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Nernst Equation

The Nernst Equation is especially used in electrochemistry to determine the equilibrium reduction potential for the half cell reaction in electrochemical cell.

The voltage and concentration of an electrochemical cell is also determine by the Nernst Equation.

The Nernst Equation is represented as

Ecell = E0cell - (RT/nF) lnQ

Where

  • Ecell = potential of cell in nonstandard conditions in volt,
  • E0cell = potential of cell in standard conditions,
  • R is the gas constant which is equal to 8.31 (volt-coulomb)/ (mol-K),
  • T is the temperature in Kelvin, n stands for number of moles of electrons which get exchanged in the electrochemical reaction,
  • F is Faraday's constant which equals to 96500 coulombs/mol, and Q is the reaction quotient which stands for equilibrium expression.


Th Nernst equation can also be expressed in the different form which is show below.

Ecell = E0cell - (2.303 x RT/nF) logQ

If the temperature is equal to 298K then equation is

Ecell = E0cell - (0.0591 V/n) log Q

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Nernst Equation Derivation

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The Nernst equation relates the concentration and electrical gradient. The chemical concentration gradient is the cause for the movement of ions from the high concentration area to low concentration region while this movement of charge is opposed by the electrical gradient.

But at the equilibrium situation, both balanced with each other. For deriving the Nernst Equation, let the cell reaction takes place at the metal surface of the electrochemical cell in equilibrium is considered;

Az+ + (z/2) H2 = A + zH+

The two half equations of the above equilibrium chemical reaction are written as

Az+ + ze- = A
And (z/2) H2= zH+ + ze-

The reaction (2) shows the equilibrium between atoms of A which present on a metal surface and ions Az+ present in solution. The equilibrium stands for equivalence between the rate of forward and backward reaction or reverse reaction.
For the above mentioned reaction, the change in free energy ΔG is represented by the below mentioned equation.

ΔG = ΔGo + RTln [XA/ XoA] / [CA+/ CoA]

where XA and XAo shows the mole fraction of A in the metal in solution and in standard condition respectively. CA+ and CoA is the concentration of ions Az+ in solution and in standard condition respectively. The value of XA equals to 1 if the metal A is present in the pure state. The value of X0A and C0Az+ equals to unit mole fraction and1 mol dm-3 in the standard state for the ions. So the free energy change can be written as

ΔG = ΔGo + RTln[XA / CA+]

The free energy change is also referred as the chemical driving force and at equilibrium, this ΔG the chemical driving force equals to the electrical driving force (Ee).

So ΔG0 = -zF Ee

where F is Faraday’s constant which equals to 96485 coulombs per mole of electron and z stands for the number of exchanged electrons in reaction in moles.

In the standard conditions,
ΔG0 = -zF E0

The free energy change I standard condition ΔG0 can be expressed in terms of thermodynamic equation. So ΔG0 = -RT lnK, where K is the equilibrium constant of reaction. Thus the standard potential is related to the equilibrium constant K.

So E0 = [RT/zF] lnK

Thus by substituting the value of ΔG and ΔG0 in the equation (4), it becomes
-zF Ee
= -zF E0 + RTln[1/ CA+]

Or the equilibrium potential Ee = E0 - RT/zF ln[1/ CA+]

If the decadic log term is used then ln X = 2.303 log X, so the equation can also
be written as

Ee = E0 – [2.303RT/zF] ln[1/ CA+]

This equation is Nernst equation for simple reaction which shows the in-dependency of the equilibrium potential, Ee on the pH of the solution but Nernst equations depends on the pH of the solution for the half-cell reactions which contains H+ ions.

For example, the half-cell reaction of

MnO4- + 4H+ + 3e- MnO2 + 2H2O and
E0 = 0.588 Volt (Standard hydrogen electrode)

So the Nernst equation for the above reaction is

Ee = E0 – (2.303RT/3F) ln[(MnO2 ) (H2O )2/ (MnO4- )(H+)4]
Or
Ee = E0 – (0.1971) log [1/ (MnO4-) (H+)4]
Or Ee= 0.588 + (0.197) log [ MnO4- ] - 0.789 pH

Thus the generalized form of the Nernst equation is

Ee = E0 – (2.303RT/zF) log [reduced/ oxidized]

The term [reduced] shows the concentrations of the species that present on the reduced side of the electrode reaction while the term [oxidized] represents the concentration of the same species for the oxidized side of the electrode reaction.

