Question

If the half-life ($t_{1/2}$) for a first-order reaction is 1 minute, then the time required for 99.9% completion of the reaction is closest to:

• A. 4 minutes
• B. 5 minutes
• C. 10 minutes
• D. 2 minutes

Hint:

For a first-order reaction, the time for 99.9\% completion can be calculated using the formula: \[ t = \frac{\ln(1/0.001)}{k} \] Where \( k \) is determined from the half-life of the reaction.

Answer 1

The Correct Option is C

The integrated rate law for a first-order reaction, describing concentration \( [A] \) as a function of time \( t \), is: \[ \ln \left( \frac{[A]_0}{[A]} \right) = kt \] where \( [A]_0 \) denotes the initial concentration, \( [A] \) is the concentration at time \( t \), \( k \) is the rate constant, and \( t \) is the elapsed time. The half-life \( t_{1/2} \) for a first-order reaction is given by: \[ t_{1/2} = \frac{0.693}{k} \] Given a half-life \( t_{1/2} = 1 \) minute, the rate constant \( k \) can be determined as: \[ k = \frac{0.693}{1} = 0.693 \, \text{min}^{-1} \] To calculate the time required for 99.9% completion, we consider that this implies 0.1% of the initial concentration remains, i.e., \( [A] = 0.001 [A]_0 \). Substituting this into the first-order equation yields: \[ \ln \left( \frac{1}{0.001} \right) = k t \] \[ \ln (1000) = 0.693 \times t \] \[ 6.907 = 0.693 \times t \] Solving for \( t \): \[ t = \frac{6.907}{0.693} \approx 10 \, \text{minutes} \] Therefore, 10 minutes are needed for 99.9% completion of the reaction.