# Nernst equation

The **Nernst equation** is the fundamental electrochemical relation that gives the equilibrium electrode potential of a redox couple from its standard (or formal) potential, the temperature, and the activities of the reacting species; in the IUPAC form it reads \( E_{eq} = E^{\circ} - \dfrac{RT}{zF} \sum_i \nu_i \ln(a_i) \), where \( R \) is the gas constant, \( T \) the thermodynamic temperature, \( F \) the [Faraday constant](https://www.edgepedia.org/faraday-constant), \( z \) the electron number, \( \nu_i \) the stoichiometric coefficients (positive for products, negative for reactants), and \( a_i \) the activities.<sup>[1](https://goldbook.iupac.org/terms/view/09068)</sup> By convention, the potential \( E^{\circ} \) is measured relative to the standard hydrogen electrode (SHE), which is defined as the standard electromotive force of a cell where, at the left-hand electrode, molecular hydrogen at standard pressure is oxidized to solvated protons.<sup>[2](https://goldbook.iupac.org/terms/view/S05912)</sup> NIST defines a standard electrode potential as the potential in volts of a half-reaction relative to a reference electrode at a specified temperature, with all species in their standard states at unit activity, ideal gases at \( 1.01325 \times 10^{5} \) Pa and ideal solutes at unit molality.<sup>[3](https://srd.nist.gov/jpcrdreprint/1.555839.pdf)</sup>

| Key fact | Detail |
|---|---|
| Canonical form | \( E_{eq} = E^{\circ} - (RT/zF) \sum_i \nu_i \ln(a_i) \), with activities of all species<sup>[1](https://goldbook.iupac.org/terms/view/09068)</sup> |
| Working shortcut | \( E_{eq} \approx E^{\circ\prime} - (0.0592/z) \log_{10}(c_{red}/c_{ox}) \) at 25 °C; NIST gives 0.0591597 V per decade<sup>[1](https://goldbook.iupac.org/terms/view/09068)</sup><sup> • </sup><sup>[3](https://srd.nist.gov/jpcrdreprint/1.555839.pdf)</sup> |
| Nernst slope | 59.16 mV per tenfold activity change at 25 °C for \( z = +1 \); 59/n mV generally<sup>[4](https://www.metrohm.com/content/dam/metrohm/shared/documents/monographs/81085088EN.pdf)</sup><sup> • </sup><sup>[5](https://chem.libretexts.org/Bookshelves/General_Chemistry/Chem1_(Lower)/16%3A_Electrochemistry/16.04%3A_The_Nernst_Equation)</sup> |
| Equilibrium link | \( E^{\circ}_{cell} = (0.0257\ \mathrm{V}/n) \ln K = (0.0592\ \mathrm{V}/n) \log K \) at 298.15 K<sup>[6](https://openstax.org/books/chemistry/pages/17-4-the-nernst-equation)</sup> |
| Biological use | Equilibrium potentials of K⁺, Na⁺, and Cl⁻ across membranes; squid axon Nernst values −101 mV (K⁺) and +59 mV (Na⁺)<sup>[7](https://www.ncbi.nlm.nih.gov/books/NBK11102/)</sup> |
| Main failure modes | Activities depart from concentrations above roughly 10⁻⁴ to 10⁻⁵ M; mixed potentials and ion-exchange membranes break the simplified form<sup>[5](https://chem.libretexts.org/Bookshelves/General_Chemistry/Chem1_(Lower)/16%3A_Electrochemistry/16.04%3A_The_Nernst_Equation)</sup><sup> • </sup><sup>[8](https://www.degruyterbrill.com/document/doi/10.1515/jnet-2020-0070/html?lang=en)</sup> |

## Derivation from thermodynamics

The equation follows from [Gibbs free energy](https://www.edgepedia.org/gibbs-free-energy). For a cell reaction, \( \Delta G = \Delta G^{\circ} + RT \ln Q \), and the electrical work at reversible conditions is \( \Delta G = -nFE_{cell} \), with \( n \) the number of moles of electrons and \( E_{cell} \) the cell potential. Substituting gives \( -nFE_{cell} = -nFE^{\circ}_{cell} + RT \ln Q \), or \( E_{cell} = E^{\circ}_{cell} - (RT/nF) \ln Q \).<sup>[6](https://openstax.org/books/chemistry/pages/17-4-the-nernst-equation)</sup> An equivalent derivation sets the chemical driving force equal to the electrical driving force at equilibrium, using \( \Delta G^{0} = -zFE_e \) with \( F = 96{,}485 \) coulombs per mole of electrons, which yields \( E^{0} = (RT/zF) \ln K \) for the standard potential of a couple with equilibrium constant \( K \).<sup>[9](https://www.doitpoms.ac.uk/tlplib/pourbaix/nersnt_detailed.php)</sup> Lecture treatments write activities as \( \gamma_i \cdot c_i / c^{\circ} \), embedding the activity coefficients directly in the reaction quotient.<sup>[10](https://www2.sci.u-szeged.hu/physchem/indexh_html_files/GYTK_Nernst_eng_2025.pdf)</sup>

