Editing Nernst equation (section)

==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.