Problem 101 - A Complete Resource Book in Chemistry for JEE Main

Question

In the first-order reaction, half of the reaction is com pleted in 100 seconds. The time for \(99 \%\) reaction to occur will be (a) \(664.64 \mathrm{~s}\) (b) \(646.6 \mathrm{~s}\) (c) \(660.9 \mathrm{~s}\) (d) \(654.5 \mathrm{~s}\)

Step-by-Step Solution

Answer

The time for 99% reaction to occur is 664.64 seconds (option a).

Step 1: Understanding the Relationship

In a first-order reaction, there's a direct relationship between the reaction time and the percentage completed, governed by the formula for half-life: \[t_{1/2} = \frac{0.693}{k}\]where \(k\) is the rate constant of the reaction.

Step 2: Calculate Rate Constant

Given that the half-life \(t_{1/2}\) is 100 seconds, we can use the half-life formula to find the rate constant:\[k = \frac{0.693}{t_{1/2}} = \frac{0.693}{100 \, s} = 0.00693 \, s^{-1}\]

Step 3: Determine Total Reaction Time for 99% Completion

Using the first-order reaction formula:\[ t = \frac{2.303}{k} \log \frac{[A]_0}{[A]}\]For 99% completion, \([A] = [A]_0 \times 0.01\). Plugging in the values:\[t = \frac{2.303}{0.00693} \log \frac{1}{0.01} = \frac{2.303}{0.00693} \times 2 = 664.5 \text{ seconds}\]

Step 4: Match with Given Options

The calculated time is approximately 664.64 seconds. Comparing with the given options, it matches option (a) exactly.

Key Concepts

Reaction Kinetics

The study of reaction kinetics helps us understand the speed of chemical reactions and the mechanisms behind them. This understanding is crucial because it enables us to predict how long a reaction will take and how modifying conditions might alter its progress. Reaction kinetics focuses on reaction rates and the factors affecting them, such as temperature, concentration, and the presence of catalysts. In a first-order reaction, the rate is proportional to the concentration of one reactant. This means as time progresses, the concentration decreases, leading to a reduction in the rate. First-order reactions follow an exponential decay curve, which can be described mathematically using a simple equation:

• \[ ext{Rate} = k[A]\]

where \[k\] is the rate constant and \[[A]\] is the concentration of the reactant. Since only one reactant concentration is involved, first-order reactions are often predictable and straightforward to analyze.

Half-Life Calculation

The concept of half-life is vital when dealing with first-order reactions. Half-life refers to the time required for half of the reactant in a chemical reaction to be consumed. It provides a simple way to gauge how quickly a reaction proceeds. For first-order reactions, the half-life is constant and does not depend on the initial concentration of the reactant. This is expressed by the formula:

• \[ t_{1/2} = rac{0.693}{k} \]

where \[ t_{1/2} \] is the half-life and \[ k \] is the rate constant. In the given exercise, the half-life is 100 seconds, making it easy to calculate the rate constant and further understand the reaction kinetics involved. Once the half-life is known, predicting the time required for any percentage of completion becomes straightforward, especially when visualizing the progress of the reaction over time. This clarity in prediction helps in practical applications, such as calculating the time required for 99% completion in the exercise.

Rate Constant

The rate constant \( k \) is a fundamental parameter in reaction kinetics. It determines the speed of the reaction and varies with different reactions. Each reaction has a unique rate constant influenced by factors such as temperature and catalysts. In first-order reactions, the rate constant is derived from the half-life formula:

• \[ k = rac{0.693}{t_{1/2}} \]

where \( t_{1/2} \) is the half-life. Given the half-life of 100 seconds in the problem, calculating \( k \) becomes straightforward:

• \[ k = rac{0.693}{100 ext{ s}} = 0.00693 ext{ s}^{-1} \]

This rate constant is then used to determine the time required for a specific completion percentage using the first-order reaction formula:

• \[ t = rac{2.303}{k} imes ext{log} rac{[A]_0}{[A]} \]

For 99% completion, this results in about 664.64 seconds, as calculated in the solution. Understanding and calculating the rate constant are key to predicting the behavior of first-order reactions accurately.