Nernst Plank Equation

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This equation describes the movement of ions in solution. The movement of ions in solution can be understood by three mechanisms for movement which are given below.
  1. Thermal Brownian motion,
  2. Drift down a concentration gradient,
  3. Ordered drift because of potential field or diffusion.
The molecules are not in stationary state but continuously move in solution. The continuously motion of ions result the collision with neighbors. The collision is higher in the high concentration area of ions than the low concentration area. But there is a collision gradient at the interface of high and low concentration area. This collision gradient moves the carriers in the low concentration area to increase the concentration. Thus the equilibrium is setup which makes the collision frequency uniform throughout the solution. The flux is associated with the drift which down the concentration gradient. The flux is denoted by;

j = -ZD d[c]/dt------------>(1)

where Z and D are the valence of the charge carrier and the diffusion constant respectively.

The movement of charge carriers is increased in the electric field with increasing the electrical attraction of the carrier. The velocity of carriers is lost after collision which is recovered by the acceleration. This acceleration is due to the attraction of the charge carrier and the potential field. Each charge feels the force in the electric field which is given by the below equation.

F= -qE = d (mv)/dt = mvd/dt,

where
  • E and q are the electric field and unit charge respectively.
  • F is the force feels by the unit charge,
  • q is the unit charge and E is the electric field.

The drift velocity is given by vd and t shows the time of the charge carriers collisions.
Thus the drift velocity of collisions due to the electric field is equal to;
vd = -qEr/m, As the mobility of the charge µ = qr/m
So the drift velocity vd = - µE

The above equation shows that the drift velocity is directly and inversely proportional to the charge and mass of the charge respectively. This equation shows the spatially uniform electric field and states that the charge moves with a fixed velocity which is defined as the drift velocity.

Now the current density j = vd ZF[C] which is related to the flow of charge in an electric field in a solution. Z and [C] are the valence and the concentration of the charge carrier. F is the number of charge in coulombs present in per mole of ion.

The drift velocity vd = - µE so the current density can be expressed as

j = µZF[C] E= σE ----------->(2)

where σ is the conductivity and it is equal to µZF[C] defined by Ohm law.
At the equilibrium condition, by combining (by equation 1 and 2) the diffusive components and electrical components of the flux, the equation becomes

-ZDd[C]/dx + µEZF[C] = 0
Or -Dd[C]/dx + µF[C]dV/dx = 0
Or Dd[C]/dx = µF[C]dV/dx


After taking integration on both the sides;

∫Dd[C]/dx = ∫µF[C]dV/dx
Or ∫Dd[C]/[C] = ∫µFdV
Or D [lnCout –Cin] = µF [Vout-Vin]


Vout-Vin = D/ µF [lnCout /Cin]

This equation is known as Nernst-Plank equation.

As D = µRT, where R stands for gas constant which is equal to 8.314 J/K mole at 27ºC, and T is the absolute temperature. The trans-membrane potential for monovalent cation or anion or single charge carrier is Vmembrane(mv)= RT/ F [Cout ]/[Cin].

Nernst Potential

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The ionic gradient is the combination of the chemical and electrical gradients. The Nernst Equation is used to determine the point at which both the chemical and electrical gradient balance each other. This is the point of ionic equilibrium. For ions, the equilibrium potential is given by below mentioned Nernst Equation.

E = (RT/zF) ln{[X]extracellular/[X]intracellular}

Where E shows the Nernst Equilibrium Potential, R and T are the gas constant and the temperature in Kelvin respectively, and z is the ionic charge. It can also be written as

E= (z61.5) log {[X]extracellular/[X]intracellular}

There are certain assumptions given by Nernst for using this Nernst Equilibrium Potential equation which are given as above. The equation can be used for only single ion at a time and the ions should be in equilibrium condition. If the equation is used for cell membrane then membrane must be permeable to that particular ion.

Nernst Equation Membrane Potential

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The membrane potential is completely different form the equilibrium potential. The membrane potential refers as the diffusion potential that generates across the cell membrane. So the membrane potential is show as Em = (RT/zF) ln{[Co]/ [Ci]}, where z is the charge on ion, R is the gas constant (1.987 cal K-1 mol-1), T is the standard temperature, F is the faraday constant (23.062 cal volt-1 mol-1) and [C]o is the inside concentration of diffusion ion, and [C]I is the outside concentration of diffusing ions.

The membrane potential is generated by the separation of both the positive and negative ions across the membrane. When the voltmeter is used to determine the voltage difference across the cell membrane of cardiomyocyte then it is observed that the inner part of cell has negative charge while the outer side has positive charge. If it is measured in resting situation then this is known as resting membrane potential.

If the proper simulation of cell is done then the negative voltage is converted into the positive voltage due to the generation of action potential. Thus the membrane potential is similar to the separation of positive and negative charges in the battery. Cardiac cells are similar to the living cells.

