Question

Half-life of first order reaction \(X \rightarrow Y + Z is 3 \text{ minutes.}\) What is the time required to reduce the concentration of X by 90% of its initial concentration?

• A. \(4.12\) minutes
• B. \(9.969\) minutes
• C. \(9.105\) minutes
• D. \(12.05\) minutes

Hint:

For a first order reaction, \(90%\) completion always corresponds to \(\log 10\) in the rate equation.

Answer 1

The Correct Option is B

Step 1: Write the half-life relation for first order reaction.
\[ t_{1/2} = \frac{0.693}{k} \] Given \(t_{1/2} = 3\) minutes.
Step 2: Calculate the rate constant.
\[ k = \frac{0.693}{3} = 0.231\,\text{min}^{-1} \]
Step 3: Use integrated rate equation.
For \(90%\) completion, \(\frac{[X]_0}{[X]} = 10\).
\[ t = \frac{2.303}{k}\log\left(\frac{[X]_0}{[X]}\right) \]
Step 4: Substitute values.
\[ t = \frac{2.303}{0.231}\log(10) = \frac{2.303}{0.231} = 9.969\,\text{minutes} \]
Step 5: Conclusion.
Thus, the time required is \(9.969\) minutes.