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*''R'' [[ثابت عمومی گازها]] (8.314472 J K<sup>-1</sup> mol<sup>-1</sup>)
*''T'' دمای مطلق. (''T''<sub>K</sub> = 273.15 + ''T''<sub>°C</sub>.)
*''a'' میزان فعالیت شیمیایی. <math>a_X = \gamma_X[X]</math> whereکه <math>\gamma_X</math> ضریب فعالیت نوع X. است. (Sinceدرحالی activityکه coefficientsضرایب tendsفعالیت toبا unityتمرکز atپایین lowبه concentrations,یک activitiesمیل inمی‌کنند، theفعالیت‌ها Nernstدر equationمعادله areنرنست frequentlyمتناوباً replacedبا byتمرکز simpleساده concentrations.جایگزین می‌شوند)
*''F'' [[ثابت فارادی]](the magnitude of the charge per mole of electrons), equal to 9.6485309×10<sup>4</sup> C mol<sup>-1</sup>
*''z'' تعداد الکترون‌هایی است که در جریان [[برقکافت]] واکنش می‌دهند
*''Q'' isخارج theقسمت [[reactionواکنش quotient]]است.
در دمای اتاق(25 °C), <i>RT/F</i> برای پایانه‌ها ثابت 25.679 mV می توان جایگزین نمود.
</math>.
معادله نرنست در [[فیزیولژی]] برای پیدا کردن [[پتانسیل الکتریکی]] [[سلول قشایی]] برحسب به یک نوع [[یون]] استفاده می‌شود
The Nernst equation is used in [[physiology]] for finding the [[electric potential]] of a [[cell membrane]] with respect to one type of [[ion]].
==Physiological application: the Nernst potential==
The Nernst potential, of a ion of charge <math> z </math>, across a membrane, is determined by the concentration ratio in and outside the cell:
:<math>E = \frac{R T}{z F} \log\frac{[\mbox{ion outside cell}]}{[\mbox{ion inside cell}]}</math>
When the membrane is in [[thermodynamic equilibrium]], i.e. no net flux of ions, the [[membrane potential]] must be equal 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]] can be determined from e.g. the [[Goldman equation]].
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==
{{Confusing|date=December 2007}}
The Nernst Equation may 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 factor]]s.
===Using Boltzmann factors===
For simplicity, we will consider a solution of redox-active molecules that undergo a one electron reversible reaction
:<math>\mathrm{Ox} + e^- \rightleftharpoons \mathrm{Red}</math>
and which have a standard potential of zero. The [[chemical potential]] <math>\mu_c</math> of this solution is the difference between the energy barriers for taking electrons from and for giving electrons to the [[Cyclic voltammetry|working electrode]] 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:
:<math>
\frac{[\mathrm{Ox}]}{[\mathrm{Red}]}
= \frac{\exp \left(-[\mbox{barrier for losing an electron}]/kT\right)}
{\exp \left(-[\mbox{barrier for gaining an electron}]/kT\right)}
= \exp \left(\mu_c / kT \right).
</math>
Taking the natural logarithm of both sides gives
:<math>
\mu_c = kT \ln \frac{[\mathrm{Ox}]}{[\mathrm{Red}]}.
</math>
If <math>\mu_c \ne 0</math> at [Ox]/[Red] = 1, we need to add in this additional
constant:
:<math>
\mu_c = \mu_c^0 + kT \ln \frac{[\mathrm{Ox}]}{[\mathrm{Red}]}.
</math>
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
<math>\mathrm{Ox} + e^- \rightarrow \mathrm{Red}</math>:
:<math>
E = E^0 + \frac{kT}{e} \ln \frac{[\mathrm{Ox}]}{[\mathrm{Red}]}
= E^0 - \frac{RT}{F} \ln \frac{[\mathrm{Red}]}{[\mathrm{Ox}]}.
</math>
===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: <math>R = kN_A</math> and
<math>F = eN_A</math>.
The entropy of a molecule is defined as
:<math>
S \ \stackrel{\mathrm{def}}{=}\ k \ln \Omega,
</math>
where <math>\Omega</math> 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
:<math>
S = k\ln \ (\mathrm{constant}\times V) = -k\ln \ (\mathrm{constant}\times c).
</math>
The change in entropy from some state 1 to another state 2 is therefore
:<math>
\Delta S = S_2 - S_1 = - k \ln \frac{c_2}{c_1},
</math>
so that the entropy of state 1 is
:<math>
S_2 = S_1 - k \ln \frac{c_2}{c_1}.
</math>
If state 1 is at standard conditions, in which <math>c_1</math> is unity (e.g.,
1 atm or 1 M), it will merely cancel the units of <math>c_2</math>. We can therefore
write the entropy of an arbitrary molecule ''A'' as
:<math>
S(A) = S^0(A) - k \ln [A], \,
</math>
where <math>S^0</math> is the entropy at standard conditions and [''A''] denotes the
concentration of ''A''.
The change in entropy for a reaction
:<math>
aA + bB \rightarrow yY + zZ
</math>
is then given by
:<math>
\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}.
</math>
We define the ratio in the last term as the reaction quotient:
:<math>
Q \ \stackrel{\mathrm{def}}{=}\ \frac{[Y]^y [Z]^z}{[A]^a [B]^b}.
</math>
In an electrochemical cell, the cell potential ''E'' is the
chemical potential available from redox reactions (<math>E = \mu_c/e</math>).
''E'' is related to
the [[Gibbs free energy]] change <math>\Delta G</math> only by a constant:
<math>\Delta G = -neE</math>, where ''n'' is the number of electrons transferred.
(There is a negative sign because a
spontaneous reaction has a negative <math>\Delta G</math> and a positive ''E''.)
The Gibbs free energy is related to the entropy by <math>G = H - TS</math>, 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,
:<math>
\Delta G = \Delta H - T \Delta S = \Delta G^0 + kT \ln Q, \,
</math>
and the cell potential,
:<math>
E = E^0 - \frac{kT}{ne} \ln Q.
</math>
This is the more general form of the Nernst equation.
For the redox reaction
<math>\mathrm{Ox} + ne^- \rightarrow \mathrm{Red},</math>
<math>Q = [\mathrm{Red}]/[\mathrm{Ox}]</math>, and we have:
:<math>
E = E^0 - \frac{kT}{ne} \ln \frac{[\mathrm{Red}]}{[\mathrm{Ox}]}
= E^0 - \frac{RT}{nF} \ln \frac{[\mathrm{Red}]}{[\mathrm{Ox}]}
= E^0 - \frac{RT}{nF} \ln Q.
</math>
The cell potential at standard conditions <math>E^0</math> is often
replaced by the formal potential <math>E^{0'}</math>, 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
:<math>
\begin{align}
0 &= E^o - \frac{RT}{nF} \ln K\\
\ln K &= \frac{nFE^o}{RT}
\end{align}
</math>
Or at [[Standard conditions for temperature and pressure |standard temperature]],
:<math>\log_{10} K = \frac{nE^o}{59.2\text{ mV}} \quad\text{at }T = 298 \text{ K}.</math>
We have thus related the [[standard electrode potential]] 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]] 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 ±infinity. 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.
==جستارهای وابسته==
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