## Forms and working equations

**Base-10 form.** Converting natural to decadic logarithms via \( \ln X = 2.303 \log X \) gives \( E_e = E^{0} - (2.303RT/zF) \log([\text{reduced}]/[\text{oxidized}]) \).<sup>[9](https://www.doitpoms.ac.uk/tlplib/pourbaix/nersnt_detailed.php)</sup> At 298.15 K the factor \( 0.0592 \approx (RT/F)/\log_{10}(e) \), so the practical form is \( E_{eq} \approx E^{\circ\prime} - (0.0592/z) \log_{10}(c_{red}/c_{ox}) \).<sup>[11](https://www.degruyterbrill.com/document/doi/10.1515/pac-2018-0109/pdf)</sup><sup> • </sup><sup>[1](https://goldbook.iupac.org/terms/view/09068)</sup> NIST's more precise value is \( E_{298} = E^{\circ}_{298} - (0.0591597/n) \log Q_{298} \).<sup>[3](https://srd.nist.gov/jpcrdreprint/1.555839.pdf)</sup> The base-10 shortcut applies only at 298 K; the \( (RT/nF) \ln Q \) form is applicable at all temperatures, and at other temperatures the constant must be recalculated.<sup>[5](https://chem.libretexts.org/Bookshelves/General_Chemistry/Chem1_(Lower)/16%3A_Electrochemistry/16.04%3A_The_Nernst_Equation)</sup><sup> • </sup><sup>[6](https://openstax.org/books/chemistry/pages/17-4-the-nernst-equation)</sup> NIST tabulates temperature coefficients \( dE^{\circ}/dT \) in mV/K relative to the SHE, whose coefficient is set to zero at all temperatures, and notes that the linear-temperature treatment of \( E^{\circ} \) is itself an approximation.<sup>[3](https://srd.nist.gov/jpcrdreprint/1.555839.pdf)</sup>

**Slope in practice.** For a simple reduction \( \mathrm{M}^{n+} + n e^- \to \mathrm{M} \), a half-cell potential changes by 59/n mV per tenfold change in the ion's activity; for two-electron processes the change is 28 mV per decade.<sup>[5](https://chem.libretexts.org/Bookshelves/General_Chemistry/Chem1_(Lower)/16%3A_Electrochemistry/16.04%3A_The_Nernst_Equation)</sup> [Instrumentation](https://www.edgepedia.org/instrumentation) references quote the Nernst slope \( U_N \) as 59.16 mV at 25 °C for univalent positively charged ions, dependent on temperature and charge.<sup>[4](https://www.metrohm.com/content/dam/metrohm/shared/documents/monographs/81085088EN.pdf)</sup> Biological texts round this to 58 mV per tenfold gradient change at room temperature, a slightly different value from the 59.16 mV figure quoted at exactly 25 °C.<sup>[7](https://www.ncbi.nlm.nih.gov/books/NBK11102/)</sup><sup> • </sup><sup>[4](https://www.metrohm.com/content/dam/metrohm/shared/documents/monographs/81085088EN.pdf)</sup>

**Concentrations versus activities.** In analytical chemistry, concentrations are often used instead of activities, giving \( E_{eq} = E^{\circ\prime} - (RT/zF) \ln(c_{red}/c_{ox}) \) with the formal potential \( E^{\circ\prime} \), which can differ from the standard electrode potential owing to pH, ionic strength, and complex-forming substances.<sup>[1](https://goldbook.iupac.org/terms/view/09068)</sup> Activity coefficients are often not determinable because analytical methods give concentrations, not individual coefficients; standard potentials are therefore determined by extrapolating the formal potential to zero concentration, and keeping the ionic strength constant is the most likely way to keep activity coefficients constant.<sup>[10](https://www2.sci.u-szeged.hu/physchem/indexh_html_files/GYTK_Nernst_eng_2025.pdf)</sup> The distinction matters quantitatively: the Nernst equation cannot accurately predict half-cell potentials when total ionic concentration exceeds about \( 10^{-3} \) M because of ion-pair formation, and activities depart from concentrations above roughly \( 10^{-4} \) to \( 10^{-5} \) M depending on ion size and charge.<sup>[5](https://chem.libretexts.org/Bookshelves/General_Chemistry/Chem1_(Lower)/16%3A_Electrochemistry/16.04%3A_The_Nernst_Equation)</sup>

