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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 Electrical conductor and an ionic conductor , and which involve electron transfer between the electrode and the electrolyte or species in solution....
, the Nernst equation is an equation which can be used (in conjunction with other information) to determine the equilibrium reduction potential
Reduction potential

Reduction potential is the tendency of a chemical species to acquire electrons and thereby be redox. Each species has its own intrinsic reduction potential; the more positive the potential, the greater the species' affinity for electrons and tendency to be reduced....
 of a half-cell in an electrochemical cell
Electrochemical cell

An electrochemical cell is a device used for generating an electromotive force and current from electrochemistry, or the reverse, inducing a chemical reaction by a flow of current....
. It can also be used to determine the total voltage
Voltage

Electrical tension is the potential difference between two points of an electrical or electronic circuit, expressed in volts. It is the measurement of the potential for an electric field to cause an electric current in an electrical conductor....
 (electromotive force
Electromotive force

Electromotive force is a term used to characterize electrical devices, such as voltaic cells, Thermoelectric effects, electrical generators and transformers, and even resistors....
) for a full electrochemical cell. It is named after the German physical chemist who first formulated it, Walther Nernst
Walther Nernst

Walther Hermann Nernst was a Germany physical chemist 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....
.

two (ultimately equivalent) equations for these two cases (half-cell, full cell) are as follows:    (half-cell reduction potential)    (total cell potential) where

At room temperature (25 °C), RT/F may be treated like a constant and replaced by 25.679 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:

The Nernst equation is used in physiology
Physiology

Physiology is the study of the mechanical, physical, and biochemical functions of living organisms. Physiology has traditionally been divided between plant physiology and animal and all living things physiology but the principles of physiology are universal, no matter what particular organism is being studied....
 for finding the electric potential
Electric potential

At a point in space, the electric potential is the potential energy per unit of electric charge that is associated with a static electric field....
 of a cell membrane
Cell membrane

The cell membrane is the interface between the cellular machinery inside the cell and the fluid outside.It is a semipermeable lipid bilayer found in all cell ....
 with respect to one type of ion
Ion

An ion is an atom or molecule which has lost or gained one or more electrons, giving it a positive or negative electrical charge. According to the Bohr_model this will be from or in the outer shield 'n'....
.

Nernst potential
The Nernst equation has a physiological application when used to calculate the potential of an ion of charge z across a membrane.






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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 Electrical conductor and an ionic conductor , and which involve electron transfer between the electrode and the electrolyte or species in solution....
, the Nernst equation is an equation which can be used (in conjunction with other information) to determine the equilibrium reduction potential
Reduction potential

Reduction potential is the tendency of a chemical species to acquire electrons and thereby be redox. Each species has its own intrinsic reduction potential; the more positive the potential, the greater the species' affinity for electrons and tendency to be reduced....
 of a half-cell in an electrochemical cell
Electrochemical cell

An electrochemical cell is a device used for generating an electromotive force and current from electrochemistry, or the reverse, inducing a chemical reaction by a flow of current....
. It can also be used to determine the total voltage
Voltage

Electrical tension is the potential difference between two points of an electrical or electronic circuit, expressed in volts. It is the measurement of the potential for an electric field to cause an electric current in an electrical conductor....
 (electromotive force
Electromotive force

Electromotive force is a term used to characterize electrical devices, such as voltaic cells, Thermoelectric effects, electrical generators and transformers, and even resistors....
) for a full electrochemical cell. It is named after the German physical chemist who first formulated it, Walther Nernst
Walther Nernst

Walther Hermann Nernst was a Germany physical chemist 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:    (half-cell reduction potential)    (total cell potential) where
  • Ered is the half-cell reduction potential
    Reduction potential

    Reduction potential is the tendency of a chemical species to acquire electrons and thereby be redox. Each species has its own intrinsic reduction potential; the more positive the potential, the greater the species' affinity for electrons and tendency to be reduced....
  • Eored is the standard half-cell reduction potential
    Standard electrode potential

    In electrochemistry, the standard electrode potential, abbreviated E? or Eo , is the measure of individual potential of a reversible electrode at standard state, which is with solutes at an effective concentration of 1 moldm-3, and gases at a pressure of 1 bar....
  • Ecell is the cell potential (electromotive force
    Electromotive force

    Electromotive force is a term used to characterize electrical devices, such as voltaic cells, Thermoelectric effects, electrical generators and transformers, and even resistors....
    )
  • Eocell is the standard cell potential
  • R is the universal gas constant: R = 8.314 472(15) J K−1 mol−1
  • T is the absolute temperature
  • a is the chemical activity
    Activity (chemistry)

    In chemical thermodynamics activity is a measure of the ?effective concentration? of a species in a mixture. By convention, it is a dimensionless quantity....
     for the relevant species. aX = ?XcX, where ?X 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 directly in terms...
     of species X. (Since activity coefficients tend to unity at low concentrations, activities in the Nernst equation are frequently replaced by simple concentrations.)
  • F is the Faraday constant
    Faraday constant

    In physics and chemistry, the Faraday constant is the magnitude of electric charge per mole of electrons. While most uses of the Faraday constant, denoted F, have been replaced by the standard SI unit, the coulomb, the Faraday is still widely used in calculations in electrochemistry....
    , the number of coulomb
    Coulomb

    The coulomb is the SI unit of electric charge. It is named after Charles-Augustin de Coulomb....
    s per mole
    Mole

    Mole may refer to:...
     of electrons: F = 9.648 533 99(24)×104 C mol−1
  • z is the number of electron
    Electron

    The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
    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....
  • Q is the reaction quotient
    Reaction quotient

    In chemistry, reaction quotient is a quantitative measure of the extent of reaction, the relative proportion of products and reactants present in the reaction mixture at some instant of time....
    .


