Nernst equation

# Nernst equation

Discussion

Encyclopedia
In electrochemistry
Electrochemistry
Electrochemistry is a branch of chemistry that studies chemical reactions which take place in a solution at the interface of an electron conductor and an ionic conductor , and which involve electron transfer between the electrode and the electrolyte or species in solution.If a chemical reaction is...

, the Nernst equation is an equation that can be used (in conjunction with other information) to determine the equilibrium reduction potential
Reduction potential
Reduction potential is a measure of the tendency of a chemical species to acquire electrons and thereby be reduced. Reduction potential is measured in volts , or millivolts...

of a half-cell in an electrochemical cell
Electrochemical cell
An electrochemical cell is a device capable of either deriving electrical energy from chemical reactions, or facilitating chemical reactions through the introduction of electrical energy. A common example of an electrochemical cell is a standard 1.5-volt "battery"...

. It can also be used to determine the total voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...

(electromotive force
Electromotive force
In physics, electromotive force, emf , or electromotance refers to voltage generated by a battery or by the magnetic force according to Faraday's Law, which states that a time varying magnetic field will induce an electric current.It is important to note that the electromotive "force" is not a...

) for a full electrochemical cell. It is named after the German physical chemist who first formulated it, Walther Nernst
Walther Nernst
Walther Hermann Nernst FRS was a German physical chemist and physicist who is known for his theories behind the calculation of chemical affinity as embodied in the third law of thermodynamics, for which he won the 1920 Nobel Prize in chemistry...

.

## Expression

The two (ultimately equivalent) equations for these two cases (half-cell, full cell) are as follows:$E_\text\left\{red\right\} = E^\left\{\ominus\right\}_\text\left\{red\right\} - \frac\left\{RT\right\}\left\{zF\right\} \ln\frac\left\{a_\text\left\{Red\right\}\right\}\left\{a_\text\left\{Ox\right\}\right\}$    (half-cell reduction potential)$E_\text\left\{cell\right\} = E^\left\{\ominus\right\}_\text\left\{cell\right\} - \frac\left\{RT\right\}\left\{zF\right\} \ln Q$    (total cell potential) whereNEWLINE
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• Ered is the half-cell reduction potential
Reduction potential
Reduction potential is a measure of the tendency of a chemical species to acquire electrons and thereby be reduced. Reduction potential is measured in volts , or millivolts...

at the temperature of interest
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• Eored is the standard half-cell reduction potential
Standard electrode potential
In electrochemistry, the standard electrode potential, abbreviated E° or E , is the measure of individual potential of a reversible electrode at standard state, which is with solutes at an effective concentration of 1 mol dm−3, and gases at a pressure of 1 atm...

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• Ecell is the cell potential (electromotive force
Electromotive force
In physics, electromotive force, emf , or electromotance refers to voltage generated by a battery or by the magnetic force according to Faraday's Law, which states that a time varying magnetic field will induce an electric current.It is important to note that the electromotive "force" is not a...

)
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• Eocell is the standard cell potential at the temperature of interest
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• R is the universal gas constant: R = 8.314 472(15) J K−1 mol−1
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• T is the absolute temperature
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• a is the chemical activity
Activity (chemistry)
In chemical thermodynamics, activity is a measure of the “effective concentration” of a species in a mixture, meaning that the species' chemical potential depends on the activity of a real solution in the same way that it would depend on concentration for an ideal solution.By convention, activity...

for the relevant species, where aRed is the reductant and aOx is the oxidant. aX = γXcX, where γX is the activity coefficient
Activity coefficient
An activity coefficient is a factor used in thermodynamics to account for deviations from ideal behaviour in a mixture of chemical substances. In an ideal mixture, the interactions between each pair of chemical species are the same and, as a result, properties of the mixtures can be expressed...

of species X. (Since activity coefficients tend to unity at low concentrations, activities in the Nernst equation are frequently replaced by simple concentrations.)
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• F is the Faraday constant, the number of coulombs per mole
Mole (unit)
The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon-12 , the isotope of carbon with atomic weight 12. This corresponds to a value...

of electrons: F = 9.648 533 99(24)×104 C mol−1
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• z is the number of moles of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s transferred in the cell reaction or half-reaction
Half-reaction
A half reaction is either the oxidation or reduction reaction component of a redox reaction. A half reaction is obtained by considering the change in oxidation states of individual substances involved in the redox reaction.-Example:...

