Nernst Equation: Definition, Expression, Equilibrium Constant, Applications

Nernst Equation: Standard cell potentials are calculated in standard conditions of temperature and pressure. However, in non-standard conditions, the Nernst equation is used to calculate cell potentials. An equation that relates the Gibbs energy to cell potential was devised by Walther Hermann Nernst, commonly known as the Nernst equation.  This equation is very helpful in determining the equilibrium constant, cell potential,  etc. It considers the values of the reaction quotient, standard electrode potentials, temperature, activity for the calculation of cell potential. Let’s explore more about it in this article.

What is Nernst Equation?

The Nernst Equation is an equation that is used to determine the cell potential under non-standard conditions. It relates the measured cell potential to the reaction quotient and accurately determines equilibrium constants (including solubility constants).

Nernst showed that for the electrode reaction, the electrode potential at any concentration measured with respect to standard hydrogen electrode could be represented by:

\({{\rm{E}}_{{\rm{(}}{{\rm{M}}^{\rm{n}}}^{\rm{ – }}{\rm{/M)}}}}{\rm{ = }}{{\rm{E}}^{\rm{0}}}_{{\rm{(}}{{\rm{M}}^{{\rm{n + }}}}{\rm{/M)}}}{\rm{ – }}\frac{{{\rm{RT}}}}{{{\rm{nF}}}}{\rm{ln}}\frac{{{\rm{[M]}}}}{{{\rm{[}}{{\rm{M}}^{{\rm{n + }}}}{\rm{]}}}}\)

Expression of Nernst Equation

For a general electrochemical reaction-

Nernst equation can be written as:

\({{\rm{E}}_{{\rm{(cell)}}}} = {{\rm{E}}^{\rm{0}}}_{{\rm{(cell)}}} – \frac{{{\rm{RT}}}}{{{\rm{nF}}}}\ln {\rm{Q}}\)

\({{\rm{E}}_{{\rm{(cell)}}}} = {{\rm{E}}^{\rm{0}}}_{{\rm{(cell)}}} – \frac{{{\rm{RT}}}}{{{\rm{nF}}}}\ln \frac{{{{\left[ {\rm{C}} \right]}^{\rm{c}}}{{\left[ {\rm{D}} \right]}^{\rm{d}}}}}{{{{\left[ {\rm{A}} \right]}^{\rm{a}}}{{\left[ {\rm{B}} \right]}^{\rm{b}}}}}\)

Where,

\({\rm{R}}\)  is the gas constant  \((8.314\,{\rm{J}}{{\rm{K}}^{{\rm{ – 1}}}}{\rm{mo}}{{\rm{l}}^{{\rm{ – 1}}}})\)

\({\rm{F}}\) is Faraday constant \((96,500\,\,{\rm{C}}\,{\rm{mo}}{{\rm{l}}^{{\rm{ – 1}}}})\)

\({\rm{T}}\) is the temperature in kelvin 

\({\rm{Q}}\) is the reaction quotient

\({\rm{n}}\) is the number of moles of electrons transferred.

Derivation of Nernst Equation

For any cell reaction, Gibbs energy is related to standard electrode potential as follows:

\(\Delta {\rm{G}}\,{\rm{ =  – nFE}}……..{\rm{Eqn(1)}}\)

Where \(\Delta {\rm{G}}\,{\rm{ = }}\) Gibbs energy,

\({\rm{n}}\) is the number of electrons transferred in the reaction, 

\({\rm{F}}\) is Faraday’s constant \((96,500\,{\rm{C/mol)}}\) and

\({\rm{E}}\) is the cell potential.

Under standard conditions, Eqn \({\rm{(1)}}\) can be written as,

\(\Delta {{\rm{G}}^0}\,{\rm{ =  – nF}}{{\rm{E}}^0}……..{\rm{Eqn(2)}}\)

Where, \(\Delta {{\rm{G}}^0}\) is standard Gibbs energy

\({{\rm{E}}^{\rm{0}}}\) is standard cell potential

The reaction is spontaneous; when \({{\rm{E}}^{\rm{0}}}\) is positive, and the reaction is non-spontaneous when \({{\rm{E}}^{\rm{0}}}\)  is negative.

According to the theory of thermodynamics, the reaction quotient, Gibbs energy, and standard Gibbs energy are related as:

\({\rm{\Delta G}} = {\rm{\Delta }}{{\rm{G}}^0} + {\rm{RT}}\,{\rm{lnQ}}………….{\rm{Eqn(3)}}\)

Where \({\rm{Q}}\) is the reaction quotient,

\({\rm{R}}\) is the universal gas constant, and;

\({\rm{T}}\,{\rm{ = }}\) Temperature in Kelvin.

