Problem 59 - A Complete Resource Book in Chemistry for JEE Main

Question

\(75 \%\) of a first-order reaction was completed in 32 min. When was \(50 \%\) of the reaction completed? (a) \(24 \mathrm{~min}\) (b) \(16 \mathrm{~min}\) (c) \(8 \mathrm{~min}\) (d) \(64 \mathrm{~min}\)

Step-by-Step Solution

Answer

The reaction was 50% complete in 16 minutes (option b).

Step 1: Understanding First-Order Reactions

In a first-order reaction, the rate of reaction is directly proportional to the concentration of one reactant. The formula used to determine the time for a certain completion percentage is derived from the equation: \[t = \frac{1}{k} \ln \left(\frac{[A]_0}{[A]}\right)\]where \([A]_0\) is the initial concentration, \([A]\) is the concentration at time \( t \), and \(k\) is the rate constant. The percentage completion gives us the fraction of reactant remaining. For instance, if the reaction is 75% complete, \([A]\) would be 25% of \([A]_0\).

Step 2: Calculate Rate Constant, k, from 75% Completion

Given: 75% of the reaction is completed in 32 minutes, meaning only 25% of \([A]_0\) remains.Plug the given values into the first-order reaction equation:\[k = \frac{1}{t} \ln \left(\frac{[A]_0}{[A]_{t}}\right) = \frac{1}{32} \ln \left(\frac{100}{25}\right)\]Compute \( \ln \left(\frac{100}{25}\right) = \ln 4 \). Then solve for \( k \):\[k = \frac{1}{32} \times \ln 4 \approx \frac{1}{32} \times 1.3863 \approx 0.0433 \text{ min}^{-1}\]

Step 3: Calculate Time for 50% Completion

For the 50% completion, 50% of \([A]_0\) remains. Use the equation:\[t = \frac{1}{k} \ln \left(\frac{[A]_0}{[A]_{t}}\right)\]Substitute \([A]_0/[A]_{t} = 2\) and the computed \(k\):\[t = \frac{1}{0.0433} \ln 2 \]Compute \( \ln 2 \approx 0.693 \):\[t = \frac{0.693}{0.0433} \approx 16 \text{ min}\]

Step 4: Verify the Answer

The obtained time for 50% completion is 16 minutes. This corresponds to option (b) in the multiple-choice answers. Thus, the calculation aligns with one of the provided options.

Key Concepts

Rate Constant

In a first-order reaction, understanding the concept of the rate constant, denoted by "k," is crucial. The rate constant is a proportionality factor that relates the rate of the reaction to the concentration of the reactants. In simpler terms, it tells us how quickly a reaction proceeds.

For first-order reactions, the rate constant has the units of time inverse, typically expressed as \( ext{min}^{-1} \) or \( ext{s}^{-1} \). This means it gives an idea of how many molecules react per unit of time.

When calculating the rate constant, one needs to understand that it remains constant for a given reaction at a specific temperature. Therefore, even if the concentration of the reactants changes, the rate constant does not. It solely depends on the conditions such as temperature and catalysts.

The equation for a first-order reaction is:

• \( t = \frac{1}{k} \ln \left(\frac{[A]_0}{[A]}\right) \)

Here \( t \) is the time required for the reaction to reach a particular stage, \( [A]_0 \) is the initial concentration, and \( [A] \) is the concentration at time \( t \). By manipulating this equation, we can determine the rate constant or the time required for certain completion percentages.

Reaction Completion

Reaction completion is a term used to express how much of a reaction has occurred over a given time. Whether a reaction is 50%, 75%, or some other percentage complete, this metric allows chemists to quantify the process precisely.

For first-order reactions, the percentage completion can be translated into the fraction of the original concentration that remains. For example:

• If 75% of the reaction is complete, then 25% of the initial reactant concentration remains.
• If 50% is complete, then half of the initial concentration remains.

This information is pivotal when using the first-order reaction formula to compute how long it takes to reach a certain completion stage.

Knowing the percentage of completion helps predict and control chemical reactions in practical applications, such as in industrial processes, or even in laboratory experiments to ensure reactions proceed as desired.

Half-Life

The half-life of a reaction is a special concept that refers to the time it takes for half of the reactant to be consumed. In first-order reactions, the half-life is particularly convenient because it remains constant throughout the reaction.

This means regardless of how much reactant you have at the beginning, the time it will take for half of it to react is always the same as long as the conditions don't change. For first-order reactions, the half-life is given by:

• \( t_{1/2} = \frac{0.693}{k} \)

This formula shows that the half-life depends on the rate constant. A larger value of \( k \) means a shorter half-life, indicating a faster reaction.

The concept of half-life is useful not only in laboratory settings but also in real-world applications, such as pharmaceuticals and radioactive decay, where knowing how long it takes for half a substance to diminish is crucial.

Chemical Kinetics

Chemical kinetics is a branch of physical chemistry that deals with understanding the speed or rate at which chemical reactions occur. It provides insights into the reaction mechanisms and is essential for controlling reactions.

By studying chemical kinetics, scientists can determine factors that affect reaction rates, such as:

• Concentration of reactants
• Temperature
• Presence of catalysts
• Surface area of reactants

This field helps us understand the dynamic nature of reactions and how they can be manipulated to favor certain products, speed them up for industrial efficiency, or control them for safety reasons.

Understanding chemical kinetics is also key in creating simulations and models that predict how reactions proceed over time, thus allowing chemists and engineers to optimize processes for desired outcomes efficiently.