Problem 9 - A Complete Resource Book in Chemistry for JEE Main

Question

The first-order rate constant for the decomposition of \(\mathrm{N}_{2} \mathrm{O}_{5}\) is \(6.2 \times 10^{-4} \mathrm{~s}^{-1}\). The half-life for this decomposition is (a) \(1177.7 \mathrm{~s}\) (b) \(1117.7\) (c) \(6.077 \mathrm{~s}\) (d) \(110.77\)

Step-by-Step Solution

Answer

The half-life for the decomposition of \(\mathrm{N}_{2}\mathrm{O}_{5}\) is 1117.7 s, which corresponds to option (b).

Step 1: Understand the Definition

The problem involves the decomposition of a chemical compound with a given rate constant. We are looking for the half-life, which is the time required for the concentration of the reactant to reduce to half of its initial value in a first-order reaction.

Step 2: Identify the Formula for Half-life of First-order Reaction

For a first-order reaction, the half-life formula is given by \[ t_{1/2} = \frac{0.693}{k} \]where \(k\) is the rate constant. This formula relates the rate constant to the half-life in first-order reactions.

Step 3: Substitute the Given Rate Constant

Substitute the given rate constant, \(k = 6.2 \times 10^{-4} \text{s}^{-1}\), into the half-life formula:\[ t_{1/2} = \frac{0.693}{6.2 \times 10^{-4}} \]

Step 4: Calculate the Half-life

Perform the calculation:\[ t_{1/2} = \frac{0.693}{6.2 \times 10^{-4}} \approx 1117.7 \text{s} \]

Key Concepts

First-order Reaction

In chemistry, a reaction is classified by the order, which gives us an insight into how the concentration of reactants affects the rate of the reaction. For a **first-order reaction**, the rate of reaction is directly proportional to the concentration of one reactant. This means that if the concentration of the reactant is doubled, the reaction rate also doubles.
For such reactions, the formula that describes the rate is **rate = k[A]**, where:

• **k** is the rate constant.
• **[A]** is the concentration of the reactant.

What makes first-order reactions unique is the fact that the half-life (the time taken for half the reactant to be consumed) is constant, regardless of the initial concentration of reactants. Thus, as you work through problems involving first-order reactions, remember the concentration decreases exponentially with time!

Rate Constant

The **rate constant**, denoted as **k**, is a crucial part of the rate equation in chemical reactions. It provides the relationship between the concentration of reactants and the speed of the reaction.
For first-order reactions, the units of the rate constant are **s**⁻¹, indicating that it represents the fraction of reactant converted per unit of time. The value of the rate constant can be influenced by several factors, such as temperature, pressure, and the presence of catalysts.
In the context of a first-order decomposition reaction, like that of \mathrm{N}_{2} \mathrm{O}_{5}, the rate constant allows you to calculate the half-life using the specific formula: \[ t_{1/2} = \frac{0.693}{k}\]Understanding the rate constant is essential for determining how quickly a reaction proceeds and predicting the behavior of the reaction over time.

Decomposition Reaction

A **decomposition reaction** involves a single compound breaking down into two or more simpler products. For example, in the decomposition of \( \mathrm{N}_{2} \mathrm{O}_{5} \), the compound breaks down into nitrogen dioxide and oxygen:
\[2\mathrm{N}_{2} \mathrm{O}_{5} \rightarrow 4\mathrm{NO}_{2} + \mathrm{O}_{2}\]Such reactions generally require energy to break the bonds in the original compound. Decomposition reactions can occur through various forms, including thermal decomposition, where heat is the source of energy.
Understanding decomposition reactions helps in predicting the products formed and calculating reaction kinetics, including the rate constant and the half-life for first-order decompositions. They play a significant role in industrial processes, laboratory synthesis, and even natural processes within the environment.