**Equilibrium constants.** Setting \( Q = 1/K \) at equilibrium gives \( K = 10^{(n \times E^{\circ}_{cell}/0.0592\ \mathrm{V})} \) at 298.15 K; OpenStax works an example with \( n = 2 \) and \( E^{\circ}_{cell} = 1.247 \) V.<sup>[6](https://openstax.org/books/chemistry/pages/17-4-the-nernst-equation)</sup>

## Applications in electrochemistry and analysis

**pH and ion-selective electrodes.** A glass membrane pH electrode obeys \( (E_{cell})_{samp} = K - (2.303RT/F) \cdot \mathrm{pH}_{samp} \), where \( K \) collects the reference electrode potential, the glass membrane asymmetry potential, and junction potentials.<sup>[12](https://chem.libretexts.org/Courses/Sewanee%3A_The_University_of_the_South/Instrumental_Analysis_(CHEM_311)/13%3A_Electrochemical_Methods/13.03%3A_Potentiometry/13.3.07%3A_Quantitative_Potentiometry)</sup> Because all contributions to \( K \) are uncertain and may change day to day and electrode to electrode, pH electrodes are calibrated against a standard buffer before use, and the sample pH is computed as \( \mathrm{pH}_{samp} = \mathrm{pH}_{std} - \{(E_{samp} - E_{std}) \cdot F\}/(2.303RT) \).<sup>[12](https://chem.libretexts.org/Courses/Sewanee%3A_The_University_of_the_South/Instrumental_Analysis_(CHEM_311)/13%3A_Electrochemical_Methods/13.03%3A_Potentiometry/13.3.07%3A_Quantitative_Potentiometry)</sup> Manufacturers summarize the temperature dependence as \( E(T) = E^{\circ}(T) - 0.1984 \cdot T \cdot \mathrm{pH} \), with \( T \) in Kelvin.<sup>[13](https://www.emerson.com/is/content/emerson/en/measurement-instrumentation/technical/products/liquid-analysis/documents/manual-theory-and-practice-of-ph-measurement.pdf)</sup> Ion-selective electrodes generalize this: \( E = E^{0} + 2.3(RT/nF) \log a_i \), with the slope in volts or millivolts.<sup>[14](https://www.fishersci.co.uk/content/dam/fssite/eu/brands/m/mettler-toledo/22667_TheoryGuide_ISE_EN.pdf)</sup> In redox measurements the electrode reports the equilibrium position of the conjugate redox couple, described by the Peters equation, rather than a direct ion activity.<sup>[4](https://www.metrohm.com/content/dam/metrohm/shared/documents/monographs/81085088EN.pdf)</sup>

**Corrosion and batteries.** In corrosion practice the Nernst equation is used in a form where \( E^{0} \) is approximated as \( E^{0\prime} \) because both activity coefficients are close to one.<sup>[15](https://assets.palmsens.com/app/uploads/2024/06/0538-PAS23-Corrosion-handbook_digital-.pdf)</sup> In zinc–air systems, corrosion consumes active zinc and shifts the open-circuit voltage, producing a mixed potential that must be assessed with non-equilibrium thermodynamics because the system is no longer at equilibrium.<sup>[8](https://www.degruyterbrill.com/document/doi/10.1515/jnet-2020-0070/html?lang=en)</sup> Simplified Nernst derivations can give incorrect open-circuit-voltage forms for systems with ion-exchange membranes or mixed potentials; a rigorous derivation from non-equilibrium thermodynamics gives better results for vanadium redox flow and zinc–air batteries.<sup>[8](https://www.degruyterbrill.com/document/doi/10.1515/jnet-2020-0070/html?lang=en)</sup>

