Editing Nernst equation

{{Short description|Physical law in electrochemistry}}
{{Distinguish|Ernst equation}}
In [[electrochemistry]], the '''Nernst equation''' is a [[Thermodynamics#Chemical thermodynamics| chemical thermodynamical]] relationship that permits the calculation of the [[reduction potential]] of a reaction ([[half-cell]] or [[electrochemical cell|full cell]] reaction) from the [[standard electrode potential]], [[Thermodynamic temperature|absolute temperature]], the number of electrons involved in the [[redox|redox reaction]], and [[Thermodynamic activity|activities]] (often approximated by concentrations) of the [[chemical species]] undergoing reduction and [[oxidation]] respectively. It was named after  [[Walther Nernst]], a German [[physical chemist]] who formulated the equation.<ref name="isbn0-8412-1572-3">{{cite book |last1=Orna |first1=Mary Virginia |last2=Stock |first2=John |title=Electrochemistry, past and present |publisher=American Chemical Society |location=Columbus, OH |year=1989 |isbn=978-0-8412-1572-6 |oclc= 19124885}}</ref><ref name=Wahl2005>{{Cite journal | last = Wahl | year = 2005 | title = A Short History of Electrochemistry | journal = Galvanotechtnik | volume = 96 | issue = 8 | pages = 1820–1828 }}</ref>

==Expression==
===General form with chemical activities===

When an oxidized species ({{Math|Ox}}) accepts a number ''z'' of electrons ({{e-}}) to be  converted in its reduced form ({{Math|Red}}), the half-reaction is expressed as: 

: <chem>Ox + ze-  ->  Red</chem>

The [[reaction quotient]] (''{{Math|Q<sub>r</sub>}}''), also often called the ion activity product (''IAP''), is the ratio between the [[chemical activity|chemical activities]] (''a'') of the reduced form (the [[reductant]], {{Math|a<sub>Red</sub>}}) and the oxidized form (the [[oxidant]], {{Math|a<sub>Ox</sub>}}). The chemical activity of a dissolved species corresponds to its true thermodynamic concentration taking into account the electrical interactions between all ions present in solution at elevated concentrations. For a given dissolved species, its chemical activity (a) is the product of its [[activity coefficient]] (γ) by its [[Molar concentration|molar]] (mol/L solution), or [[Molality|molal]] (mol/kg water), [[concentration]] (C): a = γ C. So, if the concentration (''C'', also denoted here below with square brackets [ ]) of all the dissolved species of interest are sufficiently low and that their [[activity coefficient]]s are close to unity, their chemical activities can be approximated by their [[concentration]]s as commonly done when simplifying, or idealizing, a reaction for didactic purposes: 

: <math>Q_r = \frac{a_\text{Red}}{a_\text{Ox}} = \frac{[\operatorname{Red}]}{[\operatorname{Ox}]}</math>

At [[chemical equilibrium]], the ratio ''{{Math|Q<sub>r</sub>}}'' of the activity of the reaction product (''a''<sub>Red</sub>) by the reagent activity (''a''<sub>Ox</sub>) is equal to the [[equilibrium constant]] ''{{Math|K}}'' of the half-reaction: 

: <math>K = \frac{a_\text{Red}}{a_\text{Ox}}</math>

The standard thermodynamics also says that the actual [[Gibbs free energy]] {{math|Δ''G''}} is related to the free energy change under [[standard state]] {{math|Δ''G''{{su|p=<s>o</s>}}}} by the relationship:
<math display="block">\Delta G = \Delta G^{\ominus} + RT\ln Q_r</math>
where {{math|''Q''<sub>r</sub>}} is the [[reaction quotient]] and R is the [[gas constant|universal ideal gas constant]].
The cell potential {{mvar|E}} associated with the electrochemical reaction is defined as the decrease in Gibbs free energy per coulomb of charge transferred, which leads to the relationship <math display="block">\Delta G = -zFE.</math> The constant {{mvar|F}} (the [[Faraday constant]]) is a unit conversion factor {{math|1=''F'' = ''N''<sub>A</sub>''q''}}, where {{math|''N''<sub>A</sub>}} is the [[Avogadro constant]] and {{mvar|q}} is the fundamental [[electron]] charge. This immediately leads to the Nernst equation, which for an electrochemical half-cell is
<math display="block">E_\text{red} = E^\ominus_\text{red} - \frac{RT}{zF} \ln Q_r=E^\ominus_\text{red} - \frac{RT}{zF} \ln\frac{a_\text{Red}}{a_\text{Ox}}.</math>
For a complete electrochemical reaction (full cell), the equation can be written as
<math display="block">E_\text{cell} = E^\ominus_\text{cell} - \frac{RT}{zF} \ln Q_r</math>

where:
* {{math|''E''<sub>red</sub>}} is the half-cell [[reduction potential]] at the temperature of interest,
* {{math|''E''{{su|p=<s>o</s>|b=red}}}} is the [[standard electrode potential|''standard'' half-cell reduction potential]],
* {{math|''E''<sub>cell</sub>}} is the cell potential ([[electromotive force]]) at the temperature of interest,
* {{math|''E''{{su|p=<s>o</s>|b=cell}}}} is the [[standard cell potential]] in volts,
* {{mvar|R}} is the [[gas constant|universal ideal gas constant]]: {{math|1=''R'' = {{val|8.31446261815324|u=J K<sup>−1</sup> mol<sup>−1</sup>}}}},
* {{mvar|T}} is the temperature in [[kelvin]]s<!-- pluralized – see Kelvin#Usage conventions -->,
* {{mvar|z}} is the number of [[electron]]s transferred in the cell reaction or [[half-reaction]],
* {{mvar|F}} is Faraday's constant, the magnitude of charge (in [[coulomb]]s) per [[mole (unit)|mole]] of electrons: {{math|1=''F'' = {{val|96485.3321233100184|u=C mol<sup>−1</sup>}}}},
* {{math|''Q''<sub>r</sub>}} is the reaction quotient of the cell reaction, and,
* {{mvar|a}} is the chemical [[activity (chemistry)|activity]] for the relevant species, where {{math|''a''<sub>Red</sub>}} is the activity of the reduced form and {{math|''a''<sub>Ox</sub>}} is the activity of the oxidized form.