The cardiac cell possess different ionic concentrations of different ions across the cell membrane like for Na+, K+, Cl-, and Ca++. The cell membrane is impermeable for negatively charged proteins present in the cell. The cardiac cell has high concentration of potassium ion inside the cell then outside the cell. So the potassium ions get diffused to the outside of the cell to maintain the equilibrium. But the concentration of sodium and calcium ions is lower inside the cell and greater outside the cell.

In the cell, the membrane potential is measured by considering main three factors that are mentioned as below.
  1. The concentration of both the positive and negative ions present inside and outside of the cell,
  2. The cell membrane permeability for the both ions through the ion channels,
  3. The capacity of electrogenic pumps which maintain the concentration of ions in the membrane. The pumps can be Ca+2 transport pumps or Na+ / K+-ATP pumps.

Potassium ion-If the cell is considered in which only K+ ions can move through the potassium channels. As the concentration of potassium ions are higher in the inner side of cell so they get diffused to the outer side of cell. This diffusion gives the negatively charged ions of proteins thus the separation of charge takes place across the membrane. This separation generates the membrane potential difference across the cell membrane.

This diffusion of potassium ions from the inner side can be prevented by applying the negative charge inner region of cell membrane. The applied negative charges across the cell membrane which oppose the diffusion of potassium ions of reduce the concentration gradient of potassium ions is known as the equilibrium potential for potassium ions. This is calculated by using the Nernst Equation of potential. So the Nernst potential for K+ is

EK = (-61/z )log {[K+]i /[K+]o}, if the value of [K+]i = 150 mM, [K+]o = 4 mM, and z = 1 as mono valent of K+ ion then, EK = (-90) mV.

This value of EK (the electrical potential of mono valent potassium ion) is essential for reducing the chemical gradient of potassium ions.

If the concentration of K+ is increased up to 4mM - 40 mM in the outside of the cell then the chemical gradient of K+ becomes reduced which shows that the lower value of membrane potential is needed to manage the electrochemical equilibrium (EK less negative). So when the concentration of K+ is approx 10-fold at the outer side of the cell then the less negative voltage is needed for the K+ ions diffusion to the outside of cell.

Nernst Equation Membrane Potential
  • The net electrochemical potential is the difference between the resting potential (approx -90mv) and the membrane equilibrium potential (-96mv).
  • So the net potential is equal to approx 6mv which act on the potassium ions. Thus the membrane potential is higher than the equilibrium potential so the net driving force is acted to the outward of the cell membrane due to mono valent of K+ ions.
  • The resting cell possess finite permeability for K+ and a small net driving force acting outward on K+ which makes the diffusion of K+ slow from the cell.
  • If this process of diffusion is continued then the chemical gradient of potassium ions are completely lost which brings back with the use of Na+ / K+-ATP pumps.
  • Sodium and calcium ions-The mono valent sodium ions are diffused inside the cell with reduce the chemical gradient due to low concentration of Na+ in the inside region of cell membrane.
  • Thus by applying the positive charge inside the cell, the diffusion of sodium ions can be prevented.
  • The electrochemical equilibrium is reached when the positive charge balances with the chemical diffusion force which drives Na+ into the cell and stops the movement of Na+ into the cell.
  • Thus the membrane potential which is needed to generate the electrochemical equilibrium is known as the equilibrium potential for sodium ions Na+. It is expressed as ENa.

This equilibrium potential for sodium is measured by below mentioned equation

ENa = (-61/z) log {[Na+]i /[Na+]o}, if the value of [Na+]i = 20mM, [Na+]o = 145 mM, and z = 1 as monovalent of Na+ ion then ENa = 52 mV

This amount of ENa is required to stop the diffusion of Na+ into the cell. The resting potential is equal to approx -90mV which shows the high value of chemical and electrical gradient is acted on the sodium ions. Thus the net electrochemical driving force is approx (-142mV).

The cardiac cell has very low permeability for the Na+ ions so in resting membrane potential, the small amount of sodium ions leaks in the cell. But the cation potential increases the permeability of cell membrane for sodium ions which also increases the movement of sodium ions in the cell. This movement is done through the sodium channels. The Ca2+ ions have higher differences of concentration across the cell membrane like the sodium ions. So the calcium ions also get diffused through the calcium channels inside the cell membrane.

So the equilibrium potential for calcium ions is

ENa = (-61/z) log {[Ca+2]i /[Ca+2]o}, if the value of [Ca+2]i = 0.0001 mM, [Ca+2]o = 2.5 mM, and z = 2 as divalent of Ca+2 ion
then ECa = 134 mV


This value shows that the value of the equilibrium potential for calcium is highly positive than the resting membrane potential. So the net electrochemical force is required for the movement of Ca2+ inside the cell.

