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: Understand the Problem

This is a first-order reaction. We need to find the rate constant. We know that a first-order reaction follows the equation: \[ k = \frac{0.693}{t_{1/2}} \]where \( t_{1/2} \) is the half-life of the reaction.

Step 2: Identify Given Values

The exercise states that the reaction is \(50 \%\) completed in 30 minutes, which means the half-life \( t_{1/2} \) is 30 minutes.

Step 3: Apply the First Order Rate Equation

Plug the known value into the equation:\[ k = \frac{0.693}{30 \text{ minutes}} \]This gives us the calculation needed to find \( k \).

Step 4: Calculate the Rate Constant

Perform the division:\[ k = \frac{0.693}{30} \approx 0.0231 \text{ min}^{-1} \]This is equivalent to \(2.31 \times 10^{-2} \text{ min}^{-1}\).

Step 5: Confirm the Answer

Compare the calculated rate constant to the provided options. The answer matches option (a).

Key Concepts

Rate Constant Calculation

First-order reactions depend on the concentration of a single reactant. The rate constant, denoted as \(k\), is an important value in reaction kinetics. The rate constant helps scientists understand how quickly a reaction proceeds. It is calculated using the half-life of the reaction.

• For first-order reactions, the formula used is:\[ k = \frac{0.693}{t_{1/2}} \]where \(t_{1/2}\) is the half-life of the reaction.
• The constant 0.693 originates from the natural logarithm of 2 (\(\ln(2)\)), which is a mathematical constant used during the derivation of the rate law for first-order kinetics.

The problem tells us that the reaction is half-complete in 30 minutes. This means, \(t_{1/2} = 30\) minutes. By substituting this into the formula, we find:\[ k = \frac{0.693}{30} \approx 0.0231 \, \text{min}^{-1} \]This value is crucial for determining the overall speed of the reaction.

Half-Life Equation

The half-life equation is a critical concept when analyzing first-order reactions. The term half-life describes the time taken for half of a reactant to be consumed in a reaction. It's especially useful for understanding how quickly substances change over time in chemical processes.

• For a first-order reaction, the half-life \(t_{1/2}\) is constant throughout the reaction.
• This uniqueness allows us to easily calculate other parameters using the half-life equation. Unlike zero-order or second-order reactions, first-order reaction half-lives do not depend on the initial concentration of the reactants.

Having a constant half-life simplifies calculations greatly, making it easy to predict how long the reaction will take to reach a certain milestone.

Reaction Kinetics

Reaction kinetics is the study of the rates at which chemical reactions occur. It helps in understanding the steps and factors affecting those rates. In the case of first-order reactions, kinetics focus on how the concentration of a single reactant modifies over time.

• The rate of a first-order reaction is directly proportional to the concentration of one reactant.
• Understanding kinetics involves exploring various factors such as temperature, reactant concentration, and catalysts, which can all affect the reaction rate.

Kinetic studies provide valuable insights into optimizing conditions for industrial chemical processes and predicting reaction behavior in natural systems. Learning about kinetics enables us to control and utilize reactions more effectively in a variety of applications.