===Thermal voltage===

At room temperature (25&nbsp;°C), the [[thermal voltage]] <math>V_T=\frac{RT}{F}</math> is approximately 25.693&nbsp;mV. The Nernst equation is frequently expressed in terms of base-10 [[logarithms]] (''i.e.'', [[common logarithm]]s) rather than [[natural logarithms]], in which case it is written:

<math display="block">E = E^\ominus - \frac{V_T}{z} \ln\frac{a_\text{Red}}{a_\text{Ox}} = E^\ominus - \frac{\lambda V_T}{z} \log_{10}\frac{a_\text{Red}}{a_\text{Ox}}.</math>

where ''λ'' = ln(10) ≈ 2.3026 and ''λV<sub>T</sub>'' ≈ 0.05916 Volt.

===Form with activity coefficients and concentrations===

Similarly to equilibrium constants, activities are always measured with respect to the [[standard state]] (1&nbsp;mol/L for solutes, 1&nbsp;atm for gases, and T = 298.15 K, ''i.e.'', 25 °C or 77 °F). The chemical activity of a species {{math|i}}, {{math|''a''<sub>i</sub>}}, is related to the measured concentration {{math|''C''<sub>i</sub>}} via the relationship {{math|1=''a''<sub>i</sub> = ''γ''<sub>i</sub> ''C''<sub>i</sub>}}, where {{math|''γ''<sub>i</sub>}} is the [[activity coefficient]] of the species {{math|i}}. Because activity coefficients tend to unity at low concentrations, or are unknown or difficult to determine at medium and high concentrations, activities in the Nernst equation are frequently replaced by simple concentrations and then, formal standard reduction potentials <math>E^{\ominus'}_\text{red}</math> used.  

Taking into account the activity coefficients (<math>\gamma</math>) the Nernst equation becomes: 

<math display="block">E_\text{red} = E^\ominus_\text{red} - \frac{RT}{zF} \ln\left(\frac{\gamma_\text{Red}}{\gamma_\text{Ox}}\frac{C_\text{Red}}{C_\text{Ox}}\right)</math>

<math display="block">E_\text{red} = E^\ominus_\text{red} - \frac{RT}{zF} \left(\ln\frac{\gamma_\text{Red}}{\gamma_\text{Ox}} + \ln\frac{C_\text{Red}}{C_\text{Ox}}\right)</math>

<math display="block">E_\text{red} = \underbrace{\left(E^\ominus_\text{red} - \frac{RT}{zF} \ln\frac{\gamma_\text{Red}}{\gamma_\text{Ox}}\right)}_{E^{\ominus '}_\text{red}} - \frac{RT}{zF} \ln\frac{C_\text{Red}}{C_\text{Ox}}</math>

Where the first term including the [[activity coefficient]]s (<math>\gamma</math>) is denoted <math>E^{\ominus '}_\text{red}</math> and called the formal standard reduction potential, so that <math>E_\text{red}</math> can be directly expressed as a function of <math>E^{\ominus '}_\text{red}</math> and the concentrations in the simplest form of the Nernst equation:
 
<math display="block">E_\text{red}=E^{\ominus '}_\text{red} - \frac{RT}{zF} \ln\frac{C_\text{Red}}{C_\text{Ox}}</math>

===Formal standard reduction potential===
{{See also|Standard electrode potential}}

When wishing to use simple concentrations in place of activities, but that the activity coefficients are far from unity and can no longer be neglected and are unknown or too difficult to determine, it can be convenient to introduce the notion of the "so-called" standard formal reduction potential (<math>E^{\ominus '}_\text{red}</math>) which is related to the standard reduction potential as follows:<ref name="Bard_Faultner">{{Cite book| last1 = Bard| first1 = Allen J.| last2 = Faulkner| first2 = Larry R. | date = 2001| title = Electrochemical methods: Fundamentals and applications| edition = 2| publisher = John Wiley & Sons| location = New York| chapter = Chapter 2. Potentials and Thermodynamics of Cells – See: 2.1.6 Formal Potentials| page = 52}}</ref>
<math display="block">E^{\ominus '}_\text{red}=E^{\ominus}_\text{red}-\frac{RT}{zF}\ln\frac{\gamma_\text{Red}}{\gamma_\text{Ox}}</math>
So that the Nernst equation for the half-cell reaction can be correctly formally written in terms of concentrations as:
<math display="block">E_\text{red}=E^{\ominus '}_\text{red} - \frac{RT}{zF} \ln\frac{C_\text{Red}}{C_\text{Ox}}</math>
and likewise for the full cell expression.

According to Wenzel (2020),<ref name="Wenzel_2020">{{Cite web |title=4. Table of Standard State Electrochemical Potentials |last=Wenzel |first=Thomas |work=Chemistry LibreTexts |date=2020-06-09 |access-date=2021-11-24 |url= https://chem.libretexts.org/Bookshelves/Analytical_Chemistry/Supplemental_Modules_(Analytical_Chemistry)/Analytical_Sciences_Digital_Library/Active_Learning/In_Class_Activities/Electrochemical_Methods_of_Analysis/02_Text/4._Table_of_Standard_State_Electrochemical_Potentials}}</ref> a formal reduction potential <math>E^{\ominus '}_\text{red}</math> is the reduction potential that applies to a half reaction under a set of specified conditions such as, e.g., [[pH]], [[ionic strength]], or the concentration of [[complexing agent]]s.

The formal reduction potential <math>E^{\ominus '}_\text{red}</math> is often a more convenient, but conditional, form of the standard reduction potential, taking into account activity coefficients and specific conditions characteristics of the reaction medium. Therefore, its value is a conditional value, ''i.e.'', that it depends on the experimental conditions and because the ionic strength affects the activity coefficients, <math>E^{\ominus '}_\text{red}</math> will vary from medium to medium.<ref name="Bard_Faultner" /> Several definitions of the formal reduction potential can be found in the literature, depending on the pursued objective and the experimental constraints imposed by the studied system. The general definition of <math>E^{\ominus '}_\text{red}</math> refers to its value determined when <math>\frac{C_\text{red}} {C_\text{ox}} = 1</math>. A more particular case is when <math>E^{\ominus '}_\text{red}</math> is also determined at pH 7, as e.g. for redox reactions important in biochemistry or biological systems. 