## Nernst equation in biology

For an ion X crossing a membrane, the Nernst equation predicts the equilibrium potential \( E_X \) from the ratio of its concentrations on the two sides, using \( R \), \( T \), the ion's charge \( z \), and \( F \).<sup>[7](https://www.ncbi.nlm.nih.gov/books/NBK11102/)</sup> With the cell exterior defined as zero potential, a K⁺-permeable membrane with higher internal K⁺ shows an inside-negative potential; a Na⁺-selective membrane with tenfold more Na⁺ outside equilibrates at +58 mV. A tenfold change in the K⁺ gradient shifts the potential by 58/z mV; raising K⁺ on one side to 100 mM gives −116 mV.<sup>[7](https://www.ncbi.nlm.nih.gov/books/NBK11102/)</sup>

Real membranes are permeable to several ions at once. The [Goldman equation](https://www.edgepedia.org/goldman-equation), developed by David Goldman in 1943, is an extended version of the Nernst equation that accounts for the relative permeabilities of K⁺, Na⁺, and Cl⁻.<sup>[16](https://www.ncbi.nlm.nih.gov/books/NBK11111/)</sup> In the Goldman–Hodgkin–Katz form,

\[ V_{rest} = \frac{RT}{F} \ln \frac{P_K[K^+]_{out} + P_{Na}[Na^+]_{out} + P_{Cl}[Cl^-]_{in}}{P_K[K^+]_{in} + P_{Na}[Na^+]_{in} + P_{Cl}[Cl^-]_{out}} \]

When the membrane is permeable only to K⁺ the expression collapses back to the Nernst equation: −58 mV for K⁺ alone, +58 mV for Na⁺ alone, and, for the concentration ratios considered, 0 mV if the two are equally permeant.<sup>[16](https://www.ncbi.nlm.nih.gov/books/NBK11111/)</sup> For the squid giant axon at rest, permeability ratios \( P_K : P_{Na} : P_{Cl} = 1 : 0.03 : 0.1 \) yield a resting potential of about −101 mV, matching the K⁺ Nernst potential computed from 124 mM inside versus 2.25 mM outside; the Na⁺ Nernst potential is +59 mV (10.4 mM inside, 109 mM outside).<sup>[7](https://www.ncbi.nlm.nih.gov/books/NBK11102/)</sup><sup> • </sup><sup>[17](https://sites.pitt.edu/~phase/bard/bardware/classes/compneuro/passive3.pdf)</sup> The resting neuronal membrane is far more permeable to K⁺ than to Na⁺, giving a negative resting potential; a transient rise in Na⁺ permeability drives the action potential, after which K⁺ permeability is restored and the potential returns to rest.<sup>[16](https://www.ncbi.nlm.nih.gov/books/NBK11111/)</sup>

## Relation to Pourbaix diagrams and kinetic treatments

A [Pourbaix diagram](https://www.edgepedia.org/pourbaix-diagram) plots the equilibrium potential between a metal and its oxidized species as a function of pH, constructed from the relevant Nernst equations. For a simple metal/metal-ion reaction the equilibrium potential is independent of pH, whereas half-cell reactions containing H⁺ ions have pH-dependent Nernst equations.<sup>[9](https://www.doitpoms.ac.uk/tlplib/pourbaix/nersnt_detailed.php)</sup> For the hydrogen electrode \( E = -0.059 \cdot \mathrm{pH} \) and for oxygen \( E = 1.23 - 0.059 \cdot \mathrm{pH} \), giving the water stability window used in corrosion analysis.<sup>[5](https://chem.libretexts.org/Bookshelves/General_Chemistry/Chem1_(Lower)/16%3A_Electrochemistry/16.04%3A_The_Nernst_Equation)</sup> Because the Nernst equation is derived entirely from thermodynamics, a Pourbaix diagram shows which species is thermodynamically stable at a given \( E \) and pH but gives no information about the kinetics of the corrosion process.<sup>[18](https://www.doitpoms.ac.uk/tlplib/pourbaix/printall.php)</sup> Kinetic behavior under current is treated by separate relations such as Butler–Volmer kinetics, which appear alongside Nernst-based transport in modern nanoscale models.<sup>[19](https://www.pnas.org/doi/10.1073/pnas.2530250123)</sup>

