Question

\(t_{100\%}\) is the time required for 100% completion of a reaction, while \(t_{1/2}\) is the time required for 50% completion of the reaction. Which of the following correctly represents the relation between \(t_{100\%}\) and \(t_{1/2}\) for zero order and first order reactions respectively

• A. \(t_{100\%} = 2t_{1/2}, \quad t_{100\%} = \infty\)
• B. \(t_{100\%} = 2t_{1/2} \quad \text{and} \quad t_{100\%} = \infty\)
• C. \(t_{100\%} = 2t_{1/2} \quad \text{and} \quad t_{100\%} = (2t_{1/2})^2\)
• D. \(t_{100\%} = \infty \quad \text{and} \quad t_{100\%} = 2t_{1/2}\)

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
For a chemical reaction, the rate laws define how concentration changes with time.
The time for 100% completion (\(t_{100%}\)) is the time when the final concentration \([A] = 0\).
Step 2: Key Formula or Approach:
1. Zero order: \([A]_0 - [A] = kt\).
2. First order: \(\ln([A]_0 / [A]) = kt\).
Step 3: Detailed Explanation:
1. For Zero Order Reaction:
- At \(t = t_{1/2}\), \([A] = [A]_0 / 2 \implies [A]_0 - [A]_0 / 2 = k \cdot t_{1/2} \implies t_{1/2} = \frac{[A]_0}{2k}\).
- At \(t = t_{100%}\), \([A] = 0 \implies [A]_0 - 0 = k \cdot t_{100%} \implies t_{100%} = \frac{[A]_0}{k}\).
- Clearly, \(t_{100%} = 2 \cdot t_{1/2}\).

2. For First Order Reaction:
- At \(t = t_{1/2}\), \(t_{1/2} = \frac{\ln 2}{k}\) (a finite constant).
- At \(t = t_{100%}\), \([A] = 0 \implies \ln([A]_0 / 0) = k \cdot t_{100%} \implies \ln(\infty) = k \cdot t_{100%} \implies t_{100%} = \infty\).
- In mathematical notation for half-lives, this is represented as \((t_{1/2})^{\infty}\), meaning it takes an infinite number of half-lives for the concentration to reach zero.
Step 4: Final Answer:
The relations are \(t_{100%} = 2t_{1/2}\) (Zero order) and \(t_{100%} = (t_{1/2})^{\infty}\) (First order).