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 solve the problem, we need to evaluate the Assertion (A) and Reason (R) regarding the effect of doubling the reactant concentration on the half-life of a first order reaction and determine their validity and relationship.

1. Analyzing the Assertion (A):
Assertion (A) states that in a first order reaction, if the concentration of the reactant is doubled, its half-life is also doubled. For a first order reaction, the rate law 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 given by the formula:

$ t_{1/2} = \frac{\ln(2)}{k} \approx \frac{0.693}{k} $
This formula shows that the half-life depends only on the rate constant k and is independent of the initial concentration [A]\(_0\). If the initial concentration is doubled, the half-life remains the same because k does not change. Therefore, the assertion that doubling the concentration doubles the half-life is false, as the half-life stays constant.

2. Analyzing the Reason (R):
Reason (R) states that the half-life of a first order reaction does not depend upon the initial concentration of the reactant. As derived above, the half-life formula $ t_{1/2} = \frac{\ln(2)}{k} $ confirms that t\(_{1/2}\) is a function of k only and does not involve [A]\(_0\). This is a characteristic feature of first order reactions, unlike zero or second order reactions where half-life depends on concentration. Thus, the reason is true.

3. Evaluating the Relationship:
The assertion is false because doubling the reactant concentration does not affect the half-life in a first order reaction. The reason is true and directly contradicts the assertion, as it correctly states that the half-life is independent of initial concentration. In assertion-reason questions, if the assertion is false, the reason’s truth does not make the assertion true. However, the reason explains why the assertion is incorrect, as the independence of half-life from concentration means doubling the concentration cannot double the half-life.

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

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