coefficient $\alpha$) and non-faradaic terms (uncompensated resistance, $R_u$, and double layer capacitance, $C_{dl}$). We then employ hierarchical Bayesian methods to recover the underlying "hyperdistribution" of the faradaic and non-faradaic parameters, showing that in general the variation between the experimental data sets is significantly greater than suggested by individual experiments, except for $\alpha$ where the inter-experiment variation was relatively minor. Correlations between pairs of parameters are provided, and for example, reveal a weak link between $k_0$ and $C_{dl}$ (surface activity of a glassy carbon electrode surface). Finally, we discuss the implications of our findings for voltammetric experiments more generally.