Incorporating the value of \({\rm{\Delta G}}\) and \({\rm{\Delta }}{{\rm{G}}^0}\)  in Eqn \({\rm{(3)}}\),

we get: \( – {\rm{nFE}}\, = \, – {\rm{nF}}{{\rm{E}}^0} + {\rm{RT}}\,{\rm{lnQ}}\)

Dividing both sides by \({\rm{nF}}\) we get- 

\({\rm{E}}\, = \,{{\rm{E}}^0} – \left( {\frac{{{\rm{RT}}}}{{{\rm{nF}}}}} \right)\ln {\rm{Q}}\,…………{\rm{Eqn(4)}}\)

\({\rm{Eqn(4)}}\) can also be written in the form of \({\rm{lo}}{{\rm{g}}_{10}}\) as- 

\({\rm{E}}\, = \,{{\rm{E}}^0} – \left( {\frac{{2.303{\rm{RT}}}}{{{\rm{nF}}}}} \right){\log _{10}}{\rm{Q}}……{\rm{Eqn}}(5)\)

At standard temperature \({\rm{T}} = 298\,{\rm{K}},\), the \(\frac{{2.303{\rm{RT}}}}{{\rm{F}}}\) equals \(0.0592\) Hence,

\({\rm{E}}{\mkern 1mu} = {\mkern 1mu} {{\rm{E}}^0} – \left( {\frac{{0.0592}}{{\rm{n}}}} \right){\log _{10}}{\rm{Q}}………..{\rm{Eqn}}(6)\)

\({\rm{Eqn}}(6)\) indicates that the electrical potential of a cell depends upon the reaction quotient \({\rm{Q}}\) of the reaction. As the reaction proceeds, reactants are consumed; thus, the concentration of reactants decreases and the concentration of products increases. Consequently, cell potential gradually decreases until the reaction is at equilibrium.

Determining Equilibrium Constant with Nernst Equation

At equilibrium, \(\Delta {\rm{G}}\, = \,0\) Hence, cell potential, \({\rm{E}}\, = 0\) and the reaction quotient

\({\rm{Q}}\, = \,{\rm{K}}\) Thus, the Nernst equation, \({\rm{Eqn}}\;(6)\) can be modified to:

\({\rm{0}}\,{\rm{ = }}\,{{\rm{E}}^0} – \left( {\frac{{2.303{\rm{RT}}}}{{{\rm{nF}}}}} \right){\log _{10}}{{\rm{K}}_{\rm{c}}}\)

\({E^0}\, = \,\left( {\frac{{2.303{\rm{RT}}}}{{{\rm{nF}}}}} \right){\log _{10}}{{\rm{K}}_{\rm{c}}}\,\)

Thus, the above equation gives us a relation between the standard electrode potential of the cell and the equilibrium constant.

The above equation indicates that the equilibrium constant \({{\rm{K}}_{\rm{c}}}\,\) is proportional to the standard potential of the reaction. When,

1. \({{\rm{K}}_0} > 1,\,{{\rm{E}}^0} > 0\) reaction favours products formation.
2. \({{\rm{K}}_0} < 1,\,{{\rm{E}}^0} < 0\) reaction favors reactants formation.

Daniel Cell and Nernst Equation

In Daniell cell, the electrode potential for any given concentration of \({\rm{C}}{{\rm{u}}^{2 + }}\) and

\({\rm{Z}}{{\rm{n}}^{2 + }}\) ions, we write-

For cathode-

\({{\rm{E}}_{({\rm{C}}{{\rm{u}}^{2 + }}/{\rm{Cu}})}} = {{\rm{E}}^0}_{({\rm{C}}{{\rm{u}}^{2 +}}/{\rm{Cu}})} – \frac{{{\rm{RT}}}}{{2{\rm{F}}}}\ln \frac{1}{{\left[ {{\rm{C}}{{\rm{u}}^{2 + }}({\rm{aq}})} \right]}}\)

For anode-

\({{\rm{E}}_{({\rm{Z}}{{\rm{n}}^{2 + }}/{\rm{Zn}})}} = {{\rm{E}}^0}_{({\rm{Z}}{{\rm{n}}^{2 + }}/{\rm{Zn)}}} – \frac{{{\rm{RT}}}}{{2{\rm{F}}}}\ln \frac{1}{{\left[ {{\rm{Z}}{{\rm{n}}^{2 + }}({\rm{aq}})} \right]}}\)

The cell potential,

\({{\rm{E}}_{\left( {{\rm{cell}}} \right)}} = {{\rm{E}}^0}_{({\rm{C}}{{\rm{u}}^{2 + }}/{\rm{Cu}})} – {{\rm{E}}^0}_{({\rm{Z}}{{\rm{n}}^{2 + }}/{\rm{Zn}})}\)

\( = \,{{\rm{E}}^0}_{({\rm{C}}{{\rm{u}}^{2 + }}/{\rm{Cu}})} – \frac{{{\rm{RT}}}}{{2{\rm{F}}}}\ln \,\frac{1}{{\left[ {{\rm{C}}{{\rm{u}}^{2 + }}({\rm{aq}})} \right]}} – {{\rm{E}}^0}_{({\rm{Z}}{{\rm{n}}^{2 + }}/{\rm{Zn}})} + \frac{{{\rm{RT}}}}{{2{\rm{F}}}}\ln \frac{1}{{\left[ {{\rm{Z}}{{\rm{n}}^{2 + }}({\rm{aq}})} \right]}}\)