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• Q is the reaction quotient
Reaction quotient
In chemistry, a reaction quotient: Qr is a function of the activities or concentrations of the chemical species involved in a chemical reaction. In the special case that the reaction is at equilibrium the reaction quotient is equal to the equilibrium constant....

.
NEWLINE At room temperature (25 °C), RT/F may be treated like a constant and replaced by 25.693 mV for cells. The Nernst equation is frequently expressed in terms of base 10 logarithms (i.e., common logarithm
Common logarithm
The common logarithm is the logarithm with base 10. It is also known as the decadic logarithm, named after its base. It is indicated by log10, or sometimes Log with a capital L...

s) rather than natural logarithms, in which case it is written, for a cell at 25 °C: $E = E^0 - \frac\left\{0.05916\mbox\left\{ V\right\}\right\}\left\{z\right\} \log_\left\{10\right\}\frac\left\{a_\text\left\{Red\right\}\right\}\left\{a_\text\left\{Ox\right\}\right\}.$ The Nernst equation is used in physiology
Physiology
Physiology is the science of the function of living systems. This includes how organisms, organ systems, organs, cells, and bio-molecules carry out the chemical or physical functions that exist in a living system. The highest honor awarded in physiology is the Nobel Prize in Physiology or...

for finding the electric potential
Electric potential
In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...

of a cell membrane
Cell membrane
The cell membrane or plasma membrane is a biological membrane that separates the interior of all cells from the outside environment. The cell membrane is selectively permeable to ions and organic molecules and controls the movement of substances in and out of cells. It basically protects the cell...

with respect to one type of ion
Ion
An ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge. The name was given by physicist Michael Faraday for the substances that allow a current to pass between electrodes in a...

.

## Nernst potential

{{main|Reversal potential}} The Nernst equation has a physiological application when used to calculate the potential of an ion of charge z across a membrane. This potential is determined using the concentration of the ion both inside and outside the cell: $E = \frac\left\{R T\right\}\left\{z F\right\} \ln\frac\left\{\left[\text\left\{ion outside cell\right\}\right]\right\}\left\{\left[\text\left\{ion inside cell\right\}\right]\right\} = 2.303\frac\left\{R T\right\}\left\{z F\right\} \log_\left\{10\right\}\frac\left\{\left[\text\left\{ion outside cell\right\}\right]\right\}\left\{\left[\text\left\{ion inside cell\right\}\right]\right\}.$ When the membrane is in thermodynamic equilibrium
Thermodynamic equilibrium
In thermodynamics, a thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, radiative equilibrium, and chemical equilibrium. The word equilibrium means a state of balance...

(i.e., no net flux of ions), the membrane potential
Membrane potential
Membrane potential is the difference in electrical potential between the interior and exterior of a biological cell. All animal cells are surrounded by a plasma membrane composed of a lipid bilayer with a variety of types of proteins embedded in it...

must be equal to the Nernst potential. However, in physiology, due to active ion pumps, the inside and outside of a cell are not in equilibrium. In this case, the resting potential
Resting potential
The relatively static membrane potential of quiescent cells is called the resting membrane potential , as opposed to the specific dynamic electrochemical phenomena called action potential and graded membrane potential....

can be determined from the Goldman equation
Goldman equation
The Goldman–Hodgkin–Katz voltage equation, more commonly known as the Goldman equation is used in cell membrane physiology to determine the equilibrium potential across a cell's membrane taking into account all of the ions that are permeant through that membrane.The discoverers of this are David E...

