Editing Nernst equation (section)

===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>