FR17 · From a saved spectrum to a circuit

Electrochemical Impedance Inference

Can a perfect impedance fit identify the reaction pathway? A spectrum can determine an effective resistance while leaving the microscopic pathway ambiguous. Two parallel reaction channels have exactly the same linear impedance as one channel with their combined conductance, for every frequency. Separating their contributions needs an additional measurement.

Acquire a synthetic impedance spectrum and fit its resistance and capacitance. Split the reaction into two parallel pathways and see why the same spectrum cannot determine their shares.

One spectrum, several explanations

What can the spectrum tell you?

Drag the gold source dial to scan frequency across the saved spectrum. Resistance and capacitance acquire a new record; the pathway share changes its explanation.

Solution resistance

R_s/\Omega

35

Charge-transfer resistance

R_{ct}/\Omega

180

Double-layer capacitance

C_{dl}/\mu\mathrm F

18

First pathway share

p

0.5

Probe frequency

f/\mathrm{Hz}

50

Frequencies used in the fit

Advanced conditions

Impedance noise

\sigma_Z/\Omega

0.8

Noise is the standard deviation of each saved real and imaginary component, in ohms. The seed stays fixed for a reproducible comparison.

Model assumptions

A linear small-signal Randles circuit represents one active interface and an ideal counterelectrode. Independent Gaussian noise is added to the real and imaginary impedance. Fits use the saved samples. Charge-transfer resistance is the equilibrium Butler–Volmer derivative; diffusion, adsorption, roughness and distributed elements are omitted. The source dial scans frequency; animation time is rescaled to show each cycle.

49 saved synthetic frequencies · drag the source dial to scan the response

Fitted solution resistance

35\,\Omega

Fitted transfer resistance

180\,\Omega

Fitted double-layer capacitance

18\,\mu\mathrm F

Residual per component

0.8\,\Omega

Saved impedance and fitted response

Gold points are saved synthetic data; cyan and pink show coincident fitted circuits. The white ring marks the simulated source response at the dial frequency.

Residuals across the saved frequencies

Residuals are in ohms. Selecting only high frequencies leaves most of the semicircle outside the fit and weakens the resistance estimate.

Two reaction pathways with the same spectrum

At equilibrium, the two reaction conductances add. Splitting that total between two pathways changes their inferred exchange currents while preserving the entire impedance curve.

R_1=R_{ct}/p,\quad R_2=R_{ct}/(1-p),\quad R_1^{-1}+R_2^{-1}=R_{ct}^{-1}

R_1=360\,\Omega

R_2=360\,\Omega

I_0=143\,\mu\mathrm A

R_{ct}\in[179.5,180.4]\,\Omega

Local 95% Gaussian interval, conditional on the selected circuit and assigned noise.

Seed 470217 · one electron transferred per reaction · 298.15 K

Model, units and sources

The active interface has an equilibrium capacitance in parallel with a charge-transfer resistance, in series with the solution resistance. The source dial selects a 10 mV sinusoidal drive from 0.1 to 100,000 Hz. Each displayed cycle is rescaled in time; total interfacial charge and current follow this same linear circuit at the selected frequency. Geometry and ion markers are schematic.

Z(\omega)=R_s+\left(R_{ct}^{-1}+i\omega C_{dl}\right)^{-1},\qquad R_{ct}=\frac{RT}{F I_0}

This model omits diffusion and therefore has no Warburg tail. The one-electron exchange current is a total current, not a current density.

MIT 10.626 · Electrochemical Energy Systems · Continue the electrochemistry course

Start with the essentials

Focus question: Can a perfect impedance fit identify the reaction pathway?
One-sentence intuition: An effective charge-transfer resistance can be identifiable even when the two reaction pathways behind it are not.

Core mathematical model

Store charge and transfer it in parallel

Z(\omega)=R_s+\frac{R_{ct}}{1+\mathrm i\omega R_{ct}C_{dl}}

Solution resistance adds in series. At low frequency the interface contributes its charge-transfer resistance; at high frequency the capacitor provides a low-impedance path.

Linearize the reaction near equilibrium

I_F=I_0\!\left(e^{(1-\alpha)F\eta_{\rm act}/(RT)}-e^{-\alpha F\eta_{\rm act}/(RT)}\right),\qquad R_{ct}=\left.\left(\frac{dI_F}{d\eta_{\rm act}}\right)^{-1}\right|_{\eta_{\rm act}=0}=\frac{RT}{F I_0}

One electron is transferred per reaction. The exchange current is a total current. This derivative assumes a small overpotential and fixed surface concentrations.

Two pathways can produce one response

R_1=R_{ct}/p,\qquad R_2=R_{ct}/(1-p),\qquad R_1^{-1}+R_2^{-1}=R_{ct}^{-1}

Any share strictly between zero and one gives the same total conductance and therefore the same complete linear spectrum.

Common difficulties

A semicircle and a mechanism

Typical misconceptionA perfect circuit fit proves one unique reaction mechanism.

Better mental modelThe spectrum fixes combinations of linear response parameters. Parallel reaction pathways with the same total conductance remain exactly indistinguishable without another observable.

Run the experiment

01
Start from the saved data
Use A resolved semicircle and download the saved spectrum.
What to observe: The file contains every frequency, real and imaginary impedance, seed, noise level and generation parameters. The fit reads these saved points.
02
Change the interpretation
Move First pathway share from one end to the other.
What to observe: The two pathway resistances change, while the saved points, fit and residuals stay fixed. No frequency resolves that split in this model.
03
Remove the slow response
Select High frequencies only, then compare the residuals with Full saved spectrum.
What to observe: High-frequency points probe mostly solution resistance and capacitive response. The reaction resistance becomes weakly constrained, so a good local fit can extrapolate poorly.

Electrochemical Impedance Inference

Saved impedance and fitted response

Residuals across the saved frequencies