How to Derive and Remember the Nernst Equation | CHEMCA

How to Derive and Remember the Nernst Equation Easily

Published by Abhishek Sengar | CHEMCA India

One of the most common mistakes students make in JEE and NEET exams is mixing up the numerator and denominator in the log term of the Nernst equation. Did the products go on top, or was it the reactants?

If you learn how to derive the Nernst Equation from core Thermodynamics principles, you will never memorize it blindly again. Let's see how easy it is to connect the two chapters.

Video Tutorial: The 3 Core Equations

Watch Abhishek Sengar sir from CHEMCA break down the derivation using just three simple mathematical formulas.

Step-by-Step Derivation

The Key Concept: The Nernst Equation is simply a way to calculate Cell Potential (E_cell) using the Reaction Quotient (Q). By bridging Gibbs Free Energy (ΔG) with Electrical Work, we can find the exact relationship.

To begin, we need three fundamental equations:

Thermodynamics Equation: Relates Gibbs Free energy at any moment to Standard Gibbs Free Energy using the Reaction Quotient (Q).

ΔG = ΔG° + RT ln(Q) — (Eq 1)
Electrochemistry Equations: Relate Gibbs Free Energy to the electrical work done by the cell.

ΔG = −nFE_cell — (Eq 2)
ΔG° = −nFE°_cell — (Eq 3)
Substitution: Substitute Eq 2 and Eq 3 directly into Eq 1.

−nFE_cell = −nFE°_cell + RT ln(Q)
Simplify (Divide by −nF): Divide both sides by −nF to isolate E_cell.

E_cell = E°_cell − (RT / nF) · ln(Q)
Convert to Base 10 Logarithm: To make calculations easier, we convert the natural log (ln) to log base 10 by multiplying by 2.303.

E_cell = E°_cell − (2.303 RT / nF) · log₁₀(Q)

The Standard 298K Formula

For most competitive exam questions, the temperature is standardized at 298 K (25°C). When we plug in the constants (R = 8.314 \text{ J K}^{-1} \text{ mol}^{-1}, T = 298 \text{ K}, and F = 96487 \text{ C mol}^{-1}), the massive cluster of variables reduces to a single magic number: 0.0591.

E_cell = E°_cell − (0.0591 / n) · log₁₀(Q)

Writing Q Correctly: The Reaction Quotient (Q) is always [Products] / [Reactants] raised to their stoichiometric coefficients. Crucially, the concentration (activity) of pure solids and pure liquids is taken as 1.

Example: For the Daniell cell reaction: Cu²⁺(aq) + Zn(s) → Zn²⁺(aq) + Cu(s)
Q = [Zn²⁺] / [Cu²⁺] (Solids Zn and Cu are ignored).

Practice Questions for JEE & NEET

Solidify your understanding of the Reaction Quotient and Equilibrium with these tests.

Question 1: Write the proper Nernst Equation (at 298K) for the following cell reaction:
2Al(s) + 3Ni²⁺(aq) → 2Al³⁺(aq) + 3Ni(s)

Answer:

Step 1 (Find n): 2 Al atoms lose 3 electrons each (total 6). 3 Ni atoms gain 2 electrons each (total 6). Therefore, n = 6.
Step 2 (Find Q): Products over Reactants. Ignore solids. Ensure you raise them to their coefficients.
Q = [Al³⁺]² / [Ni²⁺]³
Final Equation:
E_cell = E°_cell − (0.0591 / 6) · log( [Al³⁺]² / [Ni²⁺]³ )

Question 2: What happens to the Nernst Equation when the Galvanic cell battery goes "dead" and the reaction reaches chemical equilibrium?

Answer: E_cell becomes 0, and Q becomes K_c.

Reasoning:

When equilibrium is achieved, the battery can no longer do any useful work. Hence, Gibbs Free Energy (ΔG) = 0. Since ΔG = -nFE_cell, E_cell also equals exactly 0. At this exact moment, the reaction quotient Q is perfectly balanced and is now called the Equilibrium Constant (K_c).

The equation transforms into a useful new formula to find the equilibrium constant:
0 = E°_cell − (0.0591 / n) · log(K_c)
E°_cell = (0.0591 / n) · log(K_c)

Search This Blog