Problem 83 - A Complete Resource Book in Chemistry for JEE Main

Question

A first-order reaction is \(50 \%\) completed in 30 minutes at \(27^{\circ} \mathrm{C}\). Its rate constant is (a) \(2.31 \times 10^{-2} \mathrm{~min}^{-1}\) (b) \(3.21 \times 10^{-2} \mathrm{~min}^{-1}\) (c) \(4.75 \times 10^{-2} \mathrm{~min}^{1}\) (d) \(1.33 \times 10^{-3} \mathrm{~min}^{-1}\)

Step-by-Step Solution

Answer

The rate constant is (a) \(2.31 \times 10^{-2} \text{ min}^{-1}\).

Step 1: Understanding the Problem

We have a first-order reaction that is 50% complete in 30 minutes. We need to calculate the rate constant using the first-order kinetics formula. This will help us determine which of the given options is the correct rate constant for this reaction.

Step 2: Applying First-Order Kinetics Formula

For a first-order reaction, the relationship between concentration and time is given by \[ k = \frac{0.693}{t_{1/2}} \] where \(k\) is the rate constant and \(t_{1/2}\) is the half-life of the reaction.

Step 3: Identifying Half-Life

Since the reaction is 50% complete in 30 minutes, this means that 30 minutes is the half-life (\(t_{1/2}\)) of the reaction. Thus, \(t_{1/2} = 30\text{ minutes}\).

Step 4: Calculating Rate Constant

Substitute the half-life into the first-order rate constant formula: \[ k = \frac{0.693}{30} \approx 0.0231\, \text{min}^{-1} \]

Step 5: Selecting the Correct Answer

The calculated rate constant is \(0.0231\, \text{min}^{-1}\), which corresponds to option (a): \(2.31 \times 10^{-2} \text{ min}^{-1}\).

Key Concepts

Rate Constant

The rate constant is a crucial aspect of understanding chemical reactions. It quantitatively describes the rate at which a chemical reaction proceeds. In a first-order reaction, the rate constant, denoted as \( k \), helps determine how quickly reactants are converted to products. The rate constant is specific to each reaction and can be affected by factors such as temperature. It has units of \( ext{time}^{-1} \), often expressed in minutes or seconds, reflecting the rate of the reaction per time unit.

• Higher \( k \) value typically indicates a faster reaction.
• Lower \( k \) value indicates a slower reaction.

To find the rate constant for a first-order reaction, we often rely on experimental data, such as the time taken for the reaction to reach a certain completion percentage. Knowing the rate constant allows chemists to predict how long a reaction will take to reach a given state, which is vital for both research and practical applications.

Half-Life

In the context of chemical kinetics, half-life is the time required for half of the reactants in a chemical reaction to be converted into products. In simple terms, it means the time by which 50% of the starting material is consumed. For first-order reactions, the half-life remains constant regardless of the starting concentration of reactants. This unique characteristic allows first-order reactions to have a constant rate of decay.

• Half-life is independent of the initial concentration for first-order reactions.
• It's a crucial measure to understanding how fast a reaction proceeds.

This property is mathematically expressed using the equation: \[ t_{1/2} = \frac{0.693}{k} \] Here, \( t_{1/2} \) is the half-life, and \( k \) is the rate constant. This equation signifies that the half-life is inversely proportional to the rate constant: as the rate constant increases, the half-life decreases, signifying a faster reaction.

First-Order Kinetics Formula

The first-order kinetics formula is a vital tool in the study of first-order reactions, which conform to a rate equation where the rate of reaction is directly proportional to the concentration of one reactant. The most common equation used for first-order reactions is: \[ k = \frac{0.693}{t_{1/2}} \] This tells us that the rate constant \( k \), can be easily derived if the half-life \( t_{1/2} \) is known. Another useful form of the first-order kinetics equation is: \[ ext{ln}(rac{[A]}{[A]_0}) = -kt \] Here:

• \([A]\) is the concentration of the reactant at time \( t \).
• \([A]_0\) is the initial concentration of the reactant.
• \( k \) is the rate constant.

This equation makes it possible to calculate how the concentration of reactants changes over time, further aiding in understanding the dynamics of the reaction. By substituting the measured concentrations and time intervals into this equation, one can determine the rate constant and predict future concentrations, making it a fundamental tool in kinetics studies.