Problem 24 - A Complete Resource Book in Chemistry for JEE Main

Question

For a first-order reaction, (a) The degree of dissociation is equal to \(\left(1-\mathrm{e}^{\mathrm{kt}}\right)\) (b) The pre-exponential factor in the Arrhenius equation has the dimensions of time \(t^{-1}\). (c) The time taken for the completion of \(75 \%\) reaction is thrice the t \(1 / 2\) of the reaction. (d) both (a) and (b)

Step-by-Step Solution

Answer

Correct choice: (d) both (a) and (b).

Step 1: Understand First-order Reaction Kinetics

For a first-order reaction, the rate of reaction depends linearly on the concentration of one reactant. The general form of the rate law is given by:\[\text{Rate} = k[A]\]where \(k\) is the rate constant, and \([A]\) is the concentration of the reactant.

Step 2: Analyze Option (a)

Option (a) states that the degree of dissociation \( \alpha = \left(1 - e^{-kt}\right) \). This is indeed consistent with the integrated rate equation for a first-order reaction:\[[A] = [A]_0 e^{-kt} \]Thus, \( \alpha = \frac{[A]_0 - [A]}{[A]_0} = 1 - e^{-kt} \). Option (a) is correct.

Step 3: Analyze Option (b)

Option (b) involves the pre-exponential factor (\(A\)) in the Arrhenius equation, given by:\[k = A e^{-E_a/(RT)}\]For a first-order reaction, \(k\) has dimensions of \(t^{-1}\). Therefore, \(A\) also has dimensions of \(t^{-1}\) to keep dimensional consistency in \(k = A e^{-E_a/(RT)}\). Option (b) is correct.

Step 4: Analyze Option (c)

Option (c) must verify that the time for 75% completion (75% of reaction) is thrice the half-life \((t_{1/2})\). For a first-order reaction, the half-life is:\[t_{1/2} = \frac{0.693}{k}\]For 75% completion, the time \(t\) is given by:\[0.25 = e^{-kt} \Rightarrow t = \frac{\ln 4}{k}\]\(\ln{4} \approx 1.386\), so \(t = 2 \cdot \frac{0.693}{k} = 2 \cdot t_{1/2}\), not three times. Thus, option (c) is incorrect.

Step 5: Determine the Correct Choices

Both options (a) and (b) are consistent with the characteristics of first-order reactions regarding degree of dissociation and dimensions of the pre-exponential factor. Thus, the correct answer should include both statements.

Key Concepts

Rate Law

In first-order reaction kinetics, the rate of the reaction depends on the concentration of only one reactant. This relationship is described by the rate law formula:

• Rate = k[A]

Here, \(k\) is the rate constant, and \([A]\) denotes the concentration of the reactant at a given time. The simplicity of first-order reactions makes them a cornerstone in chemical kinetics as they provide a straightforward way to understand reaction mechanisms.
The rate constant \(k\) is unique for each reaction at a specific temperature, which means even slight temperature changes can affect the rate. The linear dependence on reactant concentration emphasizes why monitoring concentration over time provides essential insights in chemical experiments.

Arrhenius Equation

The Arrhenius Equation is a formula that connects the rate constant \(k\) of a reaction with temperature. It is crucial in understanding how temperature affects reaction speed:

• \(k = A e^{-E_a/(RT)}\)

In this equation, \(A\) is the pre-exponential factor, which represents the frequency of collisions with the correct orientation. The energy of activation, \(E_a\), represents the minimum energy required for the reaction to occur, and \(R\) is the universal gas constant. The pre-exponential factor \(A\) has the dimensions of \(t^{-1}\) for first-order reactions, as the rate constant \(k\) must also have these dimensions to maintain the equation's dimensional consistency. The Arrhenius Equation elegantly showcases why reactions speed up with temperature: higher energy collisions occur more frequently.

Half-life

For first-order reactions, the half-life (\(t_{1/2}\)) is the time required for half the amount of reactant to be consumed. It is independent of concentration, a unique attribute for first-order kinetics:

• \(t_{1/2} = \frac{0.693}{k}\)

This characteristic is handy when working with kinetics, as it simplifies calculations regarding how much reactant remains over time.
In practice, it's fascinating because the half-life remains constant throughout the reaction. Therefore, predicting when a specific amount of reactant will have reacted becomes simpler. For completeness of 75% of the reaction, the equation requires the time taken to be around twice the half-life, not three times, a point to always remember to avoid errors.

Degree of Dissociation

The degree of dissociation \(\alpha\) in a first-order reaction explains how much of the initial reactant has decomposed over time. It is primarily noted by the formula:

• \(\alpha = 1 - e^{-kt}\)

This equation derives from the integrated rate law, where \([A] = [A]_0 e^{-kt}\)
The degree of dissociation reveals the proportion of the initial reactant that has turned into products in a certain time frame. This knowledge is often used to predict reaction completeness and to calculate remaining concentrations. Understanding \(\alpha\) helps scientists and students estimate how much of a reactant remains, which is fundamental in calculating yields and efficiencies in chemical processes.