Nernst equation

It has applications in cell physiology and biophysics, concerning the cell membrane potential.

Expression

General form with chemical activities

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

${\displaystyle \text{Ox}+\mathit{\text{z}}{\text{e}}^{-}⟶\text{Red}}$

The reaction quotient (Q_r), also often called the ion activity product (IAP), is the ratio between the chemical activities (a) of the reduced form (the reductant, a_Red) and the oxidized form (the oxidant, a_Ox). The chemical activity of a dissolved species is a quantity used for 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 (mol/L solution), or 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 coefficients are close to unity, their chemical activities can be approximated by their concentrations as commonly done when simplifying, or idealizing, a reaction in order to have a simple reference case:

${\displaystyle Q_r=\frac{a_{\text{Red}}}{a_{\text{Ox}}}=\frac{[\mathrm{Red}]}{[\mathrm{Ox}]}}$

At chemical equilibrium, the quotient Q_r of the activity of the reaction product (a_Red) by the reagent activity (a_Ox) is equal to the equilibrium constant K of the half-reaction:

${\displaystyle K=\frac{a_{\text{Red}}}{a_{\text{Ox}}}}$

The standard thermodynamics also says that the actual Gibbs free energy ΔG is related to the free energy change under standard state ΔG^o
by the relationship: $${\displaystyle \mathrm{\Delta }G=\mathrm{\Delta }{G}^{\ominus }+RT\ln Q_r}$$ where Q_r is the reaction quotient and R is the universal ideal gas constant. The cell potential 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 $${\displaystyle \mathrm{\Delta }G=-zFE.}$$ The constant F (the Faraday constant) is a unit conversion factor F = N_Aq, where N_A is the Avogadro constant and q is the fundamental electron charge. This immediately leads to the Nernst equation, which for an electrochemical half-cell is $${\displaystyle E_{\text{red}}=E_{\text{red}}^{\ominus }-\frac{RT}{zF}\ln Q_r=E_{\text{red}}^{\ominus }-\frac{RT}{zF}\ln \frac{a_{\text{Red}}}{a_{\text{Ox}}}.}$$ For a complete electrochemical reaction (full cell), the equation can be written as $${\displaystyle E_{\text{cell}}=E_{\text{cell}}^{\ominus }-\frac{RT}{zF}\ln Q_r}$$

where:

• E_red is the half-cell reduction potential at the temperature of interest,
• E^o
  _red is the standard half-cell reduction potential,
• E_cell is the cell potential (electromotive force) at the temperature of interest,
• E^o
  _cell is the standard cell potential in volts,
• R is the universal ideal gas constant: R = 8.31446261815324 J K⁻¹ mol⁻¹,
• T is the temperature in kelvins,
• z is the number of electrons transferred in the cell reaction or half-reaction,
• F is Faraday's constant, the magnitude of charge (in coulombs) per mole of electrons: F = 96485.3321233100184 C mol⁻¹,
• Q_r is the reaction quotient of the cell reaction, and,
• a is the chemical activity for the relevant species, where a_Red is the activity of the reduced form and a_Ox is the activity of the oxidized form.

Thermal voltage

At room temperature (25 °C), the thermal voltage ${\displaystyle V_T=\frac{RT}{F}}$ is approximately 25.693 mV. The Nernst equation is frequently expressed in terms of base-10 logarithms (i.e., common logarithms) rather than natural logarithms, in which case it is written:

$${\displaystyle E=E^{\ominus }-\frac{V_T}{z}\ln \frac{a_{\text{Red}}}{a_{\text{Ox}}}=E^{\ominus }-\frac{\lambda V_T}{z}{\log }_{10}\frac{a_{\text{Red}}}{a_{\text{Ox}}}.}$$

where λ = ln(10) ≈ 2.3026 and λV_T ≈ 0.05916 Volt.

Form with activity coefficients and concentrations

Similarly to equilibrium constants, activities are always measured with respect to the standard state (1 mol/L for solutes, 1 atm for gases, and T = 298.15 K, i.e., 25 °C or 77 °F). The chemical activity of a species i, a_i, is related to the measured concentration C_i via the relationship a_i = γ_i C_i, where γ_i is the activity coefficient of the species 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 ${\displaystyle E_{\text{red}}^{{\ominus }^{\prime }}}$ used.