: $E_\left\{m\right\} = \frac\left\{RT\right\}\left\{F\right\} \ln\left\{ \left\left( \frac\left\{ \sum_\left\{i\right\}^\left\{N\right\} P_\left\{M^\left\{+\right\}_\left\{i\right\}\right\}\left[M^\left\{+\right\}_\left\{i\right\}\right]_\mathrm\left\{out\right\} + \sum_\left\{j\right\}^\left\{M\right\} P_\left\{A^\left\{-\right\}_\left\{j\right\}\right\}\left[A^\left\{-\right\}_\left\{j\right\}\right]_\mathrm\left\{in\right\}\right\}\left\{ \sum_\left\{i\right\}^\left\{N\right\} P_\left\{M^\left\{+\right\}_\left\{i\right\}\right\}\left[M^\left\{+\right\}_\left\{i\right\}\right]_\mathrm\left\{in\right\} + \sum_\left\{j\right\}^\left\{M\right\} P_\left\{A^\left\{-\right\}_\left\{j\right\}\right\}\left[A^\left\{-\right\}_\left\{j\right\}\right]_\mathrm\left\{out\right\}\right\} \right\right) \right\}$ NEWLINE
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• $E_\left\{m\right\}$ = The membrane potential (in volt
Volt
The volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...

s, equivalent to joule
Joule
The joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...

s per coulomb)
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• $P_\mathrm\left\{ion\right\}$ = the permeability for that ion (in meters per second)
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• $\left[ion\right]_\mathrm\left\{out\right\}$ = the extracellular concentration of that ion (in moles
Mole (unit)
The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon-12 , the isotope of carbon with atomic weight 12. This corresponds to a value...

per cubic meter, to match the other SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

units, though the units strictly don't matter, as the ion concentration terms become a dimensionless ratio)
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• $\left[ion\right]_\mathrm\left\{in\right\}$ = the intracellular concentration of that ion (in moles per cubic meter)
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• $R$ = The ideal gas constant (joules per kelvin
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

per mole)
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• $T$ = The temperature in kelvin
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

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• $F$ = Faraday's constant (coulombs per mole)
NEWLINE The potential across the cell membrane that exactly opposes net diffusion of a particular ion through the membrane is called the Nernst potential for that ion. As seen above, the magnitude of the Nernst potential is determined by the ratio of the concentrations of that specific ion on the two sides of the membrane. The greater this ratio the greater the tendency for the ion to diffuse in one direction, and therefore the greater the Nernst potential required to prevent the diffusion.

### Using Boltzmann factors

For simplicity, we will consider a solution of redox-active molecules that undergo a one-electron reversible reaction $\text\left\{Ox\right\} + e^- \rightleftharpoons \text\left\{Red\right\}\,$ and that have a standard potential of zero. The chemical potential
Chemical potential
Chemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system...

$\mu_c$ of this solution is the difference between the energy barriers for taking electrons from and for giving electrons to the working electrode
Cyclic voltammetry
Cyclic voltammetry or CV is a type of potentiodynamic electrochemical measurement. In a cyclic voltammetry experiment the working electrode potential is ramped linearly versus time like linear sweep voltammetry. Cyclic voltammetry takes the experiment a step further than linear sweep voltammetry...

that is setting the solution's electrochemical potential. The ratio of oxidized to reduced molecules, [Ox]/[Red], is equivalent to the probability of being oxidized (giving electrons) over the probability of being reduced (taking electrons), which we can write in terms of the Boltzmann factors for these processes: $\frac\left\{\left[\mathrm\left\{Ox\right\}\right]\right\}\left\{\left[\mathrm\left\{Red\right\}\right]\right\} \frac\left\{\exp \left\left(-\left[\mbox\left\{barrier for losing an electron\right\}\right]/kT\right\right)\right\} \left\{\exp \left\left(-\left[\mbox\left\{barrier for gaining an electron\right\}\right]/kT\right\right)\right\} \exp \left\left(\mu_c / kT \right\right).$ Taking the natural logarithm of both sides gives $\mu_c = kT \ln \frac\left\{\left[\mathrm\left\{Ox\right\}\right]\right\}\left\{\left[\mathrm\left\{Red\right\}\right]\right\}.$ If $\mu_c \ne 0$ at [Ox]/[Red] = 1, we need to add in this additional constant: $\mu_c = \mu_c^0 + kT \ln \frac\left\{\left[\mathrm\left\{Ox\right\}\right]\right\}\left\{\left[\mathrm\left\{Red\right\}\right]\right\}.$ Dividing the equation by e to convert from chemical potentials to electrode potentials, and remembering that kT/e = RT/F, we obtain the Nernst equation for the one-electron process $\mathrm\left\{Ox\right\} + e^- \rightarrow \mathrm\left\{Red\right\}$: $E = E^0 + \frac\left\{kT\right\}\left\{e\right\} \ln \frac\left\{\left[\mathrm\left\{Ox\right\}\right]\right\}\left\{\left[\mathrm\left\{Red\right\}\right]\right\}$

