Nernst Equation: Expression, Derivation and Equations

The Nernst Equation was given by German Chemist Walther Hermann Nernst. The relationship between electrode potential and the concentration of electrolyte solution is known as Nernst Equation. Nernst Equation is used to determine the cell potential of an electrochemical cell at a given temperature, pressure and reactant concentration. The cell potential of electrochemical cells can be determined with the help of the Nernst Equation even under non - standard conditions. 

Key Terms: Nernst Equation, Standard Electrode potential, Temperature, Electricity, Metals, Electrochemistry

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Derivation of Nernst Equation

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For general electrode reaction, 

Mⁿ⁺ + ne^- → M(s)

The electrode potential can be represented by 

E(Mⁿ⁺| M) = E ⁰ (Mⁿ⁺| M) - RT/nF ln [M(s)] / [Mⁿ⁺(aq)]

E(Mⁿ⁺| M) = E ⁰ (Mⁿ⁺| M) - 2.303RT/nF log [M(s)] / [Mⁿ⁺(aq)]

Where, E(Mⁿ⁺| M) = Electrode potential 

E ⁰ (Mⁿ⁺| M) = Standard Electrode potential

R = gas constant 

T = temperature 

n = number of electrons gained during the reaction

F = Faraday of electricity 

[M(s)] = molar concentration of metal

[Mⁿ⁺(aq)] = molar concentration of ions

Substituting the values of R = 8.314JK^-1mol^-1, T = 298K or 25 ⁰C, F = 96500C in above equation 

E(Mⁿ⁺| M) = E ⁰ (Mⁿ⁺| M) - 2.303x 8.314 x298 / n96500 log [M(s)] / [Mⁿ⁺(aq)]

After solving, we get 

E(Mⁿ⁺| M) = E ⁰ (Mⁿ⁺| M) - 0.059/n log [M(s)] / [Mⁿ⁺(aq)]

The concentration of solid phase [M(s)] is taken as unity. The above equation may be written as:

E(Mⁿ⁺| M) = E ⁰ (Mⁿ⁺| M) - 0.059/n log 1/ [Mⁿ⁺(aq)]

At 25 ⁰C, 

E = E⁰  - 0.059/n log 1/ [Mⁿ⁺(aq)]

The video below explains this:

Nernst Equation Detailed Video Explanation:

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Applications of Nernst Equation 

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• Calculation of cell potential using Nernst Equation

The electrode reactions are:-

At anode: Zn(s) \(\rightarrow \) Zn ²⁺(aq) + 2e^-

At cathode: Cu ²⁺(aq) + 2e^- \(\rightarrow \) Cu(s)

The electrochemical cell reaction is:

Zn(s) + Cu ²⁺(aq) \(\rightarrow \) Zn ²⁺(aq) + Cu(s)

The electrode potential for the two electrodes can be calculated by using Nernst equation in which n = 2.

Electrode potential for Copper electrode 

E(Cu ²⁺| Cu) = E ⁰ (Cu ²⁺| Cu) - 2.303RT/ 2F log [Cu]/ [Cu ²⁺(aq)]

Similarly, Electrode potential for Zinc electrode 

E(Zn ²⁺| Zn) = E ⁰ (Zn ²⁺| Zn) - 2.303RT/ 2F log [Zn]/ [Zn ²⁺(aq)]

The e.m.f of the cell is 

E _cell = E _cathode - E _anode 

E _cell = E(Cu ²⁺| Cu) - E(Zn ²⁺| Zn)

Now put the values of Cu and Zn electrode in this equation 

E _cell = E ⁰ (Cu ²⁺| Cu) - 2.303RT/ 2F log [Cu]/ [Cu ²⁺(aq)] - [E ⁰ (Zn ²⁺| Zn) - 2.303RT/ 2F log [Zn]/ [Zn ²⁺(aq)]]

E _cell = E ⁰ (Cu ²⁺| Cu) - E 0 (Zn ²⁺| Zn) - 2.303RT/ 2F [log Cu/ Cu ²⁺ - log Zn/ Zn ²⁺ ]

E _cell = E ⁰ (Cu ²⁺| Cu) - E 0 (Zn ²⁺| Zn) - 2.303RT/ 2F [ log Cu/ Cu ²⁺ / log Zn/ Zn ²⁺ ]

E _cell = E ⁰ (Cu ²⁺| Cu) - E ⁰ (Zn ²⁺| Zn) - 2.303RT/ 2F log [Cu] [ Zn ²⁺ ] / log [Zn] [Cu ²⁺]

Now E ⁰ (Cu ²⁺| Cu) - E ⁰ (Zn ²⁺| Zn) = E ⁰ _cell and concentration of solids is taken as unity so that [Zn] = 1 and [Cu] = 1.

Therefore, E _cell = E ⁰ _cell - 2.303RT/ 2F log[Zn ²⁺ ]/ log[Cu ²⁺]

At T = 298K or 25 ⁰C,R = 8.314JK^-1mol^-1  , F = 96500C

E _cell = E ⁰ _cell - 0.059/ 2 log[Zn ²⁺ ]/ log[Cu ²⁺]

• Calculation of concentration of a solution of Half cell

In a galvanic cell, if all the concentrations except one are known, then the unknown concentration can be calculated by measuring the cell potential and using Nernst equation.

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Equilibrium constant from Nernst Equation 

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The e.m.f of the cell used to determine the equilibrium constant for the cell reaction. At equilibrium, the electrode potentials of two electrodes become equal so that e.m.f of the cell is zero. 

Zn(s) + Cu ²⁺(aq) \(\rightarrow \) Zn ²⁺(aq) + Cu(s)

The concentration of Zn ²⁺(aq) and Cu ²⁺(aq) are equilibrium concentration and equilibrium constant, K _c  is 

 K _c = [Cu] [ Zn ²⁺ ] / [Zn] [Cu ²⁺]

Where [Zn] = 1 and [Cu] = 1

K _c = [ Zn ²⁺ ] / [Cu ²⁺]

The Nernst equation for the above reaction is:

E _cell = E ⁰ _cell - 2.303RT/ 2F log[Zn ²⁺ ]/ log[Cu ²⁺]

At equilibrium, E _cell = 0 

E ⁰ _cell = 2.303RT/ 2F log[Zn ²⁺ ]/ log[Cu ²⁺]

E ⁰ _cell = 2.303RT/ 2F log K _c

At T = 298K, K _c = [ Zn ²⁺ ] / [Cu ²⁺]

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Electrochemical cell and Gibbs energy of the Reaction

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In electrochemical cells, the chemical energy is converted into electrical energy. The cell potential is related to Gibbs energy change. 

ΔG = maximum work

For reaction occurring in an electrochemical cell whose electrodes differ in a potential by E _cell. The work done when amount of charge nF is pushed along by the potential of the cell is given by nFE _cell so that 

Maximum work = nFE _cell

Where F is Faraday constant and n is the number of Electrons transferred. Work is done on the surroundings, because electrical energy flows through the external circuit.

ΔG ⁰  = - nFE ⁰ _cell 

where ΔG 0 is standard Gibbs energy for the reaction. 

Thus, from the measurement of E ⁰  we can calculate the change in standard Gibbs energy. From the standard Gibbs energy, we can also find the equilibrium constant of the reaction.

• Significance 

This reaction helps us to predict the feasibility of the cell reaction. For cell reaction to be spontaneous, ΔG must be negative. This means E must be positive for spontaneous cell reaction.

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