====Determination of the formal standard reduction potential when {{mvar|{{sfrac|C<sub>red</sub>|C<sub>ox</sub>}}}} {{=}} 1====
{{See also|Table of standard reduction potentials for half-reactions important in biochemistry}}
The formal standard reduction potential <math>E^{\ominus '}_\text{red}</math> can be defined as the measured reduction potential <math>E_\text{red}</math> of the half-reaction at unity concentration ratio of the oxidized and reduced species (''i.e.'', when {{mvar|{{sfrac|C<sub>red</sub>|C<sub>ox</sub>}}}} {{=}} 1) under given conditions.<ref name="Kano_2002">{{Cite journal| last = Kano| first = Kenji| year = 2002| title = Redox potentials of proteins and other compounds of bioelectrochemical interest in aqueous solutions.| journal = Review of Polarography| volume = 48| issue = 1| pages = 29–46| doi = 10.5189/revpolarography.48.29| issn = 0034-6691| eissn = 1884-7692| accessdate = 2021-12-02| url = http://www.jstage.jst.go.jp/article/revpolarography1955/48/1/48_1_29/_article| doi-access = free}}</ref> 

Indeed: 

as, <math>E_\text{red} = E^{\ominus}_\text{red}</math>, when <math>\frac{a_\text{red}} {a_\text{ox}} = 1</math>, 

: <math>E_\text{red} = E^{\ominus'}_\text{red}</math>, when <math>\frac{C_\text{red}} {C_\text{ox}} = 1</math>, 

because <math>\ln{1} = 0</math>, and that the term <math>\frac{\gamma_\text{red}} {\gamma_\text{ox}}</math> is included in <math>E^{\ominus '}_\text{red}</math>. 

The formal reduction potential makes possible to more simply work with [[molar concentration|molar]] (mol/L, M) or [[molality|molal]] (mol/kg {{H2O}}, m) concentrations in place of [[chemical activity|activities]]. Because molar and molal concentrations were once referred as [[formal concentration]]s, it could explain the origin of the adjective ''formal'' in the expression ''formal'' potential.{{cn|date= December 2021}} 

The formal potential is thus the reversible potential of an electrode at equilibrium immersed in a solution where reactants and products are at unit concentration.<ref name="Freedictionary">{{Cite web |title=Formal potential |author= |work=TheFreeDictionary.com |date= |access-date=2021-12-06 |url= https://encyclopedia2.thefreedictionary.com/Formal+potential |language=English}}</ref> If any small incremental change of potential causes a change in the direction of the reaction, ''i.e.'' from reduction to oxidation or ''vice versa'', the system is close to equilibrium, reversible and is at its formal potential. When the formal potential is measured under [[standard conditions]] (''i.e.'' the activity of each dissolved species is 1 mol/L, T = 298.15 K = 25 °C = 77 °F, {{mvar|P<sub>gas</sub>}} = 1 bar) it becomes ''de facto'' a standard potential.<ref name="PalmSens">{{Cite web |title=Origins of electrochemical potentials — PalmSens |author=PalmSens |work=PalmSens |year=2021 |access-date=2021-12-06 |url=https://www.palmsens.com/knowledgebase-article/origins-of-electrochemical-potentials/}}</ref> <br />According to Brown and Swift (1949): 

<blockquote>"A formal potential is defined as the potential of a half-cell, measured against the [[standard hydrogen electrode]], when the total concentration of each [[oxidation state]] is one [[formal concentration|formal]]".<ref name="Brown_1949">{{Cite journal| last1 = Brown| first1 = Raymond A.| last2 = Swift| first2 = Ernest H.| year = 1949| title = The formal potential of the antimonous-antimonic half cell in hydrochloric acid solutions| journal = Journal of the American Chemical Society| volume = 71| issue = 8| pages = 2719–2723| doi = 10.1021/ja01176a035| issn = 0002-7863|quote = <u>Quote</u>: A formal potential is defined as the potential of a half-cell, measured against the standard hydrogen electrode, when the total concentration of each oxidation state is one formal.}}</ref></blockquote>

In this case, as for the standard reduction potentials, the concentrations of dissolved species remain equal to one [[molar concentration|molar]] (M) or one [[molality|molal]] (m), and so are said to be one [[formal concentration|formal]] (F). So, expressing the concentration {{mvar|C}} in [[molar concentration|molarity]] {{math|M}} (1 mol/L): 

: <math>\frac{C_\text{red}} {C_\text{ox}} = \frac{1 \, \mathrm{M}_\text{red}} {1 \, \mathrm{M}_\text{ox}} = 1</math>

The term formal concentration (F) is now largely ignored in the current literature and can be commonly assimilated to molar concentration (M), or molality (m) in case of thermodynamic calculations.<ref name="Harvey_2020">{{Cite web |last=Harvey |first=David |date=2020-06-15 |title=2.2: Concentration |work=Chemistry LibreTexts |access-date=2021-12-15 |url= https://chem.libretexts.org/Courses/BethuneCookman_University/B-CU%3A_CH-345_Quantitative_Analysis/Book%3A_Analytical_Chemistry_2.1_(Harvey)/02%3A_Basic_Tools_of_Analytical_Chemistry/2.02%3A_Concentration}}</ref> 

The formal potential is also found halfway between the two peaks in a cyclic [[Voltammetry|voltammogram]], where at this point the concentration of Ox (the oxidized species) and Red (the reduced species) at the electrode surface are equal. 

The [[activity coefficient]]s <math>\gamma_{red}</math> and <math>\gamma_{ox}</math> are included in the formal potential <math>E^{\ominus '}_\text{red}</math>, and because they depend on experimental conditions such as temperature, [[ionic strength]], and [[pH]], <math>E^{\ominus '}_\text{red}</math> cannot be referred as an immutable standard potential but needs to be systematically determined for each specific set of experimental conditions.<ref name="PalmSens" />

Formal reduction potentials are applied to simplify calculations of a considered system under given conditions and measurements interpretation. The experimental conditions in which they are determined and their relationship to the standard reduction potentials must be clearly described to avoid to confuse them with standard reduction potentials.

====Formal standard reduction potential at pH 7====
{{See also|Table of standard reduction potentials for half-reactions important in biochemistry}}
Formal standard reduction potentials (<math>E^{\ominus '}_\text{red}</math>) are also commonly used in [[biochemistry]] and [[cell biology]] for referring to [[standard reduction potential]]s measured at pH 7, a value closer to the [[pH]] of most physiological and intracellular fluids than the [[standard state]] pH of 0. The advantage is to defining a more appropriate [[redox|redox scale]] better corresponding to real conditions than the standard state. Formal standard reduction potentials (<math>E^{\ominus '}_\text{red}</math>) allow to more easily estimate if a [[redox]] reaction supposed to occur in a [[metabolic process]] or to fuel microbial activity under some conditions is feasible or not. 

