NationMaster - Encyclopedia: Nernst equation

In electrochemistry, the Nernst equation gives the electrode potential (E), relative to the standard electrode potential, (E⁰), of the electrode couple or, equivalently, of the half cells of a battery. In physiology the Nernst equation is used for finding the electric potential of a cell membrane with respect to one ion. English chemists John Daniell (left) and Michael Faraday (right), both credited to be founders of electrochemistry as known today. ... This article may be too technical for most readers to understand. ... It has been suggested that Electrode potential be merged into this article or section. ... This article or section does not cite its references or sources. ... Physiology (in Greek physis = nature and logos = word) is the study of the mechanical, physical, and biochemical functions of living organisms. ... Electric potential is the potential energy per unit charge associated with a static (time-invariant) electric field, also called the electrostatic potential, typically measured in volts. ... Drawing of a cell membrane A component of every biological cell, the selectively permeable cell membrane (or plasma membrane or plasmalemma) is a thin and structured bilayer of phospholipid and protein molecules that envelopes the cell. ...

$E = E^0 - frac{RT}{nF} lnfrac{a_{mbox{Red}}}{a_{mbox{Ox}}}$

The activities of pure solid or liquid phases are taken as unity. For a solution at room temperature (25 °C) the following is true

$E = E^{0'} - frac{0.0591}{n} logfrac{[mbox{Red}]}{[mbox{Ox}]}$ (or 0.025679 using ln).

For a cell membrane potential with respect to one positive ion - cation (for a negative ion (anion) the sign before the logarithm is changed to a minus!) Membrane potential (or transmembrane potential or transmembrane potential difference or transmembrane potential gradient), is the electrical potential difference (voltage) across a cells plasma membrane. ...

$E = E^{0'} + frac{0.0591}{n} logfrac{[mbox{ion out of cell}]}{[mbox{ion inside cell}]}$

where

$E^{0'} = E^0 - frac{RT}{nF} lnfrac{gamma_{mbox{Red}}}{gamma_{mbox{Ox}}}$

R is the universal gas constant, equal to 8.314510 J K^-1 mol^-1
T the temperature in kelvins. (Kelvins = 273.15 + °C.)
a the chemical activities on the reduced and oxidized side, respectively
F is the Faraday constant, equal to 9.6485309*10⁴ C mol^-1
n is the number of electrons transferred in the half-reaction.
[Red] is the concentration of oxidizing agent (the reduced species).
[Ox] is the concentration of reducing agent (the oxidized species).
$E 0'$ is the formal electrode potential
$γ$ is the activity coefficient

1 History
2 Derivation
- 2.1 Using Boltzmann factors
- 2.2 Using entropy and Gibbs free energy
3 Limitations
4 External links
5 See also

Molar gas constant (also known as universal gas constant, usually denoted by symbol R) is the constant occurring in the universal gas equation, i. ... Temperature is also the name of a song by Sean Paul. ... The kelvin (symbol: K) is the SI unit of temperature, and is one of the seven SI base units. ... Look up Activity in Wiktionary, the free dictionary Activity may refer to— in chemistry, the effective concentration of an ion or other solute for the purposes of chemical reactions and other mass action. ... In physics and chemistry, the Faraday constant is the amount of electric charge of one mole of electrons. ... Properties The electron is a lightweight fundamental subatomic particle that carries a negative electric charge. ... A half reaction is either the oxidation or reduction reaction component of a redox reaction. ... European Union Chemical hazard symbol for oxidizing agents Dangerous goods label for oxidising agents An oxidizing agent is a compound that oxidizes another substance in electrochemistry or redox chemical reactions. ... A reducing agent is the element or a compound in a redox (reduction-oxidation) reaction (see electrochemistry) that reduces another species. ... The Activity coefficient for chemicals in a mixture is an indicator of what the concentration of that chemical will be in a vapor of the mixture. ...

History

The Nernst equation is named after the German physical chemist Walther Nernst who was the first to formulate it. Walther Nernst. ...

Derivation

The Nernst Equation can be derived in several different ways. Chemistry textbooks frequently give the derivation in terms of entropy and the Gibbs free energy, but there is a more intuitive method for anyone familiar with Boltzmann factors. Ice melting - a classic example of entropy increasing In thermodynamics, thermodynamic entropy (or simply entropy) S is an important state function of a thermodynamic system: that is, a property depending only on the current state of the system, independent of how that state came to be achieved. ... Gibb In thermodynamics the Gibbs free energy (sometimes also known as free enthalpy) is a thermodynamic potential and is therefore a state function of a thermodynamic system. ... In physics, the Boltzmann factor is a weighting factor determining the relative probability of a system in thermodynamic equilibrium at a temperature T being in a state with energy E: (kB is Boltzmanns constant. ...

