Question

Assertion (A): In a first order reaction, if the concentration of the reactant is doubled, its half-life is also doubled. 
Reason (R): The half-life of a reaction does not depend upon the initial concentration of the reactant in a first order reaction.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

For first-order reactions, the half-life remains constant regardless of the concentration of reactants, which is a key feature of such reactions.

Answer 1

The Correct Option is D

To address the problem, we must examine Assertion (A) and Reason (R) concerning the impact of doubling reactant concentration on the half-life of a first-order reaction, and then assess their accuracy and connection.

1. Analysis of Assertion (A):
Assertion (A) posits that in a first-order reaction, doubling the reactant concentration also doubles its half-life. The rate law for a first-order reaction is Rate = k[A], where k is the rate constant and [A] is the reactant concentration. The half-life (t\(_{1/2}\)) of a first-order reaction is defined by the formula:

$ t_{1/2} = \frac{\ln(2)}{k} \approx \frac{0.693}{k} $
This equation indicates that the half-life is solely dependent on the rate constant k and is unaffected by the initial concentration [A]\(_0\). Consequently, if the initial concentration is doubled, the half-life remains unchanged because k is constant. Therefore, the assertion that doubling the concentration doubles the half-life is incorrect, as the half-life remains constant.

2. Analysis of Reason (R):
Reason (R) asserts that the half-life of a first-order reaction is not influenced by the initial reactant concentration. As demonstrated above, the half-life formula $ t_{1/2} = \frac{\ln(2)}{k} $ confirms that t\(_{1/2}\) is solely a function of k and does not include [A]\(_0\). This characteristic distinguishes first-order reactions from zero-order or second-order reactions, where the half-life is concentration-dependent. Thus, the reason is accurate.

3. Evaluation of the Relationship:
The assertion is false because doubling the reactant concentration does not alter the half-life in a first-order reaction. The reason is true and directly contradicts the assertion, accurately stating that the half-life is independent of the initial concentration. In assertion-reason questions, the truth of the reason does not validate a false assertion. However, the reason clarifies why the assertion is incorrect; the independence of half-life from concentration means that doubling the concentration cannot result in a doubled half-life.

4. Conclusion:
- Assertion (A) is false, as the half-life of a first-order reaction does not change upon doubling the reactant concentration.
- Reason (R) is true, as the half-life of a first-order reaction is independent of the initial concentration.
- The reason explains the falsity of the assertion, because the concentration independence of the half-life contradicts the claim that doubling the concentration doubles the half-life.

Final Answer:
Assertion (A) is false, Reason (R) is true.