\( = \,{{\rm{E}}^0}_{({\rm{C}}{{\rm{u}}^{2 + }}/{\rm{Cu}})} – {{\rm{E}}^0}_{({\rm{Z}}{{\rm{n}}^{2 + }}/{\rm{Zn}})} – \frac{{{\rm{RT}}}}{{2{\rm{F}}}}\left[ {\ln \,\frac{1}{{\left[ {{\rm{C}}{{\rm{u}}^{2 + }}({\rm{aq}})} \right]}} – \ln \frac{1}{{\left[ {{\rm{Z}}{{\rm{n}}^{2 + }}({\rm{aq}})} \right]}}} \right]\)

\({{\rm{E}}_{\left( {{\rm{cell}}} \right)}} = {{\rm{E}}^0}_{\left( {{\rm{cell}}} \right)} – \frac{{{\rm{RT}}}}{{2{\rm{F}}}}\ln \frac{{\left[ {{\rm{Z}}{{\rm{n}}^{2 + }}({\rm{aq}})} \right]}}{{\left[ {{\rm{C}}{{\rm{u}}^{2 + }}({\rm{aq}})} \right]}}\)

By converting the above equation natural logarithm to the base 10 and substituting the values of

\({\rm{R,}}\,{\rm{F}}\,{\rm{and}}\,{\rm{T = }}\,{\rm{298}}\,{\rm{K}}\) it reduces to:

\({{\rm{E}}_{({\rm{cell}})}} = {{\rm{E}}^0}_{({\rm{cell}})} – \left( {\frac{{{\rm{0}}{\rm{.0592}}}}{{\rm{n}}}} \right){\log _{10}}\frac{{\left[ {{\rm{Z}}{{\rm{n}}^{2 + }}({\rm{aq}})} \right]}}{{\left[ {{\rm{C}}{{\rm{u}}^{2 + }}({\rm{aq}})} \right]}}\)

\({{\rm{E}}^0} = \left( {\frac{{2.303{\rm{RT}}}}{{{\rm{nF}}}}} \right){\log _{10}}{{\rm{K}}_{\rm{c}}} = 1.1{\rm{V}}\)

\({{\rm{K}}_{\rm{c}}} = 2 \times {10^{37}}\)

Nernst Equation Applications

The Nernst equation can be used to calculate:

1. Emf of an electrochemical cell
2. Standard electrode potentials
3. Single electrode reduction or oxidation potential at any conditions
4. The pH of solutions and solubility of sparingly soluble salts.
5. Unknown ionic concentrations

Summary

The Nernst equation is an important relation to determine the cell potential of an electrochemical reaction. It gives the relationship between the cell potential and the reaction quotient of a reaction. This equation can be applied both to the potentials of individual electrodes and the potential differences across a pair of half-cells. However, it is generally more convenient to apply the Nernst equation to one electrode at a time. In this article, we learnt the expression, derivation of the Nernst equation and its relationship with the equilibrium constant. We also learnt the Nernst equation for Daniel cell and its applications.

Frequently Asked Questions (FAQs) on Nernst Equation

Q.1. Why is the Nernst equation used?
Ans: The Nernst equation can be used to determine the cell potential of an electrochemical cell. It gives a formula that relates the cell potential with equilibrium constants at standard conditions.

Q.2. What is the Nernst equation for Daniel cell?
Ans: The Nernst equation for Daniel cell is as follows-
\({{\rm{E}}_{{\rm{(cell)}}}} = {{\rm{E}}^{\rm{0}}}_{{\rm{(cell)}}} – \left( {\frac{{0.0592}}{{\rm{n}}}} \right){\log _{10}}\frac{{\left[ {{\rm{Z}}{{\rm{n}}^{\rm{2}}}{\rm{ +  (aq)}}} \right]}}{{\left[ {{\rm{C}}{{\rm{u}}^{\rm{2}}}{\rm{ +  (aq)}}} \right]}}\)

Q.3. What is the equilibrium constant from the Nernst equation?
Ans: The equilibrium constant from the Nernst equation is as follows-
\({{\rm{E}}^0} = \left( {\frac{{2.303{\rm{RT}}}}{{{\rm{nF}}}}} \right){\log _{10}}{{\rm{K}}_{\rm{c}}}\)
Where, \({{\rm{K}}_{\rm{c}}}\) is the reaction quotient,
\({{\rm{E}}^0}\) is standard cell potential
\({\rm{R}}\) is the universal gas constant, and
\({\rm{T}} = \) temperature in Kelvin.

Q.4. What is Daniel cell?
Ans: The Daniel cell is an electrochemical cell that consists of two dissimilar metal electrodes, \({\rm{Zn}}\) and \({\rm{Cu}}\). Each electrode is in contact with a solution of its own ion; Zinc sulphate and copper sulphate, respectively.

Q.5. What is the Nernst equation for EMF for a cell?
Ans: Nernst Equation for EMF of a cell relates the cell potential to the reaction quotient and allows the accurate determination of equilibrium constants.

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