Problem 121 - A Complete Resource Book in Chemistry for JEE Main

Question

The reaction \(\mathrm{X} \longrightarrow\) Product follows first-order kinetics, in 40 minutes, the concentration of \(X\) changes from \(0.1 \mathrm{M}\) to \(0.025 \mathrm{M}\), then the rate of reaction when concentration of \(\mathrm{X}\) is \(0.01 \mathrm{M}\) is? (a) \(3.47 \times 10^{-5} \mathrm{M} / \mathrm{min}\) (b) \(1.73 \times 10^{-4} \mathrm{M} / \mathrm{min}\) (c) \(1.73 \times 10^{-5} \mathrm{M} / \mathrm{min}\) (d) \(3.47 \times 10^{-4} \mathrm{M} / \mathrm{min}\)

Step-by-Step Solution

Answer

(d) \(3.47 \times 10^{-4} \, \text{M} / \text{min}\)

Step 1: Determine the rate constant

For a first-order reaction, the rate constant \( k \) can be determined using the formula for first-order kinetics: \[ k = \frac{1}{t} \ln\left(\frac{[X]_0}{[X]}\right)\]where \([X]_0 = 0.1 \, \text{M}\) is the initial concentration, \([X] = 0.025 \, \text{M}\) is the concentration after 40 minutes, and \(t = 40\) minutes. Substituting these values, we get: \[ k = \frac{1}{40} \ln\left(\frac{0.1}{0.025}\right)\]Calculate the natural logarithm and the rate constant next.

Step 2: Calculate the natural logarithm

Calculate the value of the natural logarithm: \[ \ln\left(\frac{0.1}{0.025}\right) = \ln(4)\]We know that \( \ln(4) \approx 1.386 \). Substituting this value, we have:\[ k = \frac{1.386}{40} \]Calculate the division to find \( k \).

Step 3: Calculate the rate constant \(k\)

Using the results from Step 2, compute:\[ k \approx \frac{1.386}{40} = 0.03465 \, \text{min}^{-1}\]This is the rate constant for the reaction.

Step 4: Determine the rate of reaction

For a first-order reaction, the rate is given by:\[ \text{Rate} = k [X]\]where \([X] = 0.01 \, \text{M}\). Substituting the values:\[ \text{Rate} = 0.03465 \times 0.01 = 0.0003465 \, \text{M} / \text{min}\]Convert this to scientific notation for clarity.

Step 5: Convert the rate to scientific notation

The calculated rate in standard notation is \(0.0003465 \, \text{M} / \text{min}\). In scientific notation, this becomes:\[ \text{Rate} = 3.465 \times 10^{-4} \, \text{M} / \text{min}\]Round this to \(3.47 \times 10^{-4} \, \text{M} / \text{min}\) given the significant figures.

Key Concepts

Rate Constant Calculation

In the study of first-order kinetics, determining the rate constant, often denoted as \( k \), is a crucial step. The rate constant is pivotal as it is unique to a reaction at a given temperature and can be used to predict future behavior of the reactants. For a first-order reaction, the rate constant can be calculated using the formula:\[ k = \frac{1}{t} \ln\left(\frac{[X]_0}{[X]}\right) \]In this formula:

• \( [X]_0 \) is the initial concentration of the reactant.
• \( [X] \) is the concentration of the reactant after a certain time \( t \).
• \( t \) is the time duration over which the reaction occurs.

To compute \( k \), you simply input the known concentrations and time. For example, if an initial concentration \( [X]_0 = 0.1 \, \text{M} \) changes to \( [X] = 0.025 \, \text{M} \) over 40 minutes, substituting these into the equation will yield the rate constant \( k \). By calculating, you may find \( k \approx 0.03465 \, \text{min}^{-1} \), meaning the reaction proceeds at this rate constant each minute.

Natural Logarithm in Kinetics

The natural logarithm comes into play when evaluating reaction rates, specifically in the context of first-order kinetics. It is utilized to relate the concentrations of reactants over time in a convenient mathematical form.In the rate constant formula, the portion \( \ln\left(\frac{[X]_0}{[X]}\right) \) represents the natural logarithm of the ratio of initial to final concentrations. Suppose the initial and final concentrations give the fraction 4, this corresponds to \( \ln(4) \). Knowing that \( \ln(4) \approx 1.386 \), you can see how logarithms simplify handling multiplicative changes in concentrations.The natural logarithm assists in linearizing the exponential decay pattern seen in first-order reactions, making it easier to plot and analyze data. It transforms complicated exponential equations into straight lines, which are much simpler to graph and manipulate in equation form.

Reaction Rate Determination

Understanding the reaction rate for first-order kinetics ties together several concepts. It's about assessing how swiftly a reaction progresses over time for a specific concentration.For a first-order reaction, the rate at any moment can be determined using the equation:\[ \text{Rate} = k [X] \]In this equation:

• \( k \) is the rate constant, which was calculated from prior steps.
• \( [X] \) is the concentration of the reactant at the time of interest.

For example, if the concentration \( [X] = 0.01 \, \text{M} \) and \( k \approx 0.03465 \, \text{min}^{-1} \), then the rate of reaction is \( \text{Rate} = k [X] = 0.03465 \times 0.01 = 0.0003465 \, \text{M} / \text{min} \). Converting this into scientific notation gives \( 3.47 \times 10^{-4} \, \text{M} / \text{min} \).This calculation allows you to see how quickly the concentration of the reactant reduces at a given point, providing insights into the dynamics of the reaction and helping predict future states.