Problem 68 - A Complete Resource Book in Chemistry for JEE Main

Question

For the reaction, \(\mathrm{A}+\mathrm{B} \rightleftharpoons \mathrm{C}+\mathrm{D}\), the rate constants for the forward and backward reactions are found to be \(4.2 \times 10^{-2}\) and \(3.36 \times 10^{-3} \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~s}^{-1}\) respectively. What is the equilibrium constant for the reaction? (a) \(11.5\) (b) \(12.5\) (c) \(8.0\) (d) \(6.0\)

Step-by-Step Solution

Answer

The equilibrium constant for the reaction is 12.5, so the correct choice is (b).

Step 1: Identify Given Data

We are given the rate constant for the forward reaction, \( k_f = 4.2 \times 10^{-2} \), and the rate constant for the backward reaction, \( k_r = 3.36 \times 10^{-3} \, \mathrm{mol} \, \mathrm{L}^{-1} \, \mathrm{s}^{-1} \).

Step 2: Understand Equilibrium Constant Formula

The equilibrium constant \( K_{eq} \) for a reaction \( aA + bB \rightleftharpoons cC + dD \) can be calculated using the relation \( K_{eq} = \frac{k_f}{k_r} \), where \( k_f \) is the rate constant for the forward reaction and \( k_r \) is the rate constant for the reverse reaction.

Step 3: Calculate Equilibrium Constant

Substitute the values of the rate constants into the equilibrium constant formula: \[ K_{eq} = \frac{4.2 \times 10^{-2}}{3.36 \times 10^{-3}} \approx 12.5 \].

Step 4: Select the Correct Answer

From the calculation, \( K_{eq} \approx 12.5 \). Therefore, the correct answer from the given options is (b) 12.5.

Key Concepts

Equilibrium Constant

When discussing chemical reactions, the equilibrium constant, often represented as \( K_{eq} \), is a key concept to understand. It gives us insight into the proportion of reactants and products in a reaction that has reached equilibrium. For any general reaction \( aA + bB \rightleftharpoons cC + dD \), the equilibrium constant is calculated using the concentrations of the chemicals involved. But, there's another way to compute it, using the rates of the forward and backward reactions.
The fascinating part is that \( K_{eq} \) can be expressed in terms of rate constants (\( k_f \) and \( k_r \), where \( f \) stands for forward, and \( r \) for reverse). This relation, \( K_{eq} = \frac{k_f}{k_r} \), shows us the inherent connection between the rate at which the reaction goes forward and the rate at which it reverses. In the problem above, by substituting in the given rate constants, we were able to determine that \( K_{eq} = 12.5 \), meaning the mixture heavily favors the products at equilibrium.

• Helps predict the direction of reaction's favorability
• Calculated by evaluating the concentrations or via rate constants

Rate Constants

Rate constants are essential values that define how fast a reaction occurs. In any chemical reaction, the rate constant \( k \) varies based on whether you're talking about the forward or backward process. For our example, the forward rate constant \( k_f \) is given as \( 4.2 \times 10^{-2} \), and the backward rate constant \( k_r \) is \( 3.36 \times 10^{-3} \text{ mol L}^{-1} \text{ s}^{-1} \).
These constants don't only tell us about the speed of a reaction but also carry information about the influence of temperature, as they typically follow the Arrhenius equation. Although the constants themselves are a bit abstract, they are the driving force behind how we understand dynamic chemical systems. Knowing them allows us to compute the equilibrium constant and predict whether a reaction will predominantly consist of reactants or products.

• Indicate how quickly reactants turn to products or vice versa
• Influenced by factors like temperature

Forward and Backward Reactions

In every reversible chemical reaction, there are two opposing processes: the forward and the backward (or reverse) reactions. It is important to grasp that while at equilibrium, both reactions continue to happen, and neither one truly stops.
The forward reaction converts reactants into products, which, in our example, is represented by \( A + B \rightarrow C + D \). Conversely, the backward reaction turns the products back into reactants, represented as \( C + D \rightarrow A + B \). As these reactions proceed, their respective rate constants (\( k_f \) and \( k_r \)) tell us how quickly each happens. At equilibrium, the rates of these reactions are equal, which results in a constant ratio of concentrations of products to reactants, reflected by the equilibrium constant. This dynamic equilibrium is a fundamental idea, showing that while the concentrations remain constant at equilibrium, the reactions are still actively occurring.

• Forward reaction: reactants to products
• Backward reaction: products to reactants
• Both occur at equilibrium but at equal rates