At room temperature (25 °C), RT/F may be treated like a constant and replaced by 25.679 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:

The Nernst equation is used in physiology
Physiology

Physiology is the study of the mechanical, physical, and biochemical functions of living organisms. Physiology has traditionally been divided between plant physiology and animal and all living things physiology but the principles of physiology are universal, no matter what particular organism is being studied....
 for finding the electric potential
Electric potential

At a point in space, the electric potential is the potential energy per unit of electric charge that is associated with a static electric field....
 of a cell membrane
Cell membrane

The cell membrane is the interface between the cellular machinery inside the cell and the fluid outside.It is a semipermeable lipid bilayer found in all cell ....
 with respect to one type of ion
Ion

An ion is an atom or molecule which has lost or gained one or more electrons, giving it a positive or negative electrical charge. According to the Bohr_model this will be from or in the outer shield 'n'....
.

Nernst 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:

When the membrane is in thermodynamic equilibrium
Thermodynamic equilibrium

In thermodynamics, a thermodynamics#Thermodynamic system is said to be in thermodynamic equilibrium when it is in thermal equilibrium, mechanical equilibrium, and chemical equilibrium....
 (i.e. no net flux of ions), the membrane potential
Membrane potential

Membrane potential , is the voltage difference between the interior and exterior of a cell. Because the fluid inside and outside a cell is highly conductive, whereas a cell's plasma membrane is highly resistive, the voltage change in moving from a point outside to a point inside occurs largely within the narrow width of the membrane itself...
 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

Relatively static membrane potential of quiescent cells is called resting membrane potential , as opposed to the specific dynamic electrochemical phenomenona 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 potential across a cell's membrane taking into account all of the ions that are permeant through that membrane....
.

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.

Derivation


Using Boltzmann factors


For simplicity, we will consider a solution of redox-active molecules that undergo a one electron reversible reaction

and which have a standard potential of zero. The chemical potential
Chemical potential

In thermodynamics, physics and chemistry, chemical potential, symbolized by ?, is a term introduced by the American engineer, chemist and mathematical physicist Willard Gibbs, which he defined as follows:...
  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 voltammetry electrochemistry measurement. In a cyclic voltammetry experiment the working electrode potential is ramped linearly versus time like linear sweep voltammetry....
 that is setting the solution's electrochemical potential
Electrochemical potential

In electrochemistry, the electrochemical potential, , sometimes confusingly abbreviated to ECP, is a thermodynamic measure that combines the concepts of energy stored in the form of chemical potential and electric charge....
.

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:

Taking the natural logarithm of both sides gives

If at [Ox]/[Red] = 1, we need to add in this additional constant:

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


Using entropy and Gibbs free energy


Quantities here are given per molecule, not per mole, and so Boltzmann constant
Boltzmann constant

The Boltzmann constant is the physical constant relating energy at the particle level with temperature observed at the bulk level. It is the gas constant R divided by the Avogadro constant NA:...
 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: and .

The entropy of a molecule is defined as

where 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

The change in entropy from some state 1 to another state 2 is therefore

so that the entropy of state 2 is

If state 1 is at standard conditions, in which is unity (e.g., 1 atm or 1 M), it will merely cancel the units of . We can therefore write the entropy of an arbitrary molecule A as

where is the entropy at standard conditions and [A] denotes the concentration of A. The change in entropy for a reaction

is then given by

We define the ratio in the last term as the reaction quotient:

In an electrochemical cell, the cell potential E is the chemical potential available from redox reactions . E is related to the Gibbs free energy
Gibbs free energy

In thermodynamics, the Gibbs free energy is a thermodynamic potential that measures the "useful" or process-initiating Work obtainable from an isothermal, Isobaric process thermodynamic system....
 change only by a constant: , where n is the number of electrons transferred. (There is a negative sign because a spontaneous reaction has a negative and a positive E.) The Gibbs free energy is related to the entropy by , where H is the enthalpy and T is the temperature of the system. Using these relations, we can now write the change in Gibbs free energy,

and the cell potential,

This is the more general form of the Nernst equation. For the redox reaction , and we have:

The cell potential at standard conditions is often replaced by the formal potential , 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

Or at standard temperature
Standard conditions for temperature and pressure

In physical sciences, standard conditions for temperature and pressure are standard sets of conditions for experimental measurements, to allow comparisons to be made between different sets of data....
,

We have thus related the standard electrode potential
Standard electrode potential

In electrochemistry, the standard electrode potential, abbreviated E? or Eo , is the measure of individual potential of a reversible electrode at standard state, which is with solutes at an effective concentration of 1 moldm-3, and gases at a pressure of 1 bar....
 and the equilibrium constant
Equilibrium constant

For a general chemical equilibriumthe equilibrium constant can be defined bywhere is the activity of the chemical species A etc . It is conventional to put the activities of the products in the numerator and those of the reactants in the denominator....
 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....
 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 tends to ħinfinity. 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.

See also

  • 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 potential across a cell's membrane taking into account all of the ions that are permeant through that membrane....
  • Galvanic cell
    Galvanic cell

    The Galvanic cell, named after Luigi Galvani, is a part of a Battery consisting of an electrochemical cell with two different metals connected by a salt bridge or a porous disk between the individual half-cells....
  • Concentration cell
    Concentration cell

    A Concentration cell is an electrochemical 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....


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