While, standard reduction potentials always refer to the [[standard hydrogen electrode]] (SHE), with [{{H+}}] = 1 M corresponding to a pH 0, and <math>E^{\ominus}_\text{red H+}</math> fixed arbitrarily to zero by convention, it is no longer the case at a pH of 7. Then, the reduction potential <math>E_\text{red}</math> of a hydrogen electrode operating at pH 7 is −0.413 V with respect to the [[standard hydrogen electrode]] (SHE).<ref name="Voet_2016">{{cite book |first1=Donald |last1=Voet |first2=Judith G. |last2=Voet |first3=Charlotte W. |last3=Pratt |title=Fundamentals of Biochemistry: Life at the Molecular Level |chapter=Table 14-4 Standard Reduction Potentials for Some Biochemically Import Half-Reactions |chapter-url=https://books.google.com/books?id=9T7hCgAAQBAJ&pg=PA466 |date=2016 |publisher=Wiley |isbn=978-1-118-91840-1 |pages=466 |edition=5th}}</ref>

===Expression of the Nernst equation as a function of pH===
{{See also|Pourbaix diagram}}

The <math>E_h</math> and [[pH]] of a solution are related by the Nernst equation as commonly represented by a [[Pourbaix diagram]] {{nowrap|(<math>E_h</math> – [[pH]] plot)}}. <math>E_h</math> explicitly denotes <math>E_\text{red}</math> expressed versus the [[standard hydrogen electrode]] (SHE). For a [[half cell]] equation, conventionally written as a reduction reaction (''i.e.'', electrons accepted by an oxidant on the left side):

: <math chem>a \, A + b \, B + h \, \ce{H+} + z \, e^{-} \quad \ce{<=>} \quad c \, C + d \, D</math>

The half-cell [[standard reduction potential]] <math>E^{\ominus}_\text{red}</math> is given by

: <math>E^{\ominus}_\text{red} (\text{volt}) = -\frac{\Delta G^\ominus}{zF}</math>

where <math>\Delta G^\ominus</math> is the standard [[Gibbs free energy]] change, {{mvar|z}} is the number of electrons involved, and {{mvar|F}} is the [[Faraday's constant]]. The Nernst equation relates pH and <math>E_h</math> as follows:

: <math>E_h = E_\text{red} = E^{\ominus}_\text{red} - \frac{0.05916}{z} \log\left(\frac{\{C\}^c\{D\}^d}{\{A\}^a\{B\}^b}\right) - \frac{0.05916\,h}{z} \text{pH}</math> &nbsp;{{cn|date=June 2020}}

where curly brackets indicate [[Activity (chemistry)|activities]], and exponents are shown in the conventional manner. This equation is the equation of a straight line for <math>E_\text{red}</math> as a function of pH with a slope of <math>-0.05916\,\left(\frac{h}{z}\right)</math> volt (pH has no units). 

This equation predicts lower <math>E_\text{red}</math> at higher pH values. This is observed for the reduction of O<sub>2</sub> into H<sub>2</sub>O, or OH<sup>−</sup>, and for the reduction of H<sup>+</sup> into H<sub>2</sub>. <math>E_\text{red}</math> is then often noted as <math>E_h</math> to indicate that it refers to the [[standard hydrogen electrode]] (SHE) whose <math>E_\text{red}</math> = 0 by convention under standard conditions (T = 298.15 K = 25 °C = 77 F, P<sub>gas</sub> = 1 atm (1.013 bar), concentrations = 1 M and thus pH = 0). 

====Main factors affecting the formal standard reduction potentials====

The main factor affecting the formal reduction potentials in biochemical or biological processes is most often the pH. To determine approximate values of formal reduction potentials, neglecting in a first approach changes in activity coefficients due to ionic strength, the Nernst equation has to be applied taking care to first express the relationship as a function of pH. The second factor to be considered are the values of the concentrations taken into account in the Nernst equation. To define a formal reduction potential for a biochemical reaction, the pH value, the concentrations values and the hypotheses made on the activity coefficients must always be explicitly indicated. When using, or comparing, several formal reduction potentials they must also be internally consistent. 

Problems may occur when mixing different sources of data using different conventions or approximations (''i.e.'', with different underlying hypotheses). When working at the frontier between inorganic and biological processes (e.g., when comparing abiotic and biotic processes in geochemistry when microbial activity could also be at work in the system), care must be taken not to inadvertently directly mix [[standard reduction potential]]s versus SHE (pH = 0) with formal reduction potentials (pH = 7). Definitions must be clearly expressed and carefully controlled, especially if the sources of data are different and arise from different fields (e.g., picking and mixing data from classical electrochemistry and microbiology textbooks without paying attention to the different conventions on which they are based).

====Examples with a Pourbaix diagram====
{{Main|Pourbaix diagram}}

[[File:PourbaixWater.png|thumb|300px|right|[[Pourbaix diagram]] for water, including stability regions for water, oxygen and hydrogen at [[standard temperature and pressure]] (STP). The vertical scale (ordinate) is the electrode potential relative to a [[Standard hydrogen electrode|SHE]] electrode. The horizontal scale (abscissa) is the [[pH]] of the electrolyte (otherwise non-interacting). Above the top line oxygen will bubble off of the electrode until water is totally consumed. Likewise, below the bottom line hydrogen will bubble off of the electrode until water is totally consumed.]]

To illustrate the dependency of the reduction potential on pH, one can simply consider the two [[Redox|oxido-reduction equilibria]] determining the water stability domain in a [[Pourbaix diagram]] {{nowrap|(E<sub>h</sub>–pH plot)}}. When water is submitted to [[Electrolysis of water|electrolysis]] by applying a sufficient difference of [[Galvanic cell|electrical potential]] between two [[electrode]]s immersed in water, [[hydrogen]] is produced at the [[cathode]] (reduction of water protons) while [[oxygen]] is formed at the [[anode]] (oxidation of water oxygen atoms). The same may occur if a reductant stronger than hydrogen (e.g., metallic Na) or an oxidant stronger than oxygen (e.g., F<sub>2</sub>) enters in contact with water and reacts with it. In the {{nowrap|E<sub>h</sub>–pH plot}} here beside (the simplest possible version of a Pourbaix diagram), the water stability domain (grey surface) is delimited in term of redox potential by two inclined red dashed lines: 

* Lower stability line with hydrogen gas evolution due to the proton reduction at very low E<sub>h</sub>: 
: {{math|{{chem2|2 H+ + 2 e- <-> H2}} }}(cathode: reduction)

* Higher stability line with oxygen gas evolution due to water oxygen oxidation at very high E<sub>h</sub>: 
: {{math|{{chem2|2 H2O <-> O2 + 4 H+ + 4 e-}} }}(anode: oxidation)

When solving the Nernst equation for each corresponding reduction reaction (need to revert the water oxidation reaction producing oxygen), both equations have a similar form because the number of protons and the number of electrons involved within a reaction are the same and their ratio is one (2{{H+}}/2{{e-}} for H<sub>2</sub> and 4{{H+}}/4{{e-}} with {{O2}} respectively), so it simplifies when solving the Nernst equation expressed as a function of pH. 

