Question

For a first-order reaction, (a) The degree of dissociation is equal to \(\left(1-\mathrm{e}^{-\mathrm{k}}\right)\) (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)

Short answer

The correct answer is (d) both (a) and (b).

Step-by-step solution

Understanding First-Order Reactions

In a first-order reaction, the rate of reaction depends linearly on the concentration of a single reactant. The rate law can be represented as \( \frac{-d[A]}{dt} = k[A] \), where \( k \) is the rate constant.

Degree of Dissociation

For a first-order reaction, the degree of dissociation \( \alpha \) at time \( t \) is given by \( \alpha = 1 - e^{-kt} \). Given \( \alpha = 1 - e^{-k} \) for this specific time \( t = 1 \). This confirms part (a).

Arrhenius Equation and Pre-exponential Factor

The Arrhenius equation is given by \( k = A e^{-E_a/(RT)} \), where \( A \) is the pre-exponential factor. For a first-order reaction, \( k \) has units of \( \text{time}^{-1} \), thus \( A \) also has units of \( \text{time}^{-1} \), which confirms part (b).

Half-Life Analysis

The half-life \( t_{1/2} \) of a first-order reaction is given by \( t_{1/2} = \frac{0.693}{k} \). It is mentioned that the time for 75% completion is thrice the half-life, hence for 75% completion \( t = 3 \times t_{1/2} = 3 \times \frac{0.693}{k} \). This confirms (c).

Conclusion of Analysis

Since all parts (a), (b), and (c) have been confirmed valid for first-order reactions separately, only parts (a) and (b) can be true simultaneously, as the options provided ask for one correct answer.

Key concepts

Rate Law

In the realm of chemical kinetics, the rate law offers a mathematical representation that describes the speed of a chemical reaction. For a first-order reaction, the rate of reaction is directly proportional to the concentration of its single reactant. This can be expressed through the equation \( \frac{-d[A]}{dt} = k[A] \), where \( [A] \) is the concentration of the reactant and \( k \) is the rate constant.

The negative sign indicates a decrease in reactant concentration over time. This linear dependence on the concentration suggests that as the concentration of the reactant doubles, the rate of the reaction also doubles.

• **Key points:** The rate constant \( k \) is crucial as it encompasses factors like temperature and activation energy that influence the reaction rate.

Degree of Dissociation

The degree of dissociation (\( \alpha \)) quantifies the fraction of a substance that breaks apart during a reaction. For a first-order reaction, this is elegantly described by the formula \( \alpha = 1 - e^{-kt} \).

Here, \( e \) stands for the base of natural logarithms, and \( t \) is the time.

• **Example:** At \( t = 1 \), if \( \alpha = 1 - e^{-k} \), it signifies that a significant portion of the reactant has dissociated, as outlined under part (a) of the provided problem.

The utility of this formula lies in its ability to give insights into the progression of the reaction over time, offering a clearer understanding of how much of the reactant has transitioned into the product as time progresses.

Arrhenius Equation

The Arrhenius equation unveils the relationship between the rate constant \( k \) and temperature \( T \). It is expressed as:
\[ k = A e^{-E_a/(RT)} \]
Here, \( A \) is known as the pre-exponential factor, \( E_a \) is the activation energy, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

This equation highlights how rate constants increase with temperature, as higher temperatures provide more energy for reactants to overcome the energy barrier. In a first-order reaction, since \( k \) has the dimensions of \( \text{time}^{-1} \), it implies that the pre-exponential factor \( A \) similarly shares this dimension, confirming part (b) in the exercise.

Half-Life Analysis

Half-life, denoted as \( t_{1/2} \), is a term that describes the time needed for half of the reactant to be consumed in a first-order reaction. It is determined by the equation \( t_{1/2} = \frac{0.693}{k} \).

This equation is a staple in chemical kinetics for first-order reactions because it does not depend on the initial concentration, making it simpler to analyze experimental data.

• **Additional insight:** According to the problem, the time required for the completion of 75% of the reaction is three times the half-life, equating to \( t = 3 \times t_{1/2} = 3 \times \frac{0.693}{k} \). This highlights the gradual nature of first-order reactions and showcases the practical application of using half-life to predict reaction progress.