The Nernst equation allows us to calculate the potential of an electrochemical cell when concentrations of the reactants and products are not at standard conditions.
Step 1: Nernst Equation.
The Nernst equation relates the cell potential to the concentration of the ions involved in the reaction. It is given by:
\[
E = E^0 - \frac{0.0592}{n} \log Q
\]
where:
- \( E \) is the cell potential under non-standard conditions,
- \( E^0 \) is the standard cell potential,
- \( n \) is the number of electrons transferred in the reaction,
- \( Q \) is the reaction quotient (the ratio of concentrations of products to reactants).
Step 2: Derivation of the Nernst Equation.
The Nernst equation can be derived from the Gibbs free energy equation. The change in Gibbs free energy is related to the cell potential as:
\[
\Delta G = -nFE
\]
where \( F \) is the Faraday constant, \( n \) is the number of moles of electrons, and \( E \) is the cell potential.
At standard conditions, \( \Delta G^0 = -nFE^0 \), where \( E^0 \) is the standard electrode potential.
The relationship between \( \Delta G \) and the reaction quotient \( Q \) is given by:
\[
\Delta G = \Delta G^0 + RT \ln Q
\]
Substituting the expressions for \( \Delta G \) and \( \Delta G^0 \) into the equation, we get:
\[
-nFE = -nFE^0 + RT \ln Q
\]
Rearranging the terms, we obtain the Nernst equation:
\[
E = E^0 - \frac{RT}{nF} \ln Q
\]
At room temperature (298 K), \( \frac{RT}{F} \approx 0.0592 \) V, and the Nernst equation becomes:
\[
E = E^0 - \frac{0.0592}{n} \log Q
\]
Step 3: Conclusion.
The Nernst equation is essential for calculating the electrode potential of a cell under non-standard conditions.
