Problem 61 - A Complete Resource Book in Chemistry for JEE Main

Question

The Nernst equation, \(E=E^{\circ}-(\mathrm{RT} / \mathrm{n} F)\) In \(\mathrm{Q}\) indicates that the equilibrium constant \(\mathrm{K}_{c}\) will be equal to Q when (a) \(E=\) zero (b) \(E^{\circ}=1\) (c) \(E=E^{\circ}\) (d) \(\mathrm{RT} / \mathrm{n} F=1\)

Step-by-Step Solution

Answer

The equilibrium constant \(K_c\) equals \(Q\) when (a) \(E=0\).

Step 1: Understanding the Nernst Equation

The Nernst Equation is given by \(E = E^{\circ} - (\text{RT} / nF) \ln Q\). This relates the electromotive force (EMF), \(E\), of a cell to the standard EMF, \(E^{\circ}\), the gas constant, R, temperature, T, the number of moles of electrons transferred, n, the Faraday's constant, F, and the reaction quotient, Q.

Step 2: Condition for Equilibrium

At equilibrium, the reaction quotient \(Q\) is equal to the equilibrium constant \(K_c\). The Nernst equation simplifies here because at equilibrium, the cell potential \(E\) is zero. Thus, \(0 = E^{\circ} - (\text{RT} / nF) \ln K_c\). This implies \(\ln K_c = \frac{nFE^{\circ}}{RT}\).

Step 3: Checking all given conditions with equilibrium condition

Given the Nernst equation at equilibrium \(E = 0\), let's test each option: - (a) \(E=0\) fits our equilibrium condition directly as it shows \(Q = K_c\).- (b) \(E^{\circ}=1\) is not directly related to equilibrium as equilibrium is defined by \(E = 0\).- (c) \(E=E^{\circ}\) results in \(Q = 1\), not necessarily \(K_c\).- (d) \(\text{RT} / nF = 1\) requires \(\ln Q = E^{\circ}\), not guaranteeing \(Q = K_c\).

Step 4: Conclusion

The condition \(E = 0\) ensures \(Q = K_c\). Thus, answer (a) is correct.

Key Concepts

Equilibrium Constant

The equilibrium constant, often denoted as \( K_c \), plays a crucial role in understanding chemical equilibria. It provides a measure of the concentrations of products and reactants at equilibrium. This value is pivotal because it offers a snapshot of the balance point of a reversible chemical reaction under constant temperature and pressure.

• When \( K_c \) is much greater than 1, it suggests that, at equilibrium, products predominate.
• When \( K_c \) is much less than 1, it indicates reactants are favored at equilibrium.
• When \( K_c \) equals 1, it shows neither reactants nor products are heavily favored, indicating a balanced system.

In the context of the Nernst Equation, the equilibrium constant becomes equivalent to the reaction quotient \( Q \) when the system reaches equilibrium and the cell potential \( E \) becomes zero. This means the chemical reaction has reached a state where the forward and reverse reactions occur at the same rate.

Electromotive Force

Electromotive Force (EMF), represented usually by the symbol \( E \), refers to the potential difference between the electrodes in a galvanic cell. Put simply, it represents the "push" or "pull" that drives electrons through a circuit.
At standard conditions, we use the standard EMF, denoted as \( E^{\circ} \). This is calculated when all species involved are at a concentration of 1 M, at 1 atm pressure and at a given standard temperature, often 298 K.

• A positive \( E^{\circ} \) suggests a spontaneous reaction under these conditions.
• A negative \( E^{\circ} \) suggests a non-spontaneous reaction.

In the context of equation solving, the Nernst Equation reveals how EMF varies with different concentrations of products and reactants. Specifically, if \( E = 0 \), the reaction reaches equilibrium; hence, \( Q \) equals \( K_c \), and the reaction essentially "pauses" with no net electron flow.

Reaction Quotient

The reaction quotient, \( Q \), is similar in concept to the equilibrium constant \( K_c \), but captures the ratio of product and reactant concentrations at any point in time, not just at equilibrium.

• If \( Q < K_c \), the reaction will shift towards products to reach equilibrium.
• If \( Q > K_c \), the reaction will shift towards reactants.
• If \( Q = K_c \), the system is at equilibrium.

Using the Nernst Equation, the value of \( Q \) is vital because it helps dictate how the EMF of a cell alters. If \( Q \) changes, \( E \) changes, displaying how far the system is from equilibrium. When \( E \) is zero, \( Q \) becomes \( K_c \), signifying that the reaction has reached a state of balance where forward and reverse reactions proceed at equal rates, and the system is at rest.