Problem 28 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The reaction, \(\boldsymbol{A} \rightarrow\) Product, follows first order kinetics. In 40 minutes the concentration of \(\boldsymbol{A}\) changes from \(0.1\) to \(0.025 \mathrm{M}\). The rate of reaction, when concentration of \(\boldsymbol{A}\) is \(0.01 \mathrm{M}\) is [2004S] (a) \(1.73 \times 10^{-4} \mathrm{M} \min ^{-1}\) (b) \(3.47 \times 10^{-5} \mathrm{M} \mathrm{min}^{-1}\) (c) \(3.47 \times 10^{-4} \mathrm{M} \min ^{-1}\) (d) \(1.73 \times 10^{-5} \mathrm{M} \mathrm{min}^{-1}\)

Step-by-Step Solution

Answer

The rate of reaction is approximately \(3.47 \times 10^{-4} \mathrm{M} \min^{-1}\), matching option (c).

Step 1: Identify First Order Kinetics Equation

For a first order reaction, the rate can be expressed using the rate law \( r = k[A] \), where \( k \) is the rate constant and \( [A] \) is the concentration of reactant \( A \).

Step 2: Use Integrated Rate Law for First Order

The integrated rate law for a first order reaction is \( \ln\left(\frac{[A]_0}{[A]}\right) = kt \), where \([A]_0\) is the initial concentration, \([A]\) is the concentration at time \( t \), and \( k \) is the rate constant.

Step 3: Calculate the Rate Constant (k)

Substitute \([A]_0 = 0.1 \text{ M}\), \([A] = 0.025 \text{ M}\), and \( t = 40 \text{ min} \) into the integrated rate law: \[ \ln\left(\frac{0.1}{0.025}\right) = k \times 40 \]Calculate: \[ \ln(4) = k \times 40 \]\[ k = \frac{\ln(4)}{40} \approx 0.0347 \text{ min}^{-1} \]

Step 4: Calculate Rate of Reaction at Given Concentration

Use the rate law equation \( r = k[A] \) and substitute \( k = 0.0347 \text{ min}^{-1} \) and \( [A] = 0.01 \text{ M} \): \[ r = 0.0347 \text{ min}^{-1} \times 0.01 \text{ M} \]\[ r \approx 3.47 \times 10^{-4} \text{ M min}^{-1} \]

Step 5: Compare with Given Options

The calculated rate \( 3.47 \times 10^{-4} \text{ M min}^{-1} \) matches option (c).

Key Concepts

Rate Law

In chemistry, the rate law is an equation that describes the rate of a chemical reaction as a function of the concentrations of its reactants. For a first order reaction, the rate law can be simplified to the expression \( r = k[A] \). Here, \( r \) represents the reaction rate, \( k \) is the rate constant, and \([A]\) is the concentration of reactant A. Understanding this equation helps you determine how the concentration of the reactant influences the speed of the reaction.

The beauty of the first order rate law is its simplicity. It indicates that the rate of reaction is directly proportional to the concentration of one reactant. This means, if you double the concentration of A, the reaction rate also doubles. This relationship only holds in the context of first order reactions, simplifying the process of predicting changes in reaction rates as concentrations vary.

Integrated Rate Law

The integrated rate law is a mathematical expression that links the concentration of a reactant to time. For first order reactions, this is given by the equation \( \ln\left(\frac{[A]_0}{[A]}\right) = kt \). Here, \([A]_0\) is the initial concentration, \([A]\) is the concentration at a specific time \( t \), and \( k \) is the rate constant.

This equation allows you to calculate the concentration of a reactant at any given time if you know the initial concentration and the rate constant. It's particularly useful for comparing concentrations at two different times or for calculating how long it takes for a reactant to reach a certain concentration. By rearranging and solving the integrated rate law, you gain powerful insights into reaction dynamics over time.

Rate Constant Calculation

Calculating the rate constant (\( k \)) is a crucial step in analyzing first order reactions. You can use the integrated rate law \( \ln\left(\frac{[A]_0}{[A]}\right) = kt \) for this purpose.

For example, if you know \([A]_0 = 0.1\, \text{M}\), \([A] = 0.025\, \text{M}\), and \( t = 40\, \text{min}\), you can find \( k \) by substituting these values into the equation. Solve for \( k \) by using the relationship \( \ln\left(4\right) = k \times 40 \), leading to \( k = \frac{\ln(4)}{40} \approx 0.0347 \text{ min}^{-1} \).

Having \( k \) not only aids in understanding the speed of the reaction but is also essential for predicting future concentrations and timescales for completion. Knowing the rate constant is key to mastering reaction kinetics efficiently.

Reaction Rate Determination

Determining the reaction rate at a specific concentration is straightforward once you know the rate constant (\( k \)). You use the rate law \( r = k[A] \) to find the rate \( r \) at any given concentration \([A]\).

For instance, if the rate constant is \( 0.0347 \text{ min}^{-1} \) and the concentration \([A]\) is \( 0.01 \text{ M} \), the rate \( r \) can be calculated as \( r = 0.0347 \text{ min}^{-1} \times 0.01 \text{ M} \equiv 3.47 \times 10^{-4} \text{ M min}^{-1} \).

This calculated reaction rate helps you understand how quickly the reaction progresses at that specific moment. It's vital to grasp these concepts, as they enable you to predict and control the rates of chemical processes effectively. Understanding how to determine reaction rates is fundamental in both academic settings and real-world applications, like industrial processes.