## Limitations and non-ideal behavior

The equation's known failure regimes are specific. At high ionic strength, activities depart from concentrations and ion pairs form, so concentration-based Nernst predictions fail above about \( 10^{-3} \) M total ionic concentration.<sup>[5](https://chem.libretexts.org/Bookshelves/General_Chemistry/Chem1_(Lower)/16%3A_Electrochemistry/16.04%3A_The_Nernst_Equation)</sup> Mixed potentials, as in corroding zinc–air electrodes, and ion-exchange membrane systems fall outside the equilibrium assumptions of the simplified derivation.<sup>[8](https://www.degruyterbrill.com/document/doi/10.1515/jnet-2020-0070/html?lang=en)</sup>

Nanoscale systems add further deviations. Using single-particle SECCM voltammetry on PDI organic electrodes, the average half-wave potential was observed to move by 117.6 mV with each pH unit in acidic conditions (pH < 4), far beyond the 59 mV per pH that Nernstian behavior predicts for a 2H⁺/2e⁻ process—a discrepancy ascribed to kinetics limited by proton transport—whereas at pH values above 4 the half-wave potential shows almost no dependence on pH.<sup>[20](https://pubs.rsc.org/sc/article/doi/10.1039/d6sc04619c/1362142/Resolving-ion-coupled-electron-transfer-in-bipolar)</sup> [Grand-canonical DFT](https://www.edgepedia.org/grand-canonical-dft) on M–N–C single-atom catalysts from Mn to Cu shows significant deviations from Nernstian linear potential extrapolation in oxygen-reduction intermediate energies, with effects on overpotential following no clear trend across metals, in part because the sparse electronic states of single-atom catalysts prevent continuous charging as in metals.<sup>[21](https://iopscience.iop.org/article/10.1149/1945-7111/ae18fc)</sup>

## Recent developments

[Single-entity electrochemistry](https://www.edgepedia.org/single-entity-electrochemistry), in which the response of one nanoparticle, one molecule, or one nanohole is measured, has grown into a distinct field; reviews emphasize steady-state responses and statistical treatments with analytical advantages for nanoscale electrode measurements.<sup>[22](https://pubs.acs.org/ancham/article/96/20/8036/85761/Recent-Developments-in-Single-Entity)</sup> A 2024 [Angewandte Chemie](https://www.edgepedia.org/angewandte-chemie) review summarizes five years of advances in signal processing for the field, including machine-learning-based recognition, classification, and interpretation of nanoelectrode data.<sup>[23](https://onlinelibrary.wiley.com/doi/abs/10.1002/ange.202316551)</sup> Opto-iontronic microscopy uses electric-double-layer modulation and lock-in detection to probe ion dynamics during cyclic voltammetry in nanoholes, validated against a Poisson–Nernst–Planck–Butler–Volmer model, and can be extended to HER, OER, CO₂RR, and Li-ion intercalation studies.<sup>[19](https://www.pnas.org/doi/10.1073/pnas.2530250123)</sup> Nanopore single-molecule electrochemistry, surveyed in a recent Annual Review of Biophysics, now integrates machine learning for high-fidelity ionic current analysis.<sup>[24](https://www.annualreviews.org/content/journals/10.1146/annurev-biophys-021424-125106)</sup> On the battery side, optical microscopy with clustering analysis shows that a representative volume of a composite cathode contains only a few hundred particles out of millions, connecting particle-scale electrochemistry to cell-level performance,<sup>[25](https://www.pnas.org/doi/10.1073/pnas.2520136122)</sup> and a virtual reference electrode model predicts negative-electrode potential from two-electrode cell signals with root-mean-squared and mean absolute errors of 0.023 and 0.018 V, extending cycle life by 8.25 times versus a constant-current constant-voltage baseline under the tested low negative-to-positive capacity ratio.<sup>[26](https://www.nature.com/articles/s41467-026-76535-y)</sup>