The result can be numerically expressed as follows: 

: <math>E_\text{red} = E^{\ominus}_\text{red} - 0.05916 \ pH</math>

Note that the slopes of the two water stability domain upper and lower lines are the same (−59.16 mV/pH unit), so they are parallel on a [[Pourbaix diagram]]. As the slopes are negative, at high pH, both hydrogen and oxygen evolution requires a much lower reduction potential than at low pH.

For the reduction of H<sup>+</sup> into H<sub>2</sub> the here above mentioned relationship becomes: 

: <math>E_\text{red} = - 0.05916 \ pH</math> <br />because by convention <math>E^{\ominus}_\text{red}</math> = 0 V for the [[standard hydrogen electrode]] (SHE: pH = 0). <br />So, at pH = 7, <math>E_\text{red}</math> = −0.414 V for the reduction of protons. 

For the reduction of O<sub>2</sub> into 2 H<sub>2</sub>O the here above mentioned relationship becomes: 

: <math>E_\text{red} = 1.229 - 0.05916 \ pH</math> <br />because <math>E^{\ominus}_\text{red}</math> = +1.229 V with respect to the [[standard hydrogen electrode]] (SHE: pH = 0). <br />So, at pH = 7, <math>E_\text{red}</math> = +0.815 V for the reduction of oxygen.

The offset of −414 mV in <math>E_\text{red}</math> is the same for both reduction reactions because they share the same linear relationship as a function of pH and the slopes of their lines are the same. This can be directly verified on a Pourbaix diagram. For other reduction reactions, the value of the formal reduction potential at a pH of 7, commonly referred for biochemical reactions, also depends on the slope of the corresponding line in a Pourbaix diagram ''i.e.'' on the ratio ''{{frac|h|z}}'' of the number of {{H+}} to the number of {{e-}} involved in the reduction reaction, and thus on the [[stoichiometry]] of the half-reaction. The determination of the formal reduction potential at pH = 7 for a given biochemical half-reaction requires thus to calculate it with the corresponding Nernst equation as a function of pH. One cannot simply apply an offset of −414 mV to the E<sub>h</sub> value (SHE) when the ratio ''{{frac|h|z}}'' differs from 1.

==Applications in biology==
{{See also|Table of standard reduction potentials for half-reactions important in biochemistry}}
Beside [[Table of standard reduction potentials for half-reactions important in biochemistry|important redox reactions in biochemistry and microbiology]], the Nernst equation is also used in [[physiology]] for calculating the [[electric potential]] of a [[cell membrane]] with respect to one type of [[ion]]. It can be linked to the [[acid dissociation constant]].

===Nernst potential===
{{main|Reversal potential}}
The Nernst equation has a physiological application when used to calculate the potential of an ion of charge {{math|''z''}} across a membrane. This potential is determined using the concentration of the ion both inside and outside the cell:

<math display="block">E = \frac{R T}{z F} \ln\frac{[\text{ion outside cell}]}{[\text{ion inside cell}]} = 2.3026\frac{R T}{z F} \log_{10}\frac{[\text{ion outside cell}]}{[\text{ion inside cell}]}.</math>

When the membrane is in [[thermodynamic equilibrium]] (i.e., no net flux of ions), and if the cell is permeable to only one ion, then the [[membrane potential]] must be equal to the Nernst potential for that ion.

===Goldman equation===
{{main|Goldman equation}}
When the membrane is permeable to more than one ion, as is inevitably the case, the [[resting potential]] can be determined from the Goldman equation, which is a solution of [[GHK flux equation|G-H-K influx equation]] under the constraints that total current density driven by electrochemical force is zero:

<math display="block">E_\mathrm{m} = \frac{RT}{F} \ln{ \left( \frac{ \displaystyle\sum_i^N P_{\mathrm{M}^+_i}\left[\mathrm{M}^+_i\right]_\mathrm{out} + \displaystyle\sum_j^M P_{\mathrm{A}^-_j}\left[\mathrm{A}^-_j\right]_\mathrm{in}}{ \displaystyle\sum_i^N P_{\mathrm{M}^+_i}\left[\mathrm{M}^+_i\right]_\mathrm{in} + \displaystyle\sum_j^M P_{\mathrm{A}^-_j}\left[\mathrm{A}^-_j\right]_\mathrm{out}} \right) },</math>

where

* {{math|''E''<sub>m</sub>}} is the membrane potential (in [[volt]]s, equivalent to [[joule]]s per [[coulomb]]),
* {{math|''P''<sub>ion</sub>}} is the permeability for that ion (in meters per second),
* {{math|[ion]<sub>out</sub>}} is the extracellular concentration of that ion (in [[Mole (unit)|moles]] per cubic meter, to match the other [[SI]] units, though the units strictly don't matter, as the ion concentration terms become a dimensionless ratio),
* {{math|[ion]<sub>in</sub>}} is the intracellular concentration of that ion (in moles per cubic meter),
* {{mvar|R}} is the [[ideal gas constant]] (joules per [[kelvin]] per mole),
* {{mvar|T}} is the temperature in [[kelvin]]s<!-- pluralized – see Kelvin#Usage conventions -->,
* {{mvar|F}} is the [[Faraday constant|Faraday's constant]] (coulombs per mole).

