Nernst Equation: Cell Potential Under Non-Standard Conditions

Concept Overview

This question tests the understanding and application of the Nernst equation, a fundamental tool in electrochemistry. The Nernst equation relates the cell potential of an electrochemical cell to its standard cell potential and the concentrations (or partial pressures) of the reactants and products. It is crucial for predicting how changes in these conditions will affect the cell's electromotive force (EMF).

Step 1: Understand the Nernst Equation The Nernst equation allows us to calculate the cell potential ($E$) under non-standard conditions (i.e., when concentrations are not 1 M or partial pressures are not 1 atm). The general form of the Nernst equation at temperature $T$ is:

Here, $E^\circ$ is the standard cell potential, $R$ is the ideal gas constant ($8.314 \, \text{J/K} \cdot \text{mol}$), $T$ is the temperature in Kelvin, $n$ is the number of moles of electrons transferred in the balanced redox reaction, $F$ is Faraday's constant ($96485 \, \text{C/mol}$), and $Q$ is the reaction quotient.

Step 2: Define the Reaction Quotient ($Q$) The reaction quotient, $Q$, has the same form as the equilibrium constant ($K$), but it uses the instantaneous concentrations or partial pressures of reactants and products. For a general reaction:

The reaction quotient is expressed as:

Note that pure solids and liquids are omitted from the expression for $Q$.

Step 3: Simplify the Nernst Equation at $25^\circ\text{C}$ ($298.15 \, \text{K}$) At the common temperature of $25^\circ\text{C}$ (298.15 K), the term $\frac{RT}{F}$ can be combined with the natural logarithm to use the base-10 logarithm, simplifying calculations.

Using $\ln Q = 2.303 \log_{10} Q$, the Nernst equation becomes:

At $25^\circ\text{C}$:

This simplified form is frequently used in JEE Main problems.

Step 4: Apply the Nernst Equation to a Specific Example Consider the Daniell cell reaction: $Zn(s) + Cu^{2+}(aq) \rightleftharpoons Zn^{2+}(aq) + Cu(s)$. The standard cell potential, $E^\circ$, is typically given or can be calculated from standard reduction potentials. The number of electrons transferred, $n$, is 2. The reaction quotient, $Q$, for this reaction is:

(Note: $Zn(s)$ and $Cu(s)$ are omitted as they are pure solids). If the concentrations are, for example, $[Zn^{2+}] = 0.1 \, \text{M}$ and $[Cu^{2+}] = 0.01 \, \text{M}$, and $E^\circ = 1.10 \, \text{V}$ at $25^\circ\text{C}$, we can calculate $E$:

This shows that the cell potential decreases when the concentration of the product ion ($Zn^{2+}$) increases relative to the reactant ion ($Cu^{2+}$), as predicted by Le Chatelier's principle.

Step 5: Interpret the Results The Nernst equation highlights that cell potential is dependent on the concentrations of the species involved.

• If $Q < 1$ (reactant concentrations are higher than product concentrations), $\log_{10} Q$ is negative, and $E > E^\circ$. The cell will produce more voltage.
• If $Q > 1$ (product concentrations are higher than reactant concentrations), $\log_{10} Q$ is positive, and $E < E^\circ$. The cell will produce less voltage.
• If $Q = 1$ (standard conditions), $\log_{10} Q = 0$, and $E = E^\circ$.

Key Takeaways:

• The Nernst equation ($E = E^\circ - \frac{RT}{nF} \ln Q$) quantifies cell potential under non-standard conditions.
• The reaction quotient ($Q$) reflects the ratio of product to reactant concentrations (or partial pressures) at any given time.
• At $25^\circ\text{C}$, the equation simplifies to $E = E^\circ - \frac{0.0591}{n} \log_{10} Q$.
• Increasing product concentrations or decreasing reactant concentrations (making $Q$ larger) reduces the cell potential.

Answer: The Nernst equation is used to calculate cell potential ($E$) under non-standard conditions using the formula $E = E^\circ - \frac{RT}{nF} \ln Q$, where $E^\circ$ is the standard cell potential, $R$ is the gas constant, $T$ is temperature, $n$ is the number of electrons transferred, $F$ is Faraday's constant, and $Q$ is the reaction quotient.