Problem 151 - A Complete Resource Book in Chemistry for JEE Main

Question

For a first order reaction \(\mathrm{A}(\mathrm{g}) \rightarrow \mathrm{B}(\mathrm{g})+\mathrm{C}(\mathrm{g})\), the total pressure of \(\mathrm{A}+\mathrm{B}+\mathrm{C}\) at time ' \(\mathrm{t}\) ' and \(\infty\) are \(\mathrm{P}_{2}\) and \(\mathrm{P}_{3}\) respectively. The constant \(\mathrm{k}\) of the reaction is (a) \(\frac{1}{t} \ln \frac{P_{3}}{2\left(P_{3}-P_{2}\right)}\) (b) \(\frac{1}{t} \ln \frac{2 P_{3}}{P_{3}-P_{2}}\) (c) \(\frac{1}{t} \ln \frac{P_{3}}{P_{3}-P_{2}}\) (d) \(\frac{1}{t} \ln \frac{P_{3}}{2 P_{3}-P_{2}}\)

Step-by-Step Solution

Answer

The constant \( k \) of the reaction is (a) \( \frac{1}{t} \ln \frac{P_{3}}{2(P_{3}-P_{2})} \)."

Step 1: Understand the Reaction and Pressure Relation

For a first-order reaction \( \text{A}(\text{g}) \rightarrow \text{B}(\text{g}) + \text{C}(\text{g}) \), the pressure change is related to the reaction progress. Initially, only \( \text{A} \) is present. As the reaction progresses to completion, \( \text{B} \) and \( \text{C} \) are formed, changing the total pressure.

Step 2: Initial and Final State Pressures

At the start, the total pressure is only due to \( \text{A} \) with initial partial pressure \( P_0 \). At time \( t \), the total pressure is \( P_2 \), and finally, at \( \infty \), the total pressure is \( P_3 \). Here, \( P_3 \) represents the theoretical pressure when \( \text{A} \) is entirely converted.

Step 3: Relate Pressure with Reaction Progress

The change in partial pressure of \( \text{A} \) is related to the formation of \( \text{B} \) and \( \text{C} \). The total pressure at infinity (\( P_3 \)) includes contributions from \( \text{B} \) and \( \text{C} \), which indicates \( P_3 = P_0 + P_0 \) (since \( \text{B} \) and \( \text{C} \) add their pressures to the total).

Step 4: Expression for Pressure at Time 't'

At time \( t \), if the reaction has progressed such that \( x \) moles of \( \text{A} \) have reacted, then \( P_2 = P_0 - x + 2x = P_0 + x \). This simplifies, using the completion idea: \( x = P_3 - P_2 \). So, \( P_2 = P_3 - x = P_3 - (P_3 - P_2) \).

Step 5: Rate Constant for First-Order Reaction

The rate constant \( k \) for a first-order reaction can be determined using: \( k = \frac{1}{t} \ln \frac{[\text{A}]_0}{[\text{A}]_t} \). The initial \( [\text{A}]_0 = P_3 - 2x \) and at time \( t \), \( [\text{A}]_t = P_2 \), leading to \( k = \frac{1}{t} \ln \frac{P_3 - (P_3 - P_2)}{P_2} = \frac{1}{t} \ln \frac{P_3}{2(P_3 - P_2)} \).

Step 6: Match the Expression to Given Options

Among the given options, the expression matches choice (a) \( \frac{1}{t} \ln \frac{P_3}{2(P_3 - P_2)} \). Hence, option (a) is the correct expression for the rate constant \( k \).

Key Concepts

Rate Constant Calculation

In chemical kinetics, the rate constant is a crucial factor that characterizes the speed of a reaction and depends upon the conditions under which the reaction is occurring, such as temperature. For a first-order reaction of the form \( \mathrm{A} \rightarrow \mathrm{B} + \mathrm{C} \), the rate constant \( k \) can be calculated using the formula:

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

Here, \([\mathrm{A}]_0\) is the initial concentration or pressure of reactant \( \text{A} \), and \([\mathrm{A}]_t\) is the concentration or pressure of \( \text{A} \) at a given time \( t \). For the given problem, we calculate the rate constant based on changes in pressure, not concentration, because the reaction involves gases. Initially, only \( \text{A} \) is present, accounting for the total pressure \( P_0 \). As the reaction progresses, the total pressure changes until it reaches \( P_3 \) when the reaction completes, with \( \text{B} \) and \( \text{C} \) being formed. Therefore, the equation for \( k \) becomes:\[ k = \frac{1}{t} \ln \frac{P_3}{2(P_3 - P_2)} \]This formula comes from relating the change in pressure directly to the reaction progress as more reactants form products.

Reaction Kinetics

Reaction kinetics deals with the rates at which chemical processes occur. Understanding kinetics is crucial because it allows us to predict how different conditions affect reaction speed. In the context of a first-order reaction, the rate solely depends on the concentration of one reactant—\( \text{A} \). First-order reactions have unique characteristics:

• The rate of reaction decreases over time as \( \text{A} \) is consumed.
• The half-life, \( t_{1/2} \), is constant and does not depend on the initial concentration. It is given by the formula \( t_{1/2} = \frac{0.693}{k} \).

For the reaction \( \text{A(g)} \rightarrow \text{B(g)} + \text{C(g)} \), kinetics are analyzed through pressure measurements. Initially, the pressure is high due to \( \text{A} \)'s presence but decreases as \( \text{A} \) is converted. Since this involves gases, pressures are used to infer concentrations kinetically, giving insight into the reaction rate itself.

Pressure and Reaction Progress

Pressure changes are a direct indicator of reaction progress in gas-phase reactions. For the given problem, pressure measurements at different times illustrate how much the reaction has advanced. Initially, the pressure reflects only reactant \( \text{A} \). However, as \( \text{A} \) converts to \( \text{B} \) and \( \text{C} \), the pressure increases until it stabilizes at \( P_3 \). This stabilization marks the completion of the reaction.The pressure at time \( t \), \( P_2 \), is crucial for determining the amount of \( \text{A} \) that has reacted. The formula \( P_2 = P_3 - x \) implies that "\( x \)" moles have reacted by time \( t \). From this information, and using the relationship:

• \( P_3 = P_0 + x \)

We connect pressure data with reaction progress, assisting in the calculation of the rate constant \( k \).This approach highlights how pressure measurement is a practical method for monitoring reactions, particularly when dealing with gases. It provides a clear way to visualize and quantify the progress of such reactions through direct observation of pressure changes.