The potential across the cell membrane that exactly opposes net diffusion of a particular ion through the membrane is called the Nernst potential for that ion. As seen above, the magnitude of the Nernst potential is determined by the ratio of the concentrations of that specific ion on the two sides of the membrane. The greater this ratio the greater the tendency for the ion to diffuse in one direction, and therefore the greater the Nernst potential required to prevent the diffusion. A similar expression exists that includes {{mvar|r}} (the absolute value of the transport ratio). This takes transporters with unequal exchanges into account. See: [[sodium-potassium pump]] where the transport ratio would be 2/3, so r equals 1.5 in the formula below. The reason why we insert a factor r = 1.5 here is that current density ''by electrochemical force'' J<sub>e.c.</sub>(Na<sup>+</sup>) + J<sub>e.c.</sub>(K<sup>+</sup>) is no longer zero, but rather J<sub>e.c.</sub>(Na<sup>+</sup>) + 1.5J<sub>e.c.</sub>(K<sup>+</sup>) = 0 (as for both ions flux by electrochemical force is compensated by that by the pump, i.e. J<sub>e.c.</sub> = −J<sub>pump</sub>), altering the constraints for applying GHK equation. The other variables are the same as above. The following example includes two ions: potassium (K<sup>+</sup>) and sodium (Na<sup>+</sup>). Chloride is assumed to be in equilibrium.
<math display="block">E_{m} = \frac{RT}{F} \ln{ \left( \frac{ rP_{\mathrm{K}^+}\left[\mathrm{K}^+\right]_\mathrm{out} + P_{\mathrm{Na}^+}\left[\mathrm{Na}^+\right]_\mathrm{out}}{ rP_{\mathrm{K}^+}\left[\mathrm{K}^+\right]_\mathrm{in} + P_{\mathrm{Na}^+}\left[\mathrm{Na}^+\right]_\mathrm{in}} \right) }.</math>

When chloride (Cl<sup>−</sup>) is taken into account, 

<math display="block">E_{m} = \frac{RT}{F} \ln{ \left( \frac{r P_{\mathrm{K}^+}\left[\mathrm{K}^+\right]_\mathrm{out} + P_{\mathrm{Na}^+}\left[\mathrm{Na}^+\right]_\mathrm{out} + P_{\mathrm{Cl}^-}\left[\mathrm{Cl}^-\right]_\mathrm{in}}{r P_{\mathrm{K}^+}\left[\mathrm{K}^+\right]_\mathrm{in} + P_{\mathrm{Na}^+}\left[\mathrm{Na}^+\right]_\mathrm{in} + P_{\mathrm{Cl}^-}\left[\mathrm{Cl}^-\right]_\mathrm{out}} \right) }.</math>

==Derivation==

===Using Boltzmann factor===
For simplicity, we will consider a solution of redox-active molecules that undergo a one-electron [[reversible reaction]]

: {{math|Ox + e<sup>−</sup> {{eqm}} Red}}

and that have a standard potential of zero, and in which the activities are well represented by the concentrations (i.e. unit activity coefficient). The [[chemical potential]] {{math|''μ''<sub>c</sub>}} of this solution is the difference between the energy barriers for taking electrons from and for giving electrons to the [[working electrode]] that is setting the solution's [[electrochemical potential]]. The ratio of oxidized to reduced molecules, {{sfrac|[Ox]|[Red]}}, is equivalent to the probability of being oxidized (giving electrons) over the probability of being reduced (taking electrons), which we can write in terms of the [[Boltzmann factor]] for these processes:
<math display="block">\begin{align}
\frac{[\mathrm{Red}]}{[\mathrm{Ox}]}
&= \frac{\exp \left(-[\text{barrier for gaining an electron}]/kT\right)}{\exp \left(-[\text{barrier for losing an electron}]/kT\right)}\\[6px]
&= \exp \left(\frac{\mu_\mathrm{c}}{kT} \right).
\end{align}</math>

Taking the natural logarithm of both sides gives
<math display="block">\mu_\mathrm{c} = kT \ln \frac{[\mathrm{Red}]}{[\mathrm{Ox}]}.</math>

If {{math|''μ''<sub>c</sub> ≠ 0}} at {{sfrac|[Ox]|[Red]}}&nbsp;=&nbsp;1, we need to add in this additional constant:
<math display="block">\mu_\mathrm{c} = \mu_\mathrm{c}^\ominus + kT \ln \frac{[\mathrm{Red}]}{[\mathrm{Ox}]}.</math>

Dividing the equation by {{mvar|e}} to convert from chemical potentials to electrode potentials, and remembering that {{math|1={{sfrac|''k''|''e''}} = {{sfrac|''R''|''F''}}}},<ref>{{math|1=''R'' = ''N''<sub>A</sub>''k''}}; see [[gas constant]]<br>{{math|1=''F'' = ''N''<sub>A</sub>''e''}}; see [[Faraday constant]]</ref> we obtain the Nernst equation for the one-electron process {{math|{{nowrap|Ox + e<sup>−</sup> {{eqm}} Red}} }}:

<math display="block">\begin{align}
E &= E^\ominus - \frac{kT}{e} \ln \frac{[\mathrm{Red}]}{[\mathrm{Ox}]} \\
 &= E^\ominus - \frac{RT}{F} \ln \frac{[\mathrm{Red}]}{[\mathrm{Ox}]}.
\end{align}</math>

===Using thermodynamics (chemical potential)===
Quantities here are given per molecule, not per [[Mole (unit)|mole]], and so [[Boltzmann constant]] {{math|''k''}} and the [[Elementary charge|electron charge]] {{math|''e''}} are used instead of the [[gas constant]] {{math|''R''}} and [[Faraday constant|Faraday's constant]] {{math|''F''}}. To convert to the molar quantities given in most chemistry textbooks, it is simply necessary to multiply by the [[Avogadro constant]]: {{math|1 = ''R'' = ''kN''<sub>A</sub>}} and {{math|1 = ''F'' = ''eN''<sub>A</sub>}}. The [[entropy]] of a molecule is defined as

<math display="block">S \ \stackrel{\mathrm{def}}{=}\ k \ln \Omega,</math>

where {{math|Ω}} is the number of states available to the molecule. The number of states must vary linearly with the volume {{math|''V''}} of the system (here an idealized system is considered for better understanding, so that activities are posited very close to the true concentrations). Fundamental statistical proof of the mentioned linearity goes beyond the scope of this section, but to see this is true it is simpler to consider usual [[isothermal process]] for an [[ideal gas]] where the change of entropy {{math|1=Δ''S'' = ''nR'' ln({{sfrac|''V''<sub>2</sub>|''V''<sub>1</sub>}})}} takes place. It follows from the definition of entropy and from the condition of constant temperature and quantity of gas {{mvar|n}} that the change in the number of states must be proportional to the relative change in volume {{math|{{sfrac|''V''<sub>2</sub>|''V''<sub>1</sub>}}}}. In this sense there is no difference in statistical properties of ideal gas atoms compared with the dissolved species of a solution with [[activity coefficient]]s equaling one: particles freely "hang around" filling the provided volume), which is inversely proportional to the [[Molar concentration|concentration]] {{mvar|c}}, so we can also write the entropy as
<math display="block">S = k\ln \ (\mathrm{constant}\times V) = -k\ln \ (\mathrm{constant}\times c).</math>

