Editing Nernst equation (section)

===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.