Taking into account the activity coefficients (${\displaystyle \gamma }$) the Nernst equation becomes:

$${\displaystyle E_{\text{red}}=E_{\text{red}}^{\ominus }-\frac{RT}{zF}\ln \left(\frac{{\gamma }_{\text{Red}}}{{\gamma }_{\text{Ox}}}\frac{C_{\text{Red}}}{C_{\text{Ox}}}\right)}$$

$${\displaystyle E_{\text{red}}=E_{\text{red}}^{\ominus }-\frac{RT}{zF}\left(\ln \frac{{\gamma }_{\text{Red}}}{{\gamma }_{\text{Ox}}}+\ln \frac{C_{\text{Red}}}{C_{\text{Ox}}}\right)}$$

$${\displaystyle E_{\text{red}}=\underset{E_{\text{red}}^{{\ominus }^{\prime }}}{\underbrace{\left(E_{\text{red}}^{\ominus }-\frac{RT}{zF}\ln \frac{{\gamma }_{\text{Red}}}{{\gamma }_{\text{Ox}}}\right)}}-\frac{RT}{zF}\ln \frac{C_{\text{Red}}}{C_{\text{Ox}}}}$$

Where the first term including the activity coefficients (${\displaystyle \gamma }$) is denoted ${\displaystyle E_{\text{red}}^{{\ominus }^{\prime }}}$ and called the formal standard reduction potential, so that ${\displaystyle E_{\text{red}}}$ can be directly expressed as a function of ${\displaystyle E_{\text{red}}^{{\ominus }^{\prime }}}$ and the concentrations in the simplest form of the Nernst equation:

$${\displaystyle E_{\text{red}}=E_{\text{red}}^{{\ominus }^{\prime }}-\frac{RT}{zF}\ln \frac{C_{\text{Red}}}{C_{\text{Ox}}}}$$

Formal standard reduction 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 (${\displaystyle E_{\text{red}}^{{\ominus }^{\prime }}}$) which is related to the standard reduction potential as follows: $${\displaystyle E_{\text{red}}^{{\ominus }^{\prime }}=E_{\text{red}}^{\ominus }-\frac{RT}{zF}\ln \frac{{\gamma }_{\text{Red}}}{{\gamma }_{\text{Ox}}}}$$ So that the Nernst equation for the half-cell reaction can be correctly formally written in terms of concentrations as: $${\displaystyle E_{\text{red}}=E_{\text{red}}^{{\ominus }^{\prime }}-\frac{RT}{zF}\ln \frac{C_{\text{Red}}}{C_{\text{Ox}}}}$$ and likewise for the full cell expression.

According to Wenzel (2020), a formal reduction potential ${\displaystyle E_{\text{red}}^{{\ominus }^{\prime }}}$ 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 agents.

The formal reduction potential ${\displaystyle E_{\text{red}}^{{\ominus }^{\prime }}}$ 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, ${\displaystyle E_{\text{red}}^{{\ominus }^{\prime }}}$ will vary from medium to medium. 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 ${\displaystyle E_{\text{red}}^{{\ominus }^{\prime }}}$ refers to its value determined when ${\displaystyle \frac{C_{\text{red}}}{C_{\text{ox}}}=1}$. A more particular case is when ${\displaystyle E_{\text{red}}^{{\ominus }^{\prime }}}$ 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 ⁠Cred/Cox⁠ = 1

The formal standard reduction potential ${\displaystyle E_{\text{red}}^{{\ominus }^{\prime }}}$ can be defined as the measured reduction potential ${\displaystyle E_{\text{red}}}$ of the half-reaction at unity concentration ratio of the oxidized and reduced species (i.e., when ⁠C_red/C_ox⁠ = 1) under given conditions.

Indeed:

as, ${\displaystyle E_{\text{red}}=E_{\text{red}}^{\ominus }}$, when ${\displaystyle \frac{a_{\text{red}}}{a_{\text{ox}}}=1}$,

${\displaystyle E_{\text{red}}=E_{\text{red}}^{{\ominus }^{\prime }}}$, when ${\displaystyle \frac{C_{\text{red}}}{C_{\text{ox}}}=1}$,

because ${\displaystyle \ln 1=0}$, and that the term ${\displaystyle \frac{{\gamma }_{\text{red}}}{{\gamma }_{\text{ox}}}}$ is included in ${\displaystyle E_{\text{red}}^{{\ominus }^{\prime }}}$.

The formal reduction potential makes possible to more simply work with molar (mol/L, M) or molal (mol/kg H₂O, m) 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, P_gas = 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".

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

${\displaystyle \frac{C_{\text{red}}}{C_{\text{ox}}}=\frac{1{\mathrm{M}}_{\text{red}}}{1{\mathrm{M}}_{\text{ox}}}=1}$

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.