### Using entropy and Gibbs energy

Quantities here are given per molecule, not per mole, and so Boltzmann constant k and the electron charge e are used instead of the gas constant R and Faraday's constant F. To convert to the molar quantities given in most chemistry textbooks, it is simply necessary to multiply by Avogadro's number: $R = kN_A$ and $F = eN_A$. The entropy of a molecule is defined as$S \ \stackrel\left\{\mathrm\left\{def\right\}\right\}\left\{=\right\}\ k \ln \Omega,$ where $\Omega$ is the number of states available to the molecule. The number of states must vary linearly with the volume V of the system, which is inversely proportional to the concentration c, so we can also write the entropy as$S = k\ln \ \left(\mathrm\left\{constant\right\}\times V\right) = -k\ln \ \left(\mathrm\left\{constant\right\}\times c\right).$ The change in entropy from some state 1 to another state 2 is therefore$\Delta S = S_2 - S_1 = - k \ln \frac\left\{c_2\right\}\left\{c_1\right\},$ so that the entropy of state 2 is$S_2 = S_1 - k \ln \frac\left\{c_2\right\}\left\{c_1\right\}.$ If state 1 is at standard conditions, in which $c_1$ is unity (e.g., 1 atm or 1 M), it will merely cancel the units of $c_2$. We can, therefore, write the entropy of an arbitrary molecule A as$S\left(A\right) = S^0\left(A\right) - k \ln \left[A\right], \,$ where $S^0$ is the entropy at standard conditions and [A] denotes the concentration of A. The change in entropy for a reaction$aA + bB \rightarrow yY + zZ$ is then given by$\Delta S_\mathrm\left\{rxn\right\} = \left[yS\left(Y\right) + zS\left(Z\right)\right] - \left[aS\left(A\right) + bS\left(B\right)\right] \Delta S^0_\mathrm\left\{rxn\right\} - k \ln \frac\left\{\left[Y\right]^y \left[Z\right]^z\right\}\left\{\left[A\right]^a \left[B\right]^b\right\}.$ We define the ratio in the last term as the reaction quotient
Reaction quotient
In chemistry, a reaction quotient: Qr is a function of the activities or concentrations of the chemical species involved in a chemical reaction. In the special case that the reaction is at equilibrium the reaction quotient is equal to the equilibrium constant....

:$Q = \frac\left\{\prod_j a_j^\left\{\nu_j\right\}\right\}\left\{\prod_i a_i^\left\{\nu_i\right\}\right\} \approx \frac\left\{\left[Z\right]^z \left[Y\right]^y\right\}\left\{\left[A\right]^a \left[B\right]^b\right\}.$ where the numerator is a product of reaction product activities, a j, each raised to the power of a stoichiometric coefficient, ν j, and the denominator is a similar product of reactant activities. All activities refer to a time t. Under certain circumstances (see chemical equilibrium
Chemical equilibrium
In a chemical reaction, chemical equilibrium is the state in which the concentrations of the reactants and products have not yet changed with time. It occurs only in reversible reactions, and not in irreversible reactions. Usually, this state results when the forward reaction proceeds at the same...

) each activity term such as $a_j^\left\{\nu_j\right\}$ may be replaced by a concentration term, [A]. In an electrochemical cell, the cell potential E is the chemical potential available from redox reactions ($E = \mu_c/e$). E is related to the Gibbs energy
Gibbs free energy
In thermodynamics, the Gibbs free energy is a thermodynamic potential that measures the "useful" or process-initiating work obtainable from a thermodynamic system at a constant temperature and pressure...

