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 (option b).

Step 1: Understanding the Given Values

In this exercise, you're provided the rate constants for both the forward and backward reactions: \( k_f = 4.2 \times 10^{-2} \) and \( k_b = 3.36 \times 10^{-3} \text{ mol L}^{-1} \text{ s}^{-1} \). The task is to find the equilibrium constant \( K \).

Step 2: Determining the Relationship

The equilibrium constant \( K \) for the given reaction is defined by the ratio of the forward rate constant \( k_f \) to the backward rate constant \( k_b \). The formula is \( K = \frac{k_f}{k_b} \).

Step 3: Substituting and Solving

Substitute the given values into the formula: \[ K = \frac{4.2 \times 10^{-2}}{3.36 \times 10^{-3}}.\] Doing the division yields \( K = 12.5 \).

Step 4: Final Verification

Now, verify your result by checking it corresponds to one of the given answer choices. We calculated \( K \) to be \( 12.5 \), which matches option (b).

Key Concepts

Rate Constants

In the realm of chemical reactions, rate constants are crucial. They represent the speed at which a reaction progresses in either direction. These constants are usually denoted by the symbol "k."

For every chemical reaction, there are typically two rate constants: one for the forward reaction, and one for the backward (reverse) reaction. These values are typically determined experimentally and depend on factors such as temperature and the nature of the reactants involved.

• The forward rate constant, denoted by \( k_f \), describes the rate to form products from reactants.
• The backward rate constant, \( k_b \), represents the rate at which the products revert to reactants.

Understanding these constants allows chemists to predict how a reaction behaves under different conditions. Higher values mean faster reactions.

Forward and Backward Reactions

All chemical reactions can proceed in both forward and backward directions unless they are irreversible.

In our example reaction \( \mathrm{A} + \mathrm{B} \rightleftharpoons \mathrm{C} + \mathrm{D} \), there is a constant movement as the mixture seeks a balance.

• The forward reaction is when reactants \( \mathrm{A} \) and \( \mathrm{B} \) form products \( \mathrm{C} \) and \( \mathrm{D} \).
• The backward reaction is the conversion of \( \mathrm{C} \) and \( \mathrm{D} \) back to \( \mathrm{A} \) and \( \mathrm{B} \).

Both reactions occur simultaneously, but their rates may differ based on conditions and concentrations. These opposing processes are what eventually lead to chemical equilibrium.

Chemical Equilibrium

Chemical equilibrium is a state reached when the rate of the forward reaction equals the rate of the backward reaction. At this point, the concentrations of all reactants and products remain constant over time, even though the reactions might still be occurring.

In mathematical terms, the equilibrium constant \( K \) is the ratio of the forward to the backward rate constants, represented as \( K = \frac{k_f}{k_b} \). This value gives us insight into the relative concentrations of reactants and products at equilibrium.

• A large \( K \) value indicates that the reaction favors the formation of products.
• A small \( K \) value suggests that the reactants are more favored.

Understanding equilibrium helps in predicting how a system will react to changes in concentration, temperature, or pressure.

Reaction Kinetics

The field of reaction kinetics is concerned with understanding how different factors affect the rates of chemical reactions. It explores how reactions proceed from reactants to products and identifies the steps involved in the process.

Scientists study these kinetics to develop models that can predict the behavior of reactions in different scenarios. This study often includes:

• Identifying the mechanism or series of steps that lead to product formation.
• Understanding the influence of concentration, temperature, and catalysts on reaction rates.

Kinetics is crucial in numerous fields, such as chemical engineering and pharmacology, where controlling the rate of reaction is essential for efficiency and safety. By grasping these ideas, one can better design reactions to achieve desired outcomes effectively.