The change in entropy from some state 1 to another state 2 is therefore
<math display="block">\Delta S = S_2 - S_1 = - k \ln \frac{c_2}{c_1},</math>
so that the entropy of state 2 is
<math display="block">S_2 = S_1 - k \ln \frac{c_2}{c_1}.</math>

If state 1 is at standard conditions, in which {{math|''c''<sub>1</sub>}} is unity (e.g., 1&nbsp;atm or 1&nbsp;M), it will merely cancel the units of {{math|''c''<sub>2</sub>}}. We can, therefore, write the entropy of an arbitrary molecule A as
<math display="block">S(\mathrm{A}) = S^\ominus(\mathrm{A}) - k \ln [\mathrm{A}],</math>
where <math>S^\ominus</math> is the entropy at [[Standard temperature and pressure|standard conditions]] and [A] denotes the concentration of A. The change in entropy for a reaction

{{block indent|em=1.5|text={{mvar|a}}A + {{mvar|b}}B → {{mvar|y}}Y + {{mvar|z}}Z}}
is then given by
<math display="block">
\Delta S_\mathrm{rxn} = \big(yS(\mathrm{Y}) + zS(\mathrm{Z})\big) - \big(aS(\mathrm{A}) + bS(\mathrm{B})\big) 
= \Delta S^\ominus_\mathrm{rxn} - k \ln \frac{[\mathrm{Y}]^y [\mathrm{Z}]^z}{[\mathrm{A}]^a [\mathrm{B}]^b}.
</math>

We define the ratio in the last term as the [[reaction quotient]]:
<math display="block">Q_r = \frac{\displaystyle\prod_j a_j^{\nu_j}}{\displaystyle\prod_i a_i^{\nu_i}} \approx \frac{[\mathrm{Z}]^z [\mathrm{Y}]^y}{[\mathrm{A}]^a [\mathrm{B}]^b},</math>
where the numerator is a product of reaction product [[Thermodynamic activity|activities]], {{math|''a<sub>j</sub>''}}, each raised to the power of a [[stoichiometric coefficient]], {{math|''ν<sub>j</sub>''}}, and the denominator is a similar product of reactant activities. All activities refer to a time {{math|''t''}}. Under certain circumstances (see [[chemical equilibrium]]) each activity term such as {{math|''a{{su|b=j|p=ν<sub>j</sub>}}''}} may be replaced by a concentration term, [A].In an electrochemical cell, the cell potential {{math|''E''}} is the [[chemical potential]] available from [[redox]] reactions ({{math|1=''E'' = {{sfrac|''μ''<sub>c</sub>|''e''}}}}). {{math|''E''}} is related to the [[Gibbs free energy]] change {{math|Δ''G''}} only by a constant: 
{{math|1=Δ''G'' = −''zFE''}}, where {{math|''n''}} is the number of electrons transferred and {{math|''F''}} is the [[Faraday constant]]. There is a negative sign because a spontaneous reaction has a negative [[Gibbs free energy]] {{math|Δ''G''}} and a positive potential {{math|''E''}}. The Gibbs free energy is related to the entropy by {{math|1=''G'' = ''H'' − ''TS''}}, where {{math|''H''}} is the [[enthalpy]] and {{math|''T''}} is the temperature of the system. Using these relations, we can now write the change in Gibbs free energy,
<math display="block">\Delta G = \Delta H - T \Delta S = \Delta G^\ominus + kT \ln Q_r,</math>
and the cell potential,
<math display="block">E = E^\ominus - \frac{kT}{ze} \ln Q_r.</math>

This is the more general form of the Nernst equation. 

For the redox reaction {{nowrap|Ox + {{mvar|z}} e<sup>−</sup> → Red}}, 
<math display="block">Q_r = \frac{[\mathrm{Red}]}{[\mathrm{Ox}]},</math>
and we have:
<math display="block">\begin{align}
E &= E^\ominus - \frac{kT}{ze} \ln \frac{[\mathrm{Red}]}{[\mathrm{Ox}]} \\
&= E^\ominus - \frac{RT}{zF} \ln \frac{[\mathrm{Red}]}{[\mathrm{Ox}]} \\
&= E^\ominus - \frac{RT}{zF} \ln Q_r.
\end{align}</math>

The cell potential at [[standard temperature and pressure]] (STP) <math>E^\ominus</math> is often replaced by the formal potential <math>E^{\ominus'}</math>, which includes the [[activity coefficient]]s of the dissolved species under given experimental conditions (T, P, [[ionic strength]], [[pH]], and complexing agents) and is the potential that is actually measured in an electrochemical cell.

==Relation to the chemical equilibrium==
The standard [[Gibbs free energy]] <math>\Delta G^\ominus</math> is related to the [[equilibrium constant]] {{mvar|K}} as follows:<ref name="Chem_Libre_Texts">{{Cite web| title = 20.5: Gibbs energy and redox reactions| work = Chemistry LibreTexts| date = 2014-11-18| accessdate = 2021-12-06| url = https://chem.libretexts.org/Bookshelves/General_Chemistry/Map%3A_Chemistry_-_The_Central_Science_(Brown_et_al.)/20%3A_Electrochemistry/20.5%3A_Gibbs_Energy_and_Redox_Reactions}}</ref> 

: <math>\Delta G^\ominus = -RT \ln{K}</math>

At the same time, <math>\Delta G^\ominus</math> is also equal to the product of the total charge  ({{mvar|zF}}) transferred during the reaction and the cell potential (<math>E^\ominus_{cell}</math>):

: <math>\Delta G^\ominus  = -zF E^\ominus_{cell}</math>

The sign is negative, because the considered system performs the work and thus releases energy. 

So, 

: <math>-zFE^\ominus_{cell} = -RT \ln{K}</math>

And therefore: 

: <math>E^\ominus_{cell} = \frac{RT} {zF} \ln{K}</math>

Starting from the Nernst equation, one can also demonstrate the same relationship in the reverse way. 

At [[chemical equilibrium]], or [[thermodynamic equilibrium]], the [[electrochemical potential]] {{math|1=(''E'')  = 0}} and therefore the [[reaction quotient]] ({{math|''Q<sub>r</sub>''}}) attains the special value known as the [[equilibrium constant]] ({{math|''K''<sub>eq</sub>}}): 
: {{math|1 = ''Q<sub>r</sub>'' = ''K''<sub>eq</sub>}} 

Therefore,

<math display="block">\begin{align}
0 &= E^\ominus - \frac{RT}{z F} \ln K \\
\frac{RT}{z F} \ln K & = E^\ominus \\
\ln K &= \frac{z F E^\ominus}{RT}
\end{align}</math>

Or at [[standard state]],

<math display="block">\log_{10} K = \frac{zE^\ominus}{\lambda V_T} = \frac{zE^\ominus}{0.05916\text{ V}} \quad\text{at }T = 298.15~\text{K}</math>

We have thus related the [[standard electrode potential]] and the [[equilibrium constant]] of a redox reaction.

