The values below are standard apparent reduction potentials (E°') for electro-biochemical half-reactions measured at 25 °C, 1 atmosphere and a pH of 7 in aqueous solution. The actual physiological potential depends on the ratio of the reduced (Red) and oxidized (Ox) forms according to the Nernst equation and the thermal voltage. When an oxidizer (Ox) accepts a number z of electrons ( e−) to be converted in its reduced form (Red), the half-reaction is expressed as:
Ox + z e− → Red The reaction quotient (Qr) is the ratio of the chemical activity (ai) of the reduced form (the reductant, aRed) to the activity of the oxidized form (the oxidant, aox). It is equal to the ratio of their concentrations (Ci) only if the system is sufficiently diluted and the activity coefficients (γi) are close to unity (ai = γi Ci):
Q r = a Red a Ox = C Red C Ox {\displaystyle Q_{r}={\frac {a_{\text{Red}}}{a_{\text{Ox}}}}={\frac {C_{\text{Red}}}{C_{\text{Ox}}}}}
The Nernst equation is a function of Qr and can be written as follows:
E red = E red ⊖ − R T z F ln Q r = E red ⊖ − R T z F ln a Red a Ox . {\displaystyle E_{\text{red}}=E_{\text{red}}^{\ominus }-{\frac {RT}{zF}}\ln Q_{r}=E_{\text{red}}^{\ominus }-{\frac {RT}{zF}}\ln {\frac {a_{\text{Red}}}{a_{\text{Ox}}}}.}
At chemical equilibrium, the reaction quotient Qr of the product activity (aRed) by the reagent activity (aOx) is equal to the equilibrium constant (K) of the half-reaction and in the absence of driving force (ΔG = 0) the potential (Ered) also becomes nul. The numerically simplified form of the Nernst equation is expressed as:
E red = E red ⊖ − 0.059 V z log 10 a Red a Ox {\displaystyle E_{\text{red}}=E_{\text{red}}^{\ominus }-{\frac {0.059\ V}{z}}\log _{10}{\frac {a_{\text{Red}}}{a_{\text{Ox}}}}}
Where E red ⊖ {\displaystyle E_{\text{red}}^{\ominus }} is the standard reduction potential of the half-reaction expressed versus the standard reduction potential of hydrogen. For standard conditions in electrochemistry (T = 25 °C, P = 1 atm and all concentrations being fixed at 1 mol/L, or 1 M) the standard reduction potential of hydrogen E red H+ ⊖ {\displaystyle E_{\text{red H+}}^{\ominus }} is fixed at zero by convention as it serves of reference. The standard hydrogen electrode (SHE), with [ H+] = 1 M works thus at a pH = 0. At pH = 7, when [ H+] = 10−7 M, the reduction potential E red {\displaystyle E_{\text{red}}} of H+ differs from zero because it depends on pH. Solving the Nernst equation for the half-reaction of reduction of two protons into hydrogen gas gives:
2 H+ + 2 e− ⇌ H2
E red = E red ⊖ − 0.05916 p H {\displaystyle E_{\text{red}}=E_{\text{red}}^{\ominus }-0.05916\ pH}
E red = 0 − ( 0.05916 × 7 ) = − 0.414 V {\displaystyle E_{\text{red}}=0-\left(0.05916\ {\text{×}}\ 7\right)=-0.414\ V}
In biochemistry and in biological fluids, at pH = 7, it is thus important to note that the reduction potential of the protons ( H+) into hydrogen gas H2 is no longer zero as with the standard hydrogen electrode (SHE) at 1 M H+ (pH = 0) in classical electrochemistry, but that E red = − 0.414 V {\displaystyle E_{\text{red}}=-0.414\mathrm {V} } versus the standard hydrogen electrode (SHE). The same also applies for the reduction potential of oxygen:
O2 + 4 H+ + 4 e− ⇌ 2 H2O For O2, E red ⊖ {\displaystyle E_{\text{red}}^{\ominus }} = 1.229 V, so, applying the Nernst equation for pH = 7 gives:
E red = E red ⊖ − 0.05916 p H {\displaystyle E_{\text{red}}=E_{\text{red}}^{\ominus }-0.05916\ pH}
E red = 1.229 − ( 0.05916 × 7 ) = 0.815 V {\displaystyle E_{\text{red}}=1.229-\left(0.05916\ {\text{×}}\ 7\right)=0.815\ V}
For obtaining the values of the reduction potential at pH = 7 for the redox reactions relevant for biological systems, the same kind of conversion exercise is done using the corresponding Nernst equation expressed as a function of pH. The conversion is simple, but care must be taken not to inadvertently mix reduction potential converted at pH = 7 with other data directly taken from tables referring to SHE (pH = 0).
