Question

For a first order reaction \(A \to B\),
x = _______ min. (Nearest integer)

Answer 1

Correct Answer: 7

Step 1: Understanding the Concept:
For a first-order reaction, the rate constant \(k\) is given by \(k = \frac{2.303}{t} \log \frac{[A]_0}{[A]}\). The time required for a certain percentage of completion depends only on \(k\), not the initial concentration.

Step 2: Key Formula or Approach:
Use the integrated rate law: \(k = \frac{1}{t} \ln \frac{[A]_0}{[A]}\).

Step 3: Detailed Explanation:
Step 1: Find \(k\) using the data at \(t = 20\) min.
\(k = \frac{1}{20} \ln \frac{0.6500}{0.00065} = \frac{1}{20} \ln(1000) = \frac{3 \ln 10}{20}\).
Step 2: Use \(k\) to find \(x\).
\(x = \frac{1}{k} \ln \frac{0.6500}{0.0650} = \frac{1}{k} \ln(10)\).
Substituting \(k\):
\(x = \frac{20}{3 \ln 10} \times \ln 10 = \frac{20}{3} = 6.666 \dots\)
Nearest integer = 7.

Step 4: Final Answer:
The value of x is 7 min.