Problem 134 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following statements are correct about half-life period? (1) time required for \(99.9 \%\) completion of a reaction is 100 times the half-life period (2) time required for \(75 \%\) completion of a first-order reaction is double the half-life of the reaction (3) average life \(=1.44\) times the half-life for firstorder reaction

Step-by-Step Solution

Answer

Statements 2 and 3 are correct.

Step 1: Understanding Half-Life in Reactions

The half-life (\(t_{1/2}\)) of a reaction is the time required for the concentration of a reactant to reduce to half its initial value. For first-order reactions, this half-life is independent of the initial concentration and is given by \(t_{1/2} = \frac{0.693}{k}\) where \(k\) is the rate constant.

Step 2: Analyzing Statement 1

Statement 1 claims that the time required for \(99.9 \%\) completion of a reaction is 100 times the half-life period. For a first-order reaction, the time required to reach a certain completion percentage is given by \(t = \frac{2.303}{k} \log\left(\frac{[A]_0}{[A]_{t}}\right)\). Solving for \(99.9 \%\) completion: \(\log1,000 \approx 3\), thus \(t \approx \frac{2.303}{k} \times 3\). The factor is roughly \(10\), not \(100\), so statement 1 is incorrect.

Step 3: Analyzing Statement 2

Statement 2 states that the time for \(75 \%\) completion is double the half-life for a first-order reaction. For \(75 \%\) completion, \([A]_t = \frac{1}{4}[A]_0\), thus \(t_{75\%} = \frac{2.303}{k} \log\left(\frac{[A]_0}{\frac{1}{4}[A]_0}\right)\ = \frac{2.303}{k} \times 2\log2\). This equals \(2 \times t_{1/2}\), so statement 2 is correct.

Step 4: Analyzing Statement 3

Statement 3 proposes that average life is \(1.44\) times the half-life for a first-order reaction. Average life (\(\tau\)) for a first-order reaction is given by \(\tau = \frac{1}{k}\) and \(t_{1/2} = \frac{0.693}{k}\) implying \(\tau = \frac{1.44}{0.693} t_{1/2}\), which holds true as \(\approx 1.44 t_{1/2}\). Therefore, statement 3 is correct.

Key Concepts

First-Order Reactions

In chemistry, understanding the kinetics of reactions helps us predict how the concentration of reactants changes over time. A first-order reaction is a reaction where the rate is directly proportional to the concentration of one reactant. This means, if you double the concentration of the reactant, the reaction rate also doubles. These reactions follow the rate equation:

• Rate = \(k[A]\)

where \(k\) is the reaction rate constant, and \([A]\) is the concentration of the reactant. First-order reaction kinetics are common in processes like radioactive decay, where only the substance itself dictates how quickly it transforms.
For these reactions, an important characteristic is that their half-life period is constant. This means that no matter how much of the reactant you start with, it always takes the same amount of time for that reactant's concentration to reach half of its original value.

Reaction Rate Constant

The reaction rate constant, often represented as \(k\), is a fundamental concept in chemical kinetics that quantifies the speed of a reaction. For a first-order reaction, the rate constant is crucial because it determines the half-life as well as the average life of the reaction. In mathematical terms:

• Half-life \(t_{1/2} = \frac{0.693}{k}\)
• Average life \(\tau = \frac{1}{k}\)

One interesting fact about the reaction rate constant is its dependence on temperature. As the temperature increases, \(k\) typically increases, which means the reaction proceeds faster. This is why reaction conditions often specify temperature along with concentration and pressure to control the rate at which a reaction progresses.

Average Life of Reactions

The average life of a reaction, especially in the context of first-order reactions, provides insight into the longevity of reactants before significant depletion occurs. Calculated as \(\tau = \frac{1}{k}\), the average life differs from half-life in that it offers a more comprehensive measure of how long reactants persist under given conditions.
Given the mathematical relationship between average life and half-life, for first-order reactions, it is approximately 1.44 times the half-life. This relationship shows that, while the half-life gives the time taken for the reactant's concentration to fall to half, the average life provides the average duration a molecule of reactant is expected to last before it reacts completely.
Therefore, understanding the average life helps in applications where long-term stability and reactant utilization are crucial, such as in pharmaceuticals and materials science.