change $\Delta G$ only by a constant: $\Delta G = -nFE$, where n is the number of electrons transferred and $F$ is the Faraday constant. There is a negative sign because a spontaneous reaction has a negative free energy $\Delta G$ and a positive potential E. The Gibbs energy is related to the entropy by $G = H - TS$, where H is the enthalpy and T is the temperature of the system. Using these relations, we can now write the change in Gibbs energy,$\Delta G = \Delta H - T \Delta S = \Delta G^0 + kT \ln Q, \,$ and the cell potential,$E = E^0 - \frac\left\{kT\right\}\left\{ne\right\} \ln Q.$ This is the more general form of the Nernst equation. For the redox reaction $\mathrm\left\{Ox\right\} + ne^- \rightarrow \mathrm\left\{Red\right\},$ $Q = \frac\left\{\left[\mathrm\left\{Red\right\}\right]\right\}\left\{\left[\mathrm\left\{Ox\right\}\right]\right\}$, and we have:$E = E^0 - \frac\left\{kT\right\}\left\{ne\right\} \ln \frac\left\{\left[\mathrm\left\{Red\right\}\right]\right\}\left\{\left[\mathrm\left\{Ox\right\}\right]\right\} E^0 - \frac\left\{RT\right\}\left\{nF\right\} \ln Q.$ The cell potential at standard conditions $E^0$ is often replaced by the formal potential $E^\left\{0\text{'}\right\}$, which includes some small corrections to the logarithm and is the potential that is actually measured in an electrochemical cell.

## Relation to equilibrium

At equilibrium, E = 0 and Q = K. Therefore\begin\left\{align\right\} 0 &= E^o - \frac\left\{RT\right\}\left\{nF\right\} \ln K\\ \ln K &= \frac\left\{nFE^o\right\}\left\{RT\right\} \end\left\{align\right\} Or at standard temperature
Standard conditions for temperature and pressure
Standard condition for temperature and pressure are standard sets of conditions for experimental measurements established to allow comparisons to be made between different sets of data...

,$\log_\left\{10\right\} K = \frac\left\{nE^o\right\}\left\{59.2\text\left\{ mV\right\}\right\} \quad\text\left\{at \right\}T = 298 \text\left\{ K\right\}.$ We have thus related the standard electrode potential
Standard electrode potential
In electrochemistry, the standard electrode potential, abbreviated E° or E , is the measure of individual potential of a reversible electrode at standard state, which is with solutes at an effective concentration of 1 mol dm−3, and gases at a pressure of 1 atm...

and the equilibrium constant of a redox reaction.

## Limitations

In dilute solutions, the Nernst equation can be expressed directly in terms of concentrations (since activity coefficients are close to unity). But at higher concentrations, the true activities of the ions must be used. This complicates the use of the Nernst equation, since estimation of non-ideal activities of ions generally requires experimental measurements. The Nernst equation also only applies when there is no net current flow through the electrode. The activity of ions at the electrode surface changes when there is current flow, and there are additional overpotential
Overpotential
Overpotential is an electrochemical term which refers to the potential difference between a half-reaction's thermodynamically determined reduction potential and the potential at which the redox event is experimentally observed. The term is directly related to a cell's voltage efficiency...

and resistive loss terms which contribute to the measured potential. At very low concentrations of the potential-determining ions, the potential predicted by Nernst equation approaches toward ±∞. This is physically meaningless because, under such conditions, the exchange current density becomes very low, and then other effects tend to take control of the electrochemical behavior of the system.

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• Goldman equation
Goldman equation
The Goldman–Hodgkin–Katz voltage equation, more commonly known as the Goldman equation is used in cell membrane physiology to determine the equilibrium potential across a cell's membrane taking into account all of the ions that are permeant through that membrane.The discoverers of this are David E...

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• Galvanic cell
Galvanic cell
A Galvanic cell, or Voltaic cell, named after Luigi Galvani, or Alessandro Volta respectively, is an electrochemical cell that derives electrical energy from spontaneous redox reaction taking place within the cell...

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• Concentration cell
Concentration cell
A concentration cell is a limited form of a galvanic cell that has two equivalent half-cells of the same material differing only in concentrations. One can calculate the potential developed by such a cell using the Nernst Equation. A concentration cell produces a voltage as it attempts to reach...

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• Nernst-Planck equation
Nernst-Planck equation
The Nernst–Planck equation is a conservation of mass equation used to describe the motion of chemical species in a fluid medium. It describes the flux of ions under the influence of both an ionic concentration gradient \nabla c and an electric field E=-\nabla \phi...

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