Thus the membrane potential depends on the concentration of different ions across the membrane. It becomes change with changing the concentration of ions.

Nernst Effect

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The Nernst effect is a kind of thermoelectric or thermo magnetic effect. This is observed when the electrically conducted sample is allowed for the magnetic field and the temperature gradient is perpendicular to each other. This induces a new electric field which is perpendicular to both the electrical and magnetic field.

The perpendicular electrical field is induced if the magnetic field and temperature gradient are transverse to each other and electrically charge carriers are moved. Thus this effect of induced field is measured by Nernst effect and given by the Nernst coefficient which is defined by the below mentioned equation.

N = (Ey/Bz)/ (dT/dx)

Ey and BZ are defined as the y and z component of the electric field and magnetic field respectively and dT/dx is the temperature gradient. The reverse phenomena of Nernst effect is known as the Nernst- Ettingshausen effect. The Nernst effect also present in semiconductors but not in metals. The Nernst effect is also shown by high-temperature superconductors in their two phases which are superconducting and pseudo-gap phase.

Nernst Equation Example

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Below you could see examples

1. Calculate the initial voltage of cell at temperature 298K if the zinc electrode is present in an acidic solution of Zn2+ solution with the concentration 0.80 M. This zinc electrode is connected to a salt bridge of 1.30 M Ag+ solution with a silver electrode.

From the standard reduction potential table,

Electrode potential for the half cell reaction Zn2+aq + 2e- ZnS; E0 reduced = -0.76 V
Electrode potential for the half cell reaction: Ag+aq + e- AgS; E0 reduced = +0.80 V

Ecell = E0cell - (0.0591 Volt/n) log {[Zn2+]/ [Ag+]2}

For the spontaneous reaction, the value of E0 is positive so the Zn must be oxidized which has standard electrode potential +0.76 V and silver must be reduced (+0.80 V).

So for the half cell reaction

Zns Zn2+aq + 2e- and E0oxidised = +0.76 V
2Ag+aq + 2e- 2Ags: E0reduced = +0.80 V

Thus the net chemical reaction can be written as

Zn(s) + 2Ag+aq Zn2+aq + 2Ags

with

E0 = E0oxidized + E0reduced = (0.76 + 0.80) V = 1.56 V

Thus the Nernst equation

Ecell = E0cell - (0.0591 Volt/n) log {[Zn2+]/ [Ag+]2 }
Or Ecell = E0cell - (0.0591 Volt/n) log {[Zn2+] / [Ag+]2}
Or Ecell = E0cell - (0.0591 Volt/n) log {(0.80)/ (1.30)2}
Or Ecell = E0cell - (0.0591 Volt/n) log {(0.80)/(1.69)}
Or Ecell = E0cell - (0.0591 Volt/n) log (0.47)
Or E = 1.56 V - (0.0591 / 2) log (0.47)

Thus the initial voltage E = 1.57 Volt

2. The main equation is O2 + 4H+ + 4e- = 2H2O in which water and oxygen is the reduced species and the oxidized species respectively.

The electrochemical half-equations of the above reaction are written as;

Oxidized State + ne- Reduced State

Thus, the Nernst equation for that reaction is;

Ecell = E0cell - (0.0591 Volt/n) log {[H2O]2 /[O2][H+]4}

The concentration of water is 1 mol dm-3 as it is considered in the pure state.
Now the equation becomes, Ecell = E0cell - (0.0591 Volt/n) log {1/ [O2] [H+]4}

3. The half cell reaction of aluminium is Al = Al3+ + 3e-,
so the Nernst equation is;
E = E0 – (2.303RT/3F) log {[Al]/ [Al+3]},
as E0 = (–1.66) V at 298K

and the pure aluminium activity is equal to 1, so the Nernst equation is now.
Or E = -1.66 + (0.0197) log [Al3+]

The net equilibrium potential does not depend on the pH in this reaction.

4. The half cell reaction is Fe (OH)3 + 3H+ + e- = Fe2+ + 3H2O and the value of standard hydrogen electrode is E0 = 1.060 V. So the Nernst equation can be written as.

E = E0 – [2.303RT/F] log {[Fe2+] [H2O]3 / [Fe(OH)3][H+]3}
Or E = 1.060-0.0591[log {[Fe2+]/ [H+]3}]
Or E = 1.060 – (0.0591) log [Fe2+] + (3 × 0.0591) log [H+], as pH = -log [H+],
So E = 1.060 – (0.059) log [Fe2+] + 0.177 log [H+].

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