Problem 198 - A Complete Resource Book in Chemistry for JEE Main

Question

The equilibrium constant for the reaction \(\mathrm{SO}_{3}(\mathrm{~g}) \rightleftharpoons \mathrm{SO}_{2}(\mathrm{~g})+1 / 2 \mathrm{O}_{2}(\mathrm{~g})\) is \(\mathrm{K}_{c}=4.9 \times 10^{-2} .\) the value of \(\mathrm{K}_{\mathrm{c}}\) for the reaction \(2 \mathrm{SO}_{2}\) \((\mathrm{g})+\mathrm{O}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{SO}_{3}(\mathrm{~g})\) will be (a) 416 (b) \(2.40 \times 10^{-3}\) (c) \(9.8 \times 10^{-2}\) (d) \(4.9 \times 10^{-2}\)

Step-by-Step Solution

Answer

The value of \( K_c \) for the reaction is (a) 416.

Step 1: Understand the Given Reaction and Equilibrium Constant

We are given the equilibrium constant \( K_c = 4.9 \times 10^{-2} \) for the decomposition of \( \text{SO}_3(g) \) into \( \text{SO}_2(g) \) and \( \frac{1}{2} \text{O}_2(g) \): \( \text{SO}_{3}(g) \rightleftharpoons \text{SO}_{2}(g) + \frac{1}{2} \text{O}_{2}(g) \). This constant relates to the balance between reactants and products in the reaction.

Step 2: Identify the Desired Reaction and Its Relationship

We need to find the equilibrium constant of the reverse reaction, which is \( 2 \text{SO}_2(g) + \text{O}_2(g) \rightleftharpoons 2 \text{SO}_3(g) \). Notice that this reaction is twice the magnitude of the reverse initial reaction and involves reversing the products and reactants.

Step 3: Calculate the Equilibrium Constant for the Reverse Reaction

In the reverse reaction, the initial reaction equation is reversed and doubled, thus the equilibrium constant for the reverse can be obtained by taking the reciprocal and squaring the initial \( K_c \) value: \[ K'_{c} = \left( \frac{1}{K_c} \right)^2 = \left( \frac{1}{4.9 \times 10^{-2}} \right)^2 \]

Step 4: Simplify the Expression

Calculate the reciprocal of \( 4.9 \times 10^{-2} \) first: \[ \frac{1}{4.9 \times 10^{-2}} = \frac{1}{0.049} = 20.408 \approx 20.41 \] Then, square the result to obtain \( K'_{c} \):\[ K'_{c} = (20.41)^2 = 416.8081 \approx 416 \]

Step 5: Choose the Correct Option

The calculated equilibrium constant for the reaction \( 2 \text{SO}_2(g) + \text{O}_2(g) \rightleftharpoons 2 \text{SO}_3(g) \) is \( 416 \). Comparing this with the given options, the correct choice is (a) 416.

Key Concepts

Chemical Equilibrium

Chemical equilibrium is reached in a reaction when the rate of the forward reaction equals the rate of the backward reaction.
This means that the concentrations of the reactants and products remain constant over time, although the reactions continue to occur.

• Equilibrium does not imply that the reactants and products are equal in concentration, but rather that their concentrations no longer change.
• At equilibrium, the ratio of the concentrations of products and reactants, raised to the power of their stoichiometric coefficients, is constant.

This ratio is known as the equilibrium constant, often denoted as \( K_c \) for reactions in the gaseous phase.
A larger \( K_c \) indicates that products predominate at equilibrium, whereas a smaller \( K_c \) suggests that reactants are more favored.

Reversible Reactions

Reversible reactions are those where the reactants can form products, which can in turn react to form the original reactants.
In other words, the reaction can proceed in both a forward and a backward direction.

• The symbol \( \rightleftharpoons \) is used to denote reversible reactions, showing that change can happen in both directions.
• Reversible reactions usually do not proceed to completion but reach a point of dynamic equilibrium.

In a dynamic equilibrium, reactions still occur but with no net change in the amounts of reactants and products.
Understanding reversible reactions is crucial when determining equilibrium constants because altering the direction or magnitude impacts the \( K_c \) value.

Thermodynamics in Chemistry

Thermodynamics in chemistry helps us understand how energy changes affect chemical reactions.
It examines the principles governing the energy transfers that accompany chemical reactions and physical changes.

• Key concepts include enthalpy (heat content), entropy (degree of disorder), and Gibbs free energy.
• At constant temperature and pressure, reactions tend to move towards lower Gibbs free energy. This is a measure of a reaction's spontaneity.
• Equilibrium constants \( K_c \) tie directly to Gibbs free energy. A reaction with a large \( K_c \) value is more likely to be spontaneous.

By understanding these thermodynamic principles, scientists can predict the direction and extent of chemical reactions, helping in designing industrial processes and developing new materials.