---
title: "Electrochemical Impedance Inference Interactive Lab | Sandbox Physics"
description: "Can a perfect impedance fit identify the reaction pathway? A spectrum can determine an effective resistance while leaving the microscopic pathway ambiguous."
canonical_url: "https://sandboxphysics.com/labs/impedance-inference/"
language: "en"
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---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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](https://ocw.mit.edu/courses/10-626-electrochemical-energy-systems-spring-2014/pages/lecture-notes/) · [Continue the electrochemistry course](https://sandboxphysics.com/stories/electrochemistry/)

Physics tutorial

## Electrochemical Impedance Inference

**Background** 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.

**Why it matters** A small alternating voltage probes how quickly an interface stores charge and transfers it through a reaction. The measured current contains both processes.

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 misconception** A perfect circuit fit proves one unique reaction mechanism.

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

### Run the experiment

1. 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.

2. 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.

3. 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.

Physical conclusion

Report the saved spectrum, the fitted effective parameters and the circuit assumptions together.

## Related experiments and articles

Electrochemistry and ion transport · Reading path

### From ion diffusion to charge, flow and impedance

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **When ions exchange electrons at an electrode**](https://sandboxphysics.com/stories/electrochemistry/butler-volmer/)
2. [Article **What can one impedance arc tell us?**](https://sandboxphysics.com/stories/electrochemistry/impedance-identifiability/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)