Problem 55 - A Complete Resource Book in Chemistry for JEE Main

Question

In a chemical equilibrium rate constant of forward reaction is \(7.5 \times 10^{-4}\) and the equilibrium constant is 1.5. The rate constant of backward reaction is (a) \(2.5 \times 10^{4}\) (b) \(5 \times 10^{-4}\) (c) \(2.5 \times 10^{-4}\) (d) \(5 \times 10^{4}\)

Step-by-Step Solution

Answer

The rate constant of the backward reaction is \( 5 \times 10^{-4} \).

Step 1: Understand the Relationship

To find the rate constant of the backward reaction, use the relationship between the equilibrium constant \( K \) and the rate constants. The equilibrium constant \( K \) is the ratio of the rate constant of the forward reaction \( k_f \) to the rate constant of the backward reaction \( k_b \). This can be expressed as: \( K = \frac{k_f}{k_b} \).

Step 2: Set Up the Equation

Plug in the known values into the equation derived in Step 1. We know the equilibrium constant \( K = 1.5 \) and the rate constant of the forward reaction \( k_f = 7.5 \times 10^{-4} \). Thus, set up the equation: \( 1.5 = \frac{7.5 \times 10^{-4}}{k_b} \).

Step 3: Solve for the Backward Reaction Rate Constant

Rearrange the equation from Step 2 to solve for \( k_b \). This gives us: \( k_b = \frac{7.5 \times 10^{-4}}{1.5} \). Calculate the division to find the value of \( k_b \).

Step 4: Perform the Calculation

Calculate \( k_b \) using the equation from Step 3: \( k_b = \frac{7.5 \times 10^{-4}}{1.5} = 5 \times 10^{-4} \). This is the rate constant of the backward reaction.

Step 5: Choose the Correct Answer

Compare the calculated rate constant \( k_b = 5 \times 10^{-4} \) to the given choices. The correct option is (b) \( 5 \times 10^{-4} \).

Key Concepts

Understanding Rate Constants

In the study of chemical reactions, rate constants are fundamental. They determine how quickly a reaction proceeds. Rate constants are part of the rate law equations, which describe how the rate of a chemical reaction depends on the concentration of its reactants. Every reaction has both a forward and a backward process, each characterized by its own rate constant, namely, the forward rate constant \( k_f \) and the backward rate constant \( k_b \). The forward rate constant is associated with the conversion of reactants to products, whereas the backward rate constant pertains to the reverse conversion from products back to reactants.Rate constants are specific to a given reaction and depend on factors such as temperature and the presence of a catalyst. Therefore, they are crucial for understanding how different conditions affect the speed of a reaction.

Chemical Equilibrium Fundamentals

Chemical equilibrium refers to the state in a reversible reaction where the rate of the forward reaction equals the rate of the backward reaction. At this point, the concentrations of the reactants and products remain constant over time. Although reactions continue to occur, there is no net change in the concentrations.One important measure at equilibrium is the equilibrium constant, expressed as \( K \). This constant provides insight into the ratios of concentrations of products and reactants at equilibrium. Mathematically, for a reaction \( aA + bB \rightleftharpoons cC + dD \), the equilibrium constant \( K \) can be written as:\[K = \frac{[C]^c[D]^d}{[A]^a[B]^b}\]The equilibrium constant is a crucial figure in chemistry as it reflects the position of equilibrium. A high \( K \) value means that, at equilibrium, products predominate, whereas a low \( K \) value signifies that reactants are more prevalent.

Exploring Reaction Kinetics

Reaction kinetics is the study of the rates at which chemical processes occur and the factors that influence these rates. It involves examining how different variables such as concentration, temperature, and catalysts affect the speed of a chemical reaction.When studying kinetic reactions, it's essential to understand the role of the rate law and how it incorporates rate constants to describe the relationship between reaction rate and reactant concentrations. The rate law is usually expressed in the form:\[ ext{Rate} = k[A]^m[B]^n\]Where \( k \) is the rate constant, and \( m \) and \( n \) are the reaction orders with respect to reactants \( A \) and \( B \) respectively. Understanding reaction kinetics allows chemists to predict how a reaction's rate will change under different conditions, enabling control and optimization of chemical processes.