Nernst equation

In electrochemistry, the Nernst equation is a chemical thermodynamical relationship that permits the calculation of the reduction potential of a reaction (half-cell or full cell reaction) from the standard electrode potential, absolute temperature, the number of electrons involved in the redox reaction, and activities (often approximated by concentrations) of the chemical species undergoing reduction and oxidation respectively. It was named after Walther Nernst, a German physical chemist who formulated the equation.^[1]^[2]

Expression

General form with chemical activities

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:

{\ce {Ox + ze- -> 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 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 (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 for didactic purposes:

Q_{r}={\frac {a_{\text{Red}}}{a_{\text{Ox}}}}={\frac {}{}}

At chemical equilibrium, the ratio $Q r$ 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:

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

The standard thermodynamics also says that the actual Gibbs free energy $Δ G$ is related to the free energy change under standard state $Δ G o$ by the relationship:

\Delta G=\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

\Delta G=-zFE.

The constant

F

(the Faraday constant) is a unit conversion factor

F = N A q

, 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

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

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,
$R$ is the universal ideal gas constant: $R = 8.314 46261815324 J K -1 mol -1$ ,
$T$ is the temperature in kelvins,
$z$ is the number of electrons transferred in the cell reaction or half-reaction,
$F$ is the Faraday constant, the magnitude of charge (in coulombs) per mole of electrons: $F = 96485 .332 1233100184 C mol -1$ ,
$Q r$ 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 $V_{T}={\frac {RT}{F}}$ 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:

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 $E_{\text{red}}^{\ominus '}$ used.

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

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)

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)

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

[1]

[2]