Steady-state For a
redox reaction R \leftrightarrow O + e, without mass-transfer limitation, the relationship between the current density and the electrode overpotential is given by the
Butler–Volmer equation: j_{\text{t}} = j_0 \left(\exp(\alpha_{\text{o}} \,f\, \eta)-\exp(-\alpha_{\text{r}}\,f\,\eta)\right) with \eta = E - E_{\text{eq}} ,\;f=F/(R\,T),\;\alpha_{\text{o}} + \alpha_{\text{r}} = 1. j_0 is the exchange current density and \alpha_{\text{o}} and \alpha_{\text{r}} are the symmetry factors. The curve j_{\text{t}} vs. E is not a straight line (Fig. 1), therefore a redox reaction is not a linear system.
Dynamic behavior Faradaic impedance In an electrochemical cell the
faradaic impedance of an electrolyte-electrode interface is the joint electrical resistance and capacitance at that interface. Let us suppose that the Butler-Volmer relationship correctly describes the dynamic behavior of the redox reaction: j_{\text{t}}(t) = j_{\text{t}}(\eta(t)) = j_0 \left(\exp(\alpha_{\text{o}}\,f\, \eta(t))-\exp(-\alpha_{\text{r}}\,f\,\eta(t))\right) Dynamic behavior of the redox reaction is characterized by the so-called charge transfer resistance R_{\text{ct}} defined by: R_{\text{ct}} = \frac{1}{\partial j_{\text{t}} / \partial \eta } = \frac{1}{f\, j_0\, \left(\alpha_{\text{o}} \exp(\alpha_{\text{o}}\,f\, \eta)+\alpha_{\text{r}} \exp(-\alpha_{\text{r}} \, f\, \eta) \right)} The value of the charge transfer resistance changes with the overpotential. For this simplest example the faradaic impedance is reduced to a resistance. It is worthwhile to notice that: R_{\text{ct}} = \frac{1}{f\,j_0} for \eta = 0.
Double-layer capacitance An electrode | electrolyte interface behaves like a capacitance called
electrochemical double-layer capacitance C_{\text{dl}}. The
equivalent circuit for the redox reaction in Fig. 2 includes the double-layer capacitance C_{\text{dl}} as well as the charge transfer resistance R_{\text{ct}}. Another analog circuit commonly used to model the electrochemical double-layer is called a
constant phase element. The electrical impedance of this circuit is easily obtained remembering the impedance of a capacitance which is given by: Z_{\text{dl}}(\omega) = \frac{1}{i \omega C_{\text{dl}}} where \omega is the angular frequency of a sinusoidal signal (rad/s), and i^2 = -1. It is obtained: Z(\omega) = \frac{R_{\text{t}}}{1 + R_{\text{t}} C_{\text{dl}} i \omega} Nyquist diagram of the impedance of the circuit shown in Fig. 3 is a semicircle with a diameter R_{\text{t}} and an angular frequency at the apex equal to 1/(R_{\text{t}}\,C_{\text{dc}}) (Fig. 3). Other representations such as
Bode plots can be used instead.
Ohmic resistance The ohmic resistance R_\Omega appears in series with the electrode impedance of the reaction and the Nyquist diagram is translated to the right.
Universal dielectric response Under AC conditions with varying frequency
ω, heterogeneous systems and composite materials exhibit a
universal dielectric response, in which overall admittance exhibits a region of power law scaling with frequency. Y \propto \omega^{\alpha} . ==Measurement of the impedance parameters==