# Nernst Equation

## What Is Nernst Equation?

An equation used to calculate the equilibrium potential of an ion is called the Nernst equation. The equilibrium potential of an ion is also known as the Nernst potential for that ion. It was first formulated in the year 1889 by Walther Nernst a German physical chemist. This equation is useful to determine the cell potential as well as equilibrium constant.

## Nernst Equation Derivation

The Nernst equation is derived from the changes of Gibbs free energy under standard conditions.

$E^{o}= E_{reduction}^{o} – E_{oxidation}^{o}$

Under general conditions –

$\Delta G = -nFE$

Where,

• The number of electrons transferred from a balanced reaction is “n”
• Faraday constant ‘F’ = 96,500 C/mol
• The potential difference is ‘E’.

For an electrochemical half-cell the Nernst equation is expressed as:

$E_{red} = E_{red}^{\Theta }-\frac{RT}{zF}lnQ = E_{red}^{\Theta }-\frac{RT}{zF}lnQ\frac{a_{red}}{a_{ox}}$

For a full cell or complete electrochemical reaction the Nernst equation is expressed as:

$E_{cell} = E_{cell}^{\Theta }-\frac{RT}{zF}lnQ_{r}$ (total cell potential)

where

• $E_{red}$ – half-cell reduction potential at the temperature of interest
• $E_{red}^{\Theta }$ – standard half-cell reduction potential
• $E_{cell}$ – cell potential
• $E_{cell}^{\Theta }$ – standard cell potential
• R = 8.314472 J K−1 mol−1
• T – temperature
• $a_{ox}$ is the activity of the oxidized form
• F = 9.64853399×104 C mol−1
• Qr – reaction quotient.

The complete derivation of the Nernst Equation is given here.

## Equilibrium Constant from Nernst Equation

The equation given below is an expression which gives a relation between the standard electrode potential of the cell and the equilibrium constant.

$0= E^{o} – \frac{2.303RT}{nF} log_{10}K_{eq}$

$E^{o} =\frac{2.303RT}{nF} log_{10}K_{eq}$

Check out the detailed derivation of Nernst Equation EquilibriumConstant.

## Applications Of Nernst Equation

In electrochemistry, the Nernst equation has various applications. A few uses are discussed below-

• Nernst equation is used to determine the reduction potential of a half-cell of an electrochemical cell.
• Full electrochemical cell – Nernst equation is used to determine the total voltage of the full electrochemical cell.
• Nernst equation is used to determine the electromotive force of a full electrochemical cell.
• Nernst Equation is also used in pH measurements.
• Nernst equation is used to find the cell potential at any condition.
• It is used in the in determining ion concentration.
• It is used in oxygen and the aquatic environment.
• It is also used in potentiometric titrations.
• Nernst equation is used to find the cell potential at any point of reaction.

Some important equations:

• $E_{cell}= E_{cell}^{o}- \frac{RT}{nF}lnQ$ (Nernst equation)
• $E_{cell}^{o}= \frac{0.0257V}{n}lnK = \frac{0.0592V}{n}log K$ at 298.15 K
• $E_{cell}=E_{cell}^{o}- \frac{0.0257V}{n}lnQ = \frac{0.0592V}{n}log Q$ at 298.15 K
• $E_{cell}^{o}=\frac{RT}{nF}lnK$
• $\Delta G = -nFE_{cell}$
• $W_{ele} = W_{max} = -nFE_{cell}$

## Nernst Equation Questions

1. Determine the equilibrium constant for the reaction given below:

$Cu(s)+2Ag^{+}(aq)\rightarrow Cu^{2+}(aq)+2Ag(s)$

$E_{(cell)}^{\Theta } = 0.46 V$

1. For the Zn-Cu redox reaction,

$E_{(cell)}^{\Theta } = +1.10 V$

Find the equilibrium constant for the given reversible reaction?

$Zn(s)+Cu^{2+}(aq)\rightarrow Zn^{2+}(aq)+Cu(s)$

1. For a galvanic cell find the cell potential for the below-given reduction half-reaction. Temperature = 25 degree C

$Cd^{2+}+2e^{-}\rightarrow Cd\,E^{o}= -0.403 \,V$

$Pb^{2+}+2e^{-}\rightarrow Pb\,E^{o}= -0.126 \,V$

Where $\left [ Cd^{2+} \right ] = 0.020\,M$ and

$\left [ Pb^{2+} \right ] = 0.200\,M$

Other important links:

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#### Practise This Question

Electra has mastered the basics of Nernst equation. She is now studying equilibrium constants relation with Nernst equation.

Electra was performing experiments with Nernst with a Daniel Cell. She noticed that when the reaction reaches equilibrium, the cell potential goes to 0.

She asked Nernst why this happened. His reply was