Using Boltzmann factors

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

$mathrm{Ox} + e^- rightarrow mathrm{Red}$

and which have a standard potential of zero. The chemical potential $μ c$ of this solution is the difference between the energy barriers for taking electrons from and for giving electrons to the working electrode that is setting the solution's electrochemical potential. The precise meaning of the term chemical potential depends on the context in which it is used. ... Cyclic voltammetry is a kind of electrochemical measurement. ... Electrochemical potential is a thermodynamic measure that reflects energy from entropy and electrostatics and is typically invoked in molecular processes that involve diffusion. ...

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:

$frac{[mathrm{Ox}]}{[mathrm{Red}]} = frac{exp left(-[mbox{barrier for losing an electron}]/kTright)} {exp left(-[mbox{barrier for gaining an electron}]/kTright)} = exp left(mu_c / kT right).$

Taking the natural logarithm of both sides gives

$mu_c = kT ln frac{[mathrm{Ox}]}{[mathrm{Red}]}.$

If $mu_c ne 0$ at [Ox]/[Red] = 1, we need to add in this additional constant:

$mu_c = mu_c^0 + kT ln frac{[mathrm{Ox}]}{[mathrm{Red}]}.$

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 $mathrm{Ox} + e^- rightarrow mathrm{Red}$ :

$E = E^0 + frac{kT}{e} ln frac{[mathrm{Ox}]}{[mathrm{Red}]} = E^0 - frac{RT}{F} ln frac{[mathrm{Red}]}{[mathrm{Ox}]}.$

Using entropy and Gibbs free energy

Quantities here are given per molecule, not per mole, and so Boltzmann's 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: $R = k N A$ and $F = e N A$ .

The entropy of a molecule is defined as

$S equiv k ln Omega,$

where $Ω$ 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

$S = kln (mathrm{constant}times V) = -kln (mathrm{constant}times c).$

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

$Delta S = S_2 - S_1 = - k ln frac{c_2}{c_1},$

so that the entropy of state 1 is

$S_2 = S_1 - k ln frac{c_2}{c_1}.$

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

$S(A) = S^0(A) - k ln [A], ,$

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

$Delta S_mathrm{rxn} = [yS(Y) + zS(Z)] - [aS(A) - bS(b)] = Delta S^0_mathrm{rxn} - k ln frac{[Y]^y [Z]^z}{[A]^a [B]^b}.$

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

$Q equiv frac{[Y]^y [Z]^z}{[A]^a [B]^b}.$

In an electrochemical cell, the cell potential E is the chemical potential available from redox reactions ( $E = μ c / e$ ). E is related to the Gibbs free energy change $Δ G$ only by a constant: $Δ G = - n e E$ , where n is the number of electrons transferred. (There is a negative sign because a spontaneous reaction has a negative $Δ G$ and a positive E.) The Gibbs free energy is related to the entropy by $G = H - T S$ , 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,

$Delta G = Delta H - T Delta S = Delta G^0 + kT ln Q, ,$

and the cell potential,

$E = E^0 - frac{kT}{n} ln Q.$

This is the more general form of the Nernst equation. For the redox reaction $mathrm{Ox} + ne^- rightarrow mathrm{Red},$ $Q = [Red] / [Ox]$ , and we have:

$E = E^0 - frac{kT}{n} ln frac{[mathrm{Red}]}{[mathrm{Ox}]} = E^0 - frac{RT}{nF} ln frac{[mathrm{Red}]}{[mathrm{Ox}]}.$

The cell potential at standard conditions $E 0$ is often replaced by the formal potential $E 0'$ , which includes some small corrections to the logarithm and is the potential that is actually measured in an electrochemical cell.

Limitations

When the Nernst equation is expressed in its most convenient form, the activity of the ions is assumed to be equal to their concentrations, however this assumption is only valid for low concentrations. At higher concentrations the true activities of the ions must be used, this complication makes the use of the Nernst equation difficult as estimation of the activities of ions in their non-ideal state often requires experimental analysis to have been undertaken.

The Nernst equation also only applies when there is no net current flow through the electrode. When there is current flow the activity of ions at the electrode surface changes, and there are additional overpotential and resistive loss terms to the measured potential. In electrochemistry, overpotential is the difference in the electric potential of an electrode with no current flowing through it, at equilibrium, and with a current flowing. ...

External links

Nernst/Goldman Equation Simulator
The Nernst Equation and Action Potentials in the Nervous System from www.medicalcomputing.net

Contents

Derivation

Using Boltzmann factors

Using entropy and Gibbs free energy

Limitations

External links

See also