The Nernst equation is a fundamental relation in electrochemistry that describes how an equilibrium electrode potential, or the reversible voltage of an electrochemical cell, depends on chemical composition and temperature. Named after Walther Nernst, it connects the thermodynamic driving force of a reaction with an electrical potential difference. Its application requires identifying the reaction, the relevant species, and the equilibrium conditions at the interfaces involved. (goldbook.iupac.org)
Mathematical form
For a specified oxidation–reduction reaction, the equation is commonly written
[ E=E^\circ-\frac{RT}{nF}\ln Q. ]
Here, (E) is the equilibrium electrode potential or reversible cell voltage; (E^\circ) is its standard value; (R) is the molar gas constant; (T) is absolute temperature, expressed in kelvin; (n) is the number of electrons transferred in the balanced reaction; and (F) is the Faraday constant. The dimensionless reaction quotient (Q) is constructed from the activities of the participating species. Electrode reactions are conventionally written as reductions. (goldbook.iupac.org)
At (25,^\circ\mathrm C), or (298.15,\mathrm K), conversion from the natural logarithm to a base-10 logarithm gives
[ E=E^\circ-\frac{0.05916\ \mathrm V}{n}\log_{10}Q. ]
Thus, a tenfold increase in (Q) lowers the potential by approximately (59.16/n) millivolts at this temperature. The numerical coefficient changes with temperature; it is not a universal constant independent of experimental conditions. (goldbook.iupac.org)
Thermodynamic basis
The equation follows from thermodynamics. For a reaction at a specified composition, the change in Gibbs free energy satisfies
[ \Delta_rG=\Delta_rG^\circ+RT\ln Q. ]
For a reversible electrochemical cell, the corresponding electrical relations are
[ \Delta_rG=-nFE,\qquad \Delta_rG^\circ=-nFE^\circ. ]
Substituting these expressions yields the Nernst equation. A positive cell voltage corresponds to a negative reaction Gibbs energy for the reaction direction as written. This describes thermodynamic favorability, not the speed at which the reaction occurs. (openstax.org)
At overall chemical equilibrium, the reaction Gibbs energy and cell voltage are zero, while (Q) equals the equilibrium constant (K). Consequently,
[ E^\circ=\frac{RT}{nF}\ln K. ]
This relationship allows equilibrium constants to be inferred from standard cell potentials. Overall reaction equilibrium must be distinguished from local equilibrium at individual electrodes: a cell may have equilibrated electrode interfaces while retaining a nonzero voltage between them. (openstax.org)
Activities and standard states
The rigorous equation uses thermodynamic activities, rather than uncorrected concentrations. Activity expresses a species’ chemical potential relative to a chosen standard state. For a dissolved species on a concentration scale, it can be represented as
[ a_i=\gamma_i\frac{c_i}{c^\circ}, ]
where (\gamma_i) is an activity coefficient, (c_i) is the molar concentration, and (c^\circ) is the standard concentration. Both activity and the logarithm’s argument are dimensionless. Activity coefficients account for departures from ideal-solution behavior. (goldbook.iupac.org)
In practical measurements, concentrations are often known more readily than activities. A formal electrode potential, (E^{\circ\prime}), can then be used for a defined medium. Unlike a standard potential, this conditional value depends on the electrolyte composition. Effects associated with pH, ionic strength, and complex formation may therefore be incorporated into the formal potential rather than treated explicitly through individual activities. (goldbook.iupac.org)
Reaction direction and an illustrative calculation
The reaction quotient and electron number must correspond to the same balanced chemical reaction. For the reduction
[ \mathrm{Ox}+ne^-\rightleftharpoons\mathrm{Red}, ]
with unit stoichiometric coefficients for the oxidized and reduced species,
[ E=E^\circ-\frac{RT}{nF} \ln\frac{a_{\mathrm{Red}}}{a_{\mathrm{Ox}}}. ]
Increasing the oxidized species’ activity relative to the reduced species raises the reduction potential. Increasing the reduced species’ relative activity lowers it. (goldbook.iupac.org)
For an illustrative one-electron couple at (25,^\circ\mathrm C), suppose (a_{\mathrm{Red}}/a_{\mathrm{Ox}}=100). Substitution gives
[ E-E^\circ=-0.05916\log_{10}(100) =-0.11832\ \mathrm V. ]
This calculated shift depends on the activity ratio; the absolute potential also requires the couple’s standard potential. (goldbook.iupac.org)
Ionic equilibrium across membranes
The same thermodynamic reasoning applies to an ion distributed across a selectively permeable cell membrane. Defining the membrane potential as inside minus outside, the equilibrium potential for ion (i) is
[ E_i=\frac{RT}{z_iF} \ln\frac{a_{i,\mathrm{outside}}}{a_{i,\mathrm{inside}}}, ]
where (z_i) is the signed charge number. Unlike the positive electron-transfer number (n) in the electrochemical form, (z_i) is negative for anions. Concentration ratios commonly approximate activity ratios in introductory treatments. (ncbi.nlm.nih.gov)
At this potential, the chemical and electrical contributions to the ion’s electrochemical gradient balance, so there is no net passive movement of that ion. This does not imply that microscopic ionic movement ceases. For a membrane permeable to only one ionic species, its equilibrium voltage is that species’ Nernst potential. (ncbi.nlm.nih.gov)
Real membranes can conduct several ionic species simultaneously. Their voltage therefore need not equal any individual Nernst potential. In physiology, the equation identifies each ion’s equilibrium potential rather than, by itself, determining the complete voltage of a multi-ion membrane. (ncbi.nlm.nih.gov)
Scope and limitations
The Nernst equation concerns reversible potentials and interfacial equilibrium. It is not a law of chemical kinetics and does not predict current or reaction rate. Applying it to measured cell voltages requires examining whether the electrode reactions are equilibrated and whether additional interfacial or membrane contributions are present. A thermodynamically correct expression can depend on the actual cell configuration, not merely on a formally balanced overall reaction. (arxiv.org)