Expression of the Nernst equation as a function of pH
The E h {\displaystyle E_{h}} and pH of a solution are related by the Nernst equation as commonly represented by a Pourbaix diagram ( E h {\displaystyle E_{h}} – pH plot). For a half cell equation, conventionally written as a reduction reaction (i.e., electrons accepted by an oxidant on the left side):
a A + b B + h H + + z e − ↽ − − ⇀ c C + d D {\displaystyle a\,A+b\,B+h\,{\ce {H+}}+z\,e^{-}\quad {\ce {<=>}}\quad c\,C+d\,D}
The half-cell standard reduction potential E red ⊖ {\displaystyle E_{\text{red}}^{\ominus }} is given by
E red ⊖ ( volt ) = − Δ G ⊖ z F {\displaystyle E_{\text{red}}^{\ominus }({\text{volt}})=-{\frac {\Delta G^{\ominus }}{zF}}}
where Δ G ⊖ {\displaystyle \Delta G^{\ominus }} is the standard Gibbs free energy change, z is the number of electrons involved, and F is Faraday's constant. The Nernst equation relates pH and E h {\displaystyle E_{h}} :
E h = E red = E red ⊖ − 0.05916 z log ( { C } c { D } d { A } a { B } b ) − 0.05916 h z pH {\displaystyle E_{h}=E_{\text{red}}=E_{\text{red}}^{\ominus }-{\frac {0.05916}{z}}\log \left({\frac {\{C\}^{c}\{D\}^{d}}{\{A\}^{a}\{B\}^{b}}}\right)-{\frac {0.05916\,h}{z}}{\text{pH}}} where curly braces { } indicate activities, and exponents are shown in the conventional manner.This equation is the equation of a straight line for E h {\displaystyle E_{h}} as a function of pH with a slope of − 0.05916 ( h z ) {\displaystyle -0.05916\,\left({\frac {h}{z}}\right)} volt (pH has no units). This equation predicts lower E h {\displaystyle E_{h}} at higher pH values. This is observed for the reduction of O2 into H2O, or OH−, and for reduction of H+ into H2.
Formal standard reduction potential combined with the pH dependency To obtain the reduction potential as a function of the measured concentrations of the redox-active species in solution, it is necessary to express the activities as a function of the concentrations.