## References

1. [IUPAC Gold Book – Nernst equation (09068)](https://goldbook.iupac.org/terms/view/09068)
2. [IUPAC Gold Book – standard electrode potential (S05912)](https://goldbook.iupac.org/terms/view/S05912)
3. [Standard Electrode Potentials and Temperature Coefficients, NIST Journal of Physical and Chemical Reference Data](https://srd.nist.gov/jpcrdreprint/1.555839.pdf)
4. [Metrohm Monograph: Electrodes and Measurement Practice](https://www.metrohm.com/content/dam/metrohm/shared/documents/monographs/81085088EN.pdf)
5. [Chem1/LibreTexts – The Nernst Equation](https://chem.libretexts.org/Bookshelves/General_Chemistry/Chem1_(Lower)/16%3A_Electrochemistry/16.04%3A_The_Nernst_Equation)
6. [OpenStax Chemistry 2e, §17.4 The Nernst Equation](https://openstax.org/books/chemistry/pages/17-4-the-nernst-equation)
7. [Neuroscience (NCBI Bookshelf) – The Forces that Create Membrane Potentials](https://www.ncbi.nlm.nih.gov/books/NBK11102/)
8. [Open-Circuit Voltage Comes from Non-Equilibrium Thermodynamics, Journal of Non-Equilibrium Thermodynamics](https://www.degruyterbrill.com/document/doi/10.1515/jnet-2020-0070/html?lang=en)
9. [DoITPoMS TLP – Detailed derivation of the Nernst equation, University of Cambridge](https://www.doitpoms.ac.uk/tlplib/pourbaix/nersnt_detailed.php)
10. [On the Nernst equation in electrochemistry, University of Szeged lecture notes (2025)](https://www2.sci.u-szeged.hu/physchem/indexh_html_files/GYTK_Nernst_eng_2025.pdf)
11. [IUPAC Pure and Applied Chemistry technical report – formal electrode potential](https://www.degruyterbrill.com/document/doi/10.1515/pac-2018-0109/pdf)
12. [Quantitative Potentiometry, Chemistry LibreTexts](https://chem.libretexts.org/Courses/Sewanee%3A_The_University_of_the_South/Instrumental_Analysis_(CHEM_311)/13%3A_Electrochemical_Methods/13.03%3A_Potentiometry/13.3.07%3A_Quantitative_Potentiometry)
13. [Emerson: Theory and Practice of pH Measurement](https://www.emerson.com/is/content/emerson/en/measurement-instrumentation/technical/products/liquid-analysis/documents/manual-theory-and-practice-of-ph-measurement.pdf)
14. [A Guide for Ion-Selective Electrodes, Mettler Toledo](https://www.fishersci.co.uk/content/dam/fssite/eu/brands/m/mettler-toledo/22667_TheoryGuide_ISE_EN.pdf)
15. [PalmSens Corrosion Handbook](https://assets.palmsens.com/app/uploads/2024/06/0538-PAS23-Corrosion-handbook_digital-.pdf)
16. [Electrochemical Equilibrium in an Environment with More Than One Permeant Ion, Neuroscience (NCBI Bookshelf)](https://www.ncbi.nlm.nih.gov/books/NBK11111/)
17. [University of Pittsburgh – Ionic basis of the membrane potential (Goldman–Hodgkin–Katz)](https://sites.pitt.edu/~phase/bard/bardware/classes/compneuro/passive3.pdf)
18. [The Nernst Equation and Pourbaix Diagrams, DoITPoMS, University of Cambridge](https://www.doitpoms.ac.uk/tlplib/pourbaix/printall.php)
19. [Optical voltammetry of redox processes inside a nanohole with opto-iontronic microscopy, PNAS](https://www.pnas.org/doi/10.1073/pnas.2530250123)
20. [Resolving ion-coupled electron transfer in bipolar organic electrodes by single-particle electrochemistry, Chemical Science](https://pubs.rsc.org/sc/article/doi/10.1039/d6sc04619c/1362142/Resolving-ion-coupled-electron-transfer-in-bipolar)
21. [Non-Nernstian Potential Responses of Oxygen Reduction Intermediates on Single-Atom Catalysts, J. Electrochem. Soc.](https://iopscience.iop.org/article/10.1149/1945-7111/ae18fc)
22. [Recent Developments in Single-Entity Electrochemistry, Analytical Chemistry](https://pubs.acs.org/ancham/article/96/20/8036/85761/Recent-Developments-in-Single-Entity)
23. [Emerging Data Processing Methods for Single-Entity Electrochemistry, Angewandte Chemie (2024)](https://onlinelibrary.wiley.com/doi/abs/10.1002/ange.202316551)
24. [Nanopore Single-Molecule Chemistry, Annual Review of Biophysics](https://www.annualreviews.org/content/journals/10.1146/annurev-biophys-021424-125106)
25. [On the scale of heterogeneity in composite electrodes of batteries, PNAS](https://www.pnas.org/doi/10.1073/pnas.2520136122)
26. [Battery smart sensing via a virtual reference electrode, Nature Communications](https://www.nature.com/articles/s41467-026-76535-y)

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*Topic: Encyclopedia › Physical world and mathematics › Chemistry › Chemical principles and methods › Thermodynamics and equilibrium › Chemical equilibrium › Complexation and redox equilibria*

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