==Limitations==
In dilute solutions, the Nernst equation can be expressed directly in the terms of concentrations (since activity coefficients are close to unity). But at higher concentrations, the true activities of the ions must be used. This complicates the use of the Nernst equation, since estimation of non-ideal activities of ions generally requires experimental measurements. The Nernst equation also only applies when there is no net current flow through the electrode. The activity of ions at the electrode surface changes [[electrochemical kinetics|when there is current flow]], and there are additional [[overpotential]] and resistive loss terms which contribute to the measured potential. 

At very low concentrations of the potential-determining ions, the potential predicted by Nernst equation approaches toward {{math|±∞}}. This is physically meaningless because, under such conditions, the [[exchange current density]] becomes very low, and there may be no thermodynamic equilibrium necessary for Nernst equation to hold. The electrode is called unpoised in such case. Other effects tend to take control of the electrochemical behavior of the system, like the involvement of the [[solvated electron]] in electricity transfer and electrode equilibria, as analyzed by [[Alexander Frumkin]] and B. Damaskin,<ref>[[J. Electroanal. Chem.]], 79 (1977), 259-266</ref> Sergio Trasatti, etc.

===Time dependence of the potential===
The expression of time dependence has been established by Karaoglanoff.<ref>{{citation |first=Z. |last=Karaoglanoff |author-link=Zakhari Karaoglanoff |title=Über Oxydations- und Reduktionsvorgänge bei der Elektrolyse von Eisensaltzlösungen |language=de |trans-title=On Oxidation and Reduction Processes in the Electrolysis of Iron Salt Solutions |journal=Zeitschrift für Elektrochemie |volume=12 |issue=1 |pages=5–16 |date=January 1906 |doi=10.1002/bbpc.19060120105 |url=https://zenodo.org/record/1424952 }}</ref><ref>{{citation |title=Electrochemical Dictionary |editor1-first=Allen J. |editor1-last=Bard |editor2-first=György |editor2-last=Inzelt |editor3-first=Fritz |editor3-last=Scholz |contribution=Karaoglanoff equation |pages=527&ndash;528 |url=https://books.google.com/books?id=4TBWg3dIyKQC&pg=PA527 |publisher=Springer|isbn=9783642295515 |date=2012-10-02 }}</ref><ref>{{citation |title=Introduction to Polarography and Allied Techniques |first=Kamala |last=Zutshi |pages=127–128 |url=https://books.google.com/books?id=WmiaCVH-MEIC&pg=PA127 |isbn= 9788122417913|year=2008|publisher=New Age International }}</ref><ref>{{Cite book |url=https://books.google.com/books?id=zCMSAAAAIAAJ&pg=PA316 |title=The Journal of Physical Chemistry |date=1906 |publisher=Cornell University |language=en}}</ref>

==Significance in other scientific fields==
The Nernst equation has been involved in the scientific controversy about [[cold fusion]]. Fleischmann and Pons, claiming that cold fusion could exist, calculated that a [[palladium]] [[cathode]] immersed in a [[heavy water]] electrolysis cell could achieve up to 10<sup>27</sup> atmospheres of pressure inside the [[crystal lattice]] of the metal of the cathode, enough pressure to cause spontaneous [[nuclear fusion]]. In reality, only 10,000–20,000 atmospheres were achieved. The American physicist [[John R. Huizenga]] claimed their original calculation was affected by a misinterpretation of the Nernst equation.<ref>{{cite book| last=Huizenga | first=John R. | author-link=John R. Huizenga | title=Cold Fusion: The Scientific Fiasco of the Century | edition=2 | location=Oxford and New York | publisher=Oxford University Press | year=1993 | pages=33, 47 | isbn=978-0-19-855817-0 }}</ref> He cited a paper about Pd–Zr [[Alloy|alloys]].<ref name="Huot1989">{{cite journal|last1=Huot|first1=J. Y.|title=Electrolytic Hydrogenation and Amorphization of Pd-Zr Alloys|journal=Journal of the Electrochemical Society|volume=136|issue=3|year=1989|pages=630–635|issn=0013-4651|doi=10.1149/1.2096700|bibcode=1989JElS..136..630H }}</ref>

The Nernst equation allows the calculation of the extent of reaction between two [[redox]] systems and can be used, for example, to assess whether a particular reaction will go to completion or not. At [[chemical equilibrium]], the [[electromotive force]]s (emf) of the two half cells are equal. This allows the [[equilibrium constant]]  {{math|''K''}} of the reaction to be calculated and hence the extent of the reaction.

==See also==

* [[Concentration cell]]
* [[Reduction potential#Nernst equation|Dependency of reduction potential on pH]]
* [[Electrode potential]]
* [[Galvanic cell]]
* [[Goldman equation]]
* [[Membrane potential]]
* [[Nernst–Planck equation]]
* [[Pourbaix diagram]]
* [[Reduction potential]]
* [[Solvated electron]]
* [[Standard electrode potential]]
* [[Standard electrode potential (data page)]]
* [[Table of standard reduction potentials for half-reactions important in biochemistry|Standard apparent reduction potentials in biochemistry at pH 7 (data page)]]

==References==
{{Reflist}}

==External links==

* [https://web.archive.org/web/20100808191814/http://www.nernstgoldman.physiology.arizona.edu/ Nernst/Goldman Equation Simulator]
* [http://www.physiologyweb.com/calculators/nernst_potential_calculator.html Nernst Equation Calculator]
* [http://thevirtualheart.org/GHKindex.html Interactive Nernst/Goldman Java Applet]
* [http://www.doitpoms.ac.uk/tlplib/pourbaix/index.php DoITPoMS Teaching and Learning Package- "The Nernst Equation and Pourbaix Diagrams"]
* {{Cite web| title = 20.5: Gibbs energy and redox reactions| work = Chemistry LibreTexts| accessdate = 2021-12-06| date = 2014-11-18| url = https://chem.libretexts.org/Bookshelves/General_Chemistry/Map%3A_Chemistry_-_The_Central_Science_(Brown_et_al.)/20%3A_Electrochemistry/20.5%3A_Gibbs_Energy_and_Redox_Reactions}}

[[Category:Walther Nernst]]
[[Category:Electrochemical equations]]
[[Category:Eponymous equations of physics]]