E h = E red = E red ⊖ − 0.05916 z log ( { C } c { D } d { A } a { B } b ) − 0.05916 h z pH {\displaystyle E_{h}=E_{\text{red}}=E_{\text{red}}^{\ominus }-{\frac {0.05916}{z}}\log \left({\frac {\{C\}^{c}\{D\}^{d}}{\{A\}^{a}\{B\}^{b}}}\right)-{\frac {0.05916\,h}{z}}{\text{pH}}}
Given that the chemical activity denoted here by { } is the product of the activity coefficient γ by the concentration denoted by [ ]: ai = γi·Ci, here expressed as {X} = γx [X] and {X}x = (γx)x [X]x and replacing the logarithm of a product by the sum of the logarithms (i.e., log (a·b) = log a + log b), the log of the reaction quotient ( Q r {\displaystyle Q_{r}} ) (without {H+} already isolated apart in the last term as h pH) expressed here above with activities { } becomes:
log ( { C } c { D } d { A } a { B } b ) = log ( ( γ C ) c ( γ D ) d ( γ A ) a ( γ B ) b ) + log ( [ C ] c [ D ] d [ A ] a [ B ] b ) {\displaystyle \log \left({\frac {\{C\}^{c}\{D\}^{d}}{\{A\}^{a}\{B\}^{b}}}\right)=\log \left({\frac {\left({\gamma _{\text{C}}}\right)^{c}\left({\gamma _{\text{D}}}\right)^{d}}{\left({\gamma _{\text{A}}}\right)^{a}\left({\gamma _{\text{B}}}\right)^{b}}}\right)+\log \left({\frac {\left[C\right]^{c}\left[D\right]^{d}}{\left[A\right]^{a}\left[B\right]^{b}}}\right)}
It allows to reorganize the Nernst equation as:
E h = E red = ( E red ⊖ − 0.05916 z log ( ( γ C ) c ( γ D ) d ( γ A ) a ( γ B ) b ) ) ⏟ E red ⊖ ′ − 0.05916 z log ( [ C ] c [ D ] d [ A ] a [ B ] b ) − 0.05916 h z pH {\displaystyle E_{h}=E_{\text{red}}=\underbrace {\left(E_{\text{red}}^{\ominus }-{\frac {0.05916}{z}}\log \left({\frac {\left({\gamma _{\text{C}}}\right)^{c}\left({\gamma _{\text{D}}}\right)^{d}}{\left({\gamma _{\text{A}}}\right)^{a}\left({\gamma _{\text{B}}}\right)^{b}}}\right)\right)} _{E_{\text{red}}^{\ominus '}}-{\frac {0.05916}{z}}\log \left({\frac {\left[C\right]^{c}\left[D\right]^{d}}{\left[A\right]^{a}\left[B\right]^{b}}}\right)-{\frac {0.05916\,h}{z}}{\text{pH}}}
E h = E red = E red ⊖ ′ − 0.05916 z log ( [ C ] c [ D ] d [ A ] a [ B ] b ) − 0.05916 h z pH {\displaystyle E_{h}=E_{\text{red}}=E_{\text{red}}^{\ominus '}-{\frac {0.05916}{z}}\log \left({\frac {\left[C\right]^{c}\left[D\right]^{d}}{\left[A\right]^{a}\left[B\right]^{b}}}\right)-{\frac {0.05916\,h}{z}}{\text{pH}}}
Where E red ⊖ ′ {\displaystyle E_{\text{red}}^{\ominus '}} is the formal standard potential independent of pH including the activity coefficients. Combining E red ⊖ ′ {\displaystyle E_{\text{red}}^{\ominus '}} directly with the last term depending on pH gives:
E h = E red = ( E red ⊖ ′ − 0.05916 h z pH ) − 0.05916 z log ( [ C ] c [ D ] d [ A ] a [ B ] b ) {\displaystyle E_{h}=E_{\text{red}}=\left(E_{\text{red}}^{\ominus '}-{\frac {0.05916\,h}{z}}{\text{pH}}\right)-{\frac {0.05916}{z}}\log \left({\frac {\left[C\right]^{c}\left[D\right]^{d}}{\left[A\right]^{a}\left[B\right]^{b}}}\right)}
For a pH = 7:
E h = E red = ( E red ⊖ ′ − 0.05916 h z × 7 ) ⏟ E red apparent at pH 7 ⊖ ′ − 0.05916 z log ( [ C ] c [ D ] d [ A ] a [ B ] b ) {\displaystyle E_{h}=E_{\text{red}}=\underbrace {\left(E_{\text{red}}^{\ominus '}-{\frac {0.05916\,h}{z}}{\text{× 7}}\right)} _{E_{\text{red apparent at pH 7}}^{\ominus '}}-{\frac {0.05916}{z}}\log \left({\frac {\left[C\right]^{c}\left[D\right]^{d}}{\left[A\right]^{a}\left[B\right]^{b}}}\right)}
So,
E h = E red = E red apparent at pH 7 ⊖ ′ − 0.05916 z log ( [ C ] c [ D ] d [ A ] a [ B ] b ) {\displaystyle E_{h}=E_{\text{red}}=E_{\text{red apparent at pH 7}}^{\ominus '}-{\frac {0.05916}{z}}\log \left({\frac {\left[C\right]^{c}\left[D\right]^{d}}{\left[A\right]^{a}\left[B\right]^{b}}}\right)}
