Question

In the following question two statements Assertion (A) and Reason (R) are given Mark. a. If \(\mathrm{A}\) and \(\mathrm{R}\) both are correct and \(\mathrm{R}\) is the correct explanation of \(\mathrm{A}\); b. If \(A\) and \(R\) both are correct but \(R\) is not the correct explanation of \(\mathrm{A}\); c. \(\mathrm{A}\) is true but \(\mathrm{R}\) is false; d. \(\mathrm{A}\) is false but \(\mathrm{R}\) is true, e. \(\mathrm{A}\) and \(\mathrm{R}\) both are false. (A): In first order reaction \(t_{1 / 2}\) is independent of initial concentration. \((\mathbf{R})\) : The unit of \(\mathrm{K}\) is time \(^{-1}\).

Short answer

Option (b) is correct: A and R are both true, but R is not the explanation for A.

Step-by-step solution

Analyze the Assertion

The assertion (A) states: 'In first order reaction \( t_{1/2} \) is independent of initial concentration.' For a first-order reaction, the half-life \( t_{1/2} \) does not depend on the initial concentration. This is true because the formula for half-life in a first-order reaction is \( t_{1/2} = \frac{0.693}{k} \), where \( k \) is the rate constant. This confirms that the assertion is true.

Analyze the Reason

The reason (R) states: 'The unit of \( K \) is time \(^{-1}\).' In a first-order reaction, the rate constant \( k \) indeed has units of time \(^{-1}\), for example, \( s^{-1} \) or \( min^{-1} \). This statement is true.

Determine the Relationship

Now, determine if reason (R) is the correct explanation for assertion (A). The fact that the unit of \( K \) is time \(^{-1}\) is a characteristic of the reaction order but does not directly explain why the half-life \( t_{1/2} \) is independent of initial concentration. Therefore, while both statements are correct individually, (R) is not the explanation of (A).

Choose the Correct Option

Based on the analysis, option (b) fits: 'If \( A \) and \( R \) both are correct but \( R \) is not the correct explanation of \( A \).'

Key concepts

First Order Reaction

In a first-order reaction, the rate of the reaction depends linearly on the concentration of one reactant. This means that the speed at which the reactant is being consumed is directly proportional to its quantity in the mixture. Although the concentration changes over time, the constant, known as the rate constant, remains unchanged.
The relationship for a first-order reaction can be mathematically expressed as: \[ ext{Rate} = k[A]\]where \(k\) is the rate constant and \([A]\) represents the concentration of the reactant.
This reaction order is significant in chemical kinetics because it provides a simpler means of analyzing how reactions proceed over time, especially when graphing natural logarithm plots of concentration against time.

Half-Life

Half-life, denoted as \( t_{1/2} \), is the time required for half of the reactant in a chemical reaction to be consumed. In first-order reactions, this parameter is particularly interesting because it does not depend on the initial concentration. Unlike zero or second-order reactions, where such dependence is notable, the first-order half-life formula remains:\[t_{1/2} = \frac{0.693}{k}\]Here, \(0.693\) is the natural logarithm of 2, and \(k\) is the rate constant. This makes calculations simplified, as knowing the rate constant allows us to predict the time it will take for the concentration of a substance to reach half of its original value, regardless of what that original value was.

Rate Constant

The rate constant \(k\) is a crucial parameter in chemical kinetics, giving us insight into how fast a reaction proceeds. For a first-order reaction, the rate constant has the unit of time \(^{-1}\), such as \(s^{-1}\) or \(min^{-1}\).
This unit indicates the reaction's per unit time consumption of the reactant, assuming a constant rate. It serves as an indicator of reaction speed. Kinetics research often seek to determine these rate constants experimentally, to better understand chemical behavior and predict future behavior in similar systems.
The constancy of \(k\) for a given reaction at a specified temperature allows chemists to make consistent and accurate predictions of reaction behavior in real-world applications.

Units of Reaction Rate

The units of a reaction rate can vary depending on the order of the reaction and what is being measured. For a first-order reaction, since the rate is proportional to a single reactant's concentration, the unit is typically given in terms of concentration over time, such as mol/L/s.
Understanding these units is essential when studying reaction kinetics, as they help in interpreting how quickly or slowly a system tends toward equilibrium. These units also provide critical insights when comparing rates across different reactions under similar conditions.
Distinctions in units between reactions of various orders stem from their different dependencies on reactant concentrations, providing key information on how reaction conditions need to be controlled.

Assertion Reason Questions

Assertion-reason questions are a common tool in educational assessments to gauge understanding of scientific concepts. These types of questions consist of two statements: an assertion and a reason. The task is to determine not only the truth of each statement but also the logical connection between them.
For example, in the provided problem, the assertion and reason are both correct, as they correctly describe facts about first-order reactions and rate constant units. However, the reason does not directly explain the assertion, highlighting a crucial aspect of these questions: evaluating the relationship between statements, rather than accepting them at face value.
This style of questioning encourages deeper thinking and comprehension, as students must work to understand the nuances in the relationship between scientific principles.