Nernst Equation

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Nernst Equation

:

E = E^0 - rac{RT}{nF} \lnrac{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'} - rac{0.0591}{n} \lograc{

{Link without Title} }{ {Link without Title} } (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!)

:

E = E^{0'} + rac{0.0591}{n} \lograc{out of cell}}{inside cell}}

where
:

E^{0'} = E^0 - rac{RT}{nF} \lnrac{\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 Kelvin s. (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 Electron s transferred in the Half-reaction .

'' {Link without Title} '' is the concentration of Oxidizing Agent (the reduced species).

'' {Link without Title} '' is the concentration of Reducing Agent (the oxidized species).

'' $E^{0'}$ '' is the formal electrode potential

'' $\gamma$ '' is the Activity Coefficient

HISTORY

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

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 .

Using Boltzmann factors

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

\mathrm{Ox} + e^- 
ightarrow \mathrm{Red}

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

\mu_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 ratio of oxidized to reduced molecules, {Link without Title} / {Link without Title} , 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:
:

rac{

{Link without Title} }{ {Link without Title} } = rac{\exp \left(- for losing an electron} /kT ight)} {\exp \left(- for gaining an electron} /kT ight)} = \exp \left(\mu_c / kT ight).
Taking the natural logarithm of both sides gives
:

\mu_c = kT \ln rac{

{Link without Title} }{ {Link without Title} }.
If

\mu_c 
e 0

at {Link without Title} / {Link without Title} = 1, we need to add in this additional
constant:
:

\mu_c = \mu_c^0 + kT \ln rac{

{Link without Title} }{ {Link without Title} }.
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^- 
ightarrow \mathrm{Red}

:
:

E = E^0 + rac{kT}{e} \ln rac{

{Link without Title} }{ {Link without Title} } = E^0 - rac{RT}{F} \ln rac{ {Link without Title} }{ {Link without Title} }.

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 = kN_A

and

F = eN_A

.

The entropy of a molecule is defined as
:

S \equiv 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 \ (\mathrm{constant}	imes V) = -k\ln \ (\mathrm{constant}	imes c).

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

\Delta S = S_2 - S_1 = - k \ln rac{c_2}{c_1},

so that the entropy of state 1 is
:

S_2 = S_1 - k \ln rac{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

{Link without Title} , \,
where

S^0

is the entropy at standard conditions and {Link without Title} denotes the
concentration of A.
The change in entropy for a reaction
:

aA + bB 
ightarrow yY + zZ

is then given by
:

\Delta S_\mathrm{rxn} = + zS(Z) - -  bS(b) = \Delta S^0_\mathrm{rxn} - k \ln rac{[Z^z}{[B^b}.

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

Q \equiv rac{[Z^z}{[B^b}.

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 free energy change

\Delta G

only by a constant:

\Delta G = -neE

, where n is the number of electrons transferred.
(There is a negative sign because a
spontaneous reaction has a negative

\Delta G

and a positive E.)
The Gibbs free 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 free energy,
:

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

and the cell potential,
:

E = E^0 - rac{kT}{n} \ln Q.

This is the more general form of the Nernst equation.
For the redox reaction

\mathrm{Ox} + ne^- 
ightarrow \mathrm{Red},

Q =

{Link without Title} / {Link without Title} , and we have:
:

E = E^0 - rac{kT}{n} \ln rac{

{Link without Title} }{ {Link without Title} } = E^0 - rac{RT}{nF} \ln rac{ {Link without Title} }{ {Link without Title} }.
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.

EXTERNAL LINKS

Nernst/Goldman Equation Simulator

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

SEE ALSO

Goldman Equation

Galvanic Cell