It is therefore important to know to what exact definition does refer the value of a reduction potential for a given biochemical redox process reported at pH = 7, and to correctly understand the relationship used. Is it simply:
E h = E red {\displaystyle E_{h}=E_{\text{red}}} calculated at pH 7 (with or without corrections for the activity coefficients),
E red ⊖ ′ {\displaystyle E_{\text{red}}^{\ominus '}} , a formal standard reduction potential including the activity coefficients but no pH calculations, or, is it,
E red apparent at pH 7 ⊖ ′ {\displaystyle E_{\text{red apparent at pH 7}}^{\ominus '}} , an apparent formal standard reduction potential at pH 7 in given conditions and also depending on the ratio h z = (number of involved protons) (number of exchanged electrons) {\displaystyle {\frac {h}{z}}={\frac {\text{(number of involved protons)}}{\text{(number of exchanged electrons)}}}} . This requires thus to dispose of a clear definition of the considered reduction potential, and of a sufficiently detailed description of the conditions in which it is valid, along with a complete expression of the corresponding Nernst equation. Were also the reported values only derived from thermodynamic calculations, or determined from experimental measurements and under what specific conditions? Without being able to correctly answering these questions, mixing data from different sources without appropriate conversion can lead to errors and confusion.
Determination of the formal standard reduction potential when Cred/Cox = 1
The formal standard reduction potential E red ⊖ ′ {\displaystyle E_{\text{red}}^{\ominus '}} can be defined as the measured reduction potential E red {\displaystyle E_{\text{red}}} of the half-reaction at unity concentration ratio of the oxidized and reduced species (i.e., when Cred/Cox = 1) under given conditions. Indeed: as, E red = E red ⊖ {\displaystyle E_{\text{red}}=E_{\text{red}}^{\ominus }} , when a red a ox = 1 {\displaystyle {\frac {a_{\text{red}}}{a_{\text{ox}}}}=1} ,
E red = E red ⊖ ′ {\displaystyle E_{\text{red}}=E_{\text{red}}^{\ominus '}} , when C red C ox = 1 {\displaystyle {\frac {C_{\text{red}}}{C_{\text{ox}}}}=1} , because ln 1 = 0 {\displaystyle \ln {1}=0} , and that the term γ red γ ox {\displaystyle {\frac {\gamma _{\text{red}}}{\gamma _{\text{ox}}}}} is included in E red ⊖ ′ {\displaystyle E_{\text{red}}^{\ominus '}} . The formal reduction potential makes possible to more simply work with molar or molal concentrations in place of activities. Because molar and molal concentrations were once referred as formal concentrations, it could explain the origin of the adjective formal in the expression formal potential. 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. 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, Pgas = 1 bar) it becomes de facto a standard potential. According to Brown and Swift (1949), "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". The activity coefficients γ r e d {\displaystyle \gamma _{red}} and γ o x {\displaystyle \gamma _{ox}} are included in the formal potential E red ⊖ ′ {\displaystyle E_{\text{red}}^{\ominus '}} , and because they depend on experimental conditions such as temperature, ionic strength, and pH, E red ⊖ ′ {\displaystyle E_{\text{red}}^{\ominus '}} cannot be referred as an immuable standard potential but needs to be systematically determined for each specific set of experimental conditions. Formal reduction potentials are applied to simplif