The formal potential is also found halfway between the two peaks in a cyclic 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 coefficients ${\displaystyle {\gamma }_{red}}$ and ${\displaystyle {\gamma }_{ox}}$ are included in the formal potential ${\displaystyle E_{\text{red}}^{{\ominus }^{\prime }}}$, and because they depend on experimental conditions such as temperature, ionic strength, and pH, ${\displaystyle E_{\text{red}}^{{\ominus }^{\prime }}}$ cannot be referred as an immutable standard potential but needs to be systematically determined for each specific set of experimental conditions.

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

Formal standard reduction potentials (${\displaystyle E_{\text{red}}^{{\ominus }^{\prime }}}$) are also commonly used in biochemistry and cell biology for referring to standard reduction potentials 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 scale better corresponding to real conditions than the standard state. Formal standard reduction potentials (${\displaystyle E_{\text{red}}^{{\ominus }^{\prime }}}$) 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 ${\displaystyle E_{\text{red H+}}^{\ominus }}$ fixed arbitrarily to zero by convention, it is no longer the case at a pH of 7. Then, the reduction potential ${\displaystyle E_{\text{red}}}$ of a hydrogen electrode operating at pH 7 is −0.413 V with respect to the standard hydrogen electrode (SHE).

Expression of the Nernst equation as a function of pH

The ${\displaystyle E_h}$ and pH of a solution are related by the Nernst equation as commonly represented by a Pourbaix diagram (${\displaystyle E_h}$ – pH plot). ${\displaystyle E_h}$ explicitly denotes ${\displaystyle E_{\text{red}}}$ 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):

${\displaystyle aA+bB+h{\text{H}}^{+}+z{e}^{-}\stackrel{-\rightharpoonup }{\leftharpoondown -}cC+dD}$

The half-cell standard reduction potential ${\displaystyle E_{\text{red}}^{\ominus }}$ is given by

${\displaystyle E_{\text{red}}^{\ominus }(\text{volt})=-\frac{\mathrm{\Delta }{G}^{\ominus }}{zF}}$

where ${\displaystyle \mathrm{\Delta }{G}^{\ominus }}$ is the standard Gibbs free energy change, z is the number of electrons involved, and F is the Faraday's constant. The Nernst equation relates pH and ${\displaystyle E_h}$ as follows:

${\displaystyle E_h=E_{\text{red}}=E_{\text{red}}^{\ominus }-\frac{0.05916}{z}\log \left(\frac{\{C{\}}^c\{D{\}}^d}{\{A{\}}^a\{B{\}}^b}\right)-\frac{0.05916h}{z}\text{pH}}$

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

This equation predicts lower ${\displaystyle E_{\text{red}}}$ at higher pH values. This is observed for the reduction of O₂ into H₂O, or OH⁻, and for the reduction of H⁺ into H₂. ${\displaystyle E_{\text{red}}}$ is then often noted as ${\displaystyle E_h}$ to indicate that it refers to the standard hydrogen electrode (SHE) whose ${\displaystyle E_{\text{red}}}$ = 0 by convention under standard conditions (T = 298.15 K = 25 °C = 77 F, P_gas = 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 potentials 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

To illustrate the dependency of the reduction potential on pH, one can simply consider the two oxido-reduction equilibria determining the water stability domain in a Pourbaix diagram (E_h–pH plot). When water is submitted to electrolysis by applying a sufficient difference of electrical potential between two electrodes 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., F₂) enters in contact with water and reacts with it. In the E_h–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_h:

2 H⁺ + 2 e⁻ ⇌ H₂ (cathode: reduction)

• Higher stability line with oxygen gas evolution due to water oxygen oxidation at very high E_h:

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 H₂ 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:

${\displaystyle E_{\text{red}}=E_{\text{red}}^{\ominus }-0.05916pH}$

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⁺ into H₂ the here above mentioned relationship becomes:

${\displaystyle E_{\text{red}}=-0.05916pH}$
because by convention ${\displaystyle E_{\text{red}}^{\ominus }}$ = 0 V for the standard hydrogen electrode (SHE: pH = 0).
So, at pH = 7, ${\displaystyle E_{\text{red}}}$ = −0.414 V for the reduction of protons.

For the reduction of O₂ into 2 H₂O the here above mentioned relationship becomes:

${\displaystyle E_{\text{red}}=1.229-0.05916pH}$
because ${\displaystyle E_{\text{red}}^{\ominus }}$ = +1.229 V with respect to the standard hydrogen electrode (SHE: pH = 0).
So, at pH = 7, ${\displaystyle E_{\text{red}}}$ = +0.815 V for the reduction of oxygen.

The offset of −414 mV in ${\displaystyle E_{\text{red}}}$ 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 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_h value (SHE) when the ratio h⁄z differs from 1.