Derive the Nernst equation.

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Solution

Suppose the reaction occurring in a reversible cell is represented by the equation, $A + B ⇌ C + D$ .
The decrease in free energy, $Δ G$ accompanying the process by the well, known thermodynamic equation.

$Δ G = - Δ G^{o} - R T l n J$

Where $- Δ G^{o}$ is the decreases is free energy accompanying the same process when all the reactants & products are in their standard states of unit activity & J stands for the reaction quotient of the activities of the products and reactants at any given state of the reaction.

Substituting the value of J, we have

$- Δ G = - Δ G^{o} - R T l n \frac{a_{C} \times a_{D}}{a_{A} \times a_{B}}$

If E is the emf of the cell in volts and the cell reaction involves the passage of n Faradays (i.e.) nF Coulombs, the electrical work done by the cell is in nFE volt - Coulombs, (or) Joules. Hence free energy decrease of the sys, $- Δ G$ is given by the expression $- Δ G = n F E$

$n F E = - Δ G^{o} - R T l n \frac{a_{C} \times a_{D}}{a_{A} \times a_{B}} = n F E^{o} - R T l n \frac{a_{c} \times a_{D}}{a_{A} \times a_{B}}$

$E = E^{o} - \frac{R T}{n F} l n \frac{a_{C} \times a_{D}}{a_{A} \times a_{B}}$

where $E^{o}$ is the emf of the cell in which the activity, or an approximation when the concentration ion of each reactant and each product of the cell reaction is equal to unity. $E^{o}$ is known as the standard emf of the cell.

$E = E^{o} - \frac{R T}{n F} l n \frac{a_{C} \times a_{D}}{a_{A} \times a_{B}}$

is often referred to as the Nernst equation.
Replacing activities concentration is an approximation; the Nernst equation may be written as

$E = E^{o} - \frac{R T}{n F} l n \frac{[C] \times [D]}{[A] \times [B]}$

where the quantities in parentheses represent the concentration of the species involved. Replacing.
$\frac{[C] \times [D]}{[A] \times [B]}$ as equal to K, the equilibrium in the molar concentration Units.

$E = E^{o} - \frac{R T}{n F} l n K$

This equation is known as Nernst equation.

$E = E^{o} - 2.303 \frac{R T}{n F} l o g K$ and

where $E^{o}$ stands for electron potential.
R = gas constant
T = Kelvin temperature
n = number of electrons transformed in the half-reaction.
F = Faraday of electricity
K = equilibrium constant for the half cell reactions as in equilibrium law.

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