Question

A first order reaction is half completed in 45 minutes. How long does it need for 99.9% of the reaction to be completed?

• A. 5 hours
• B. 7.5 hours
• C. 10 hours
• D. 20 hours

Hint:

For a first
order reaction, the time required for a certain percentage of the reaction to be completed can be calculated using the formula: \[ \ln \left( \frac{[A]_0}{[A]} \right) = kt. \] The half
life (\( t_{1/2} \)) is related to the rate constant \( k \) by: \[ t_{1/2} = \frac{0.693}{k}. \]

Answer 1

The Correct Option is B

Step 1: Understanding the Problem 
A first
order reaction is half completed in 45 minutes. We need to find the time required for 99.9\% of the reaction to be completed. 
Step 2: Using the Half
Life Formula for First
Order Reactions 
The half
life (\( t_{1/2} \)) of a first
order reaction is given by: \[ t_{1/2} = \frac{0.693}{k}, \] where \( k \) is the rate constant. 
Step 3: Calculating the Rate Constant \( k \) 
Given \( t_{1/2} = 45 \) minutes: \[ k = \frac{0.693}{45} \, \text{min}^{1}. \] 
Step 4: Using the First
Order Reaction Formula 
The first
order reaction formula is: \[ \ln \left( \frac{[A]_0}{[A]} \right) = kt, \] where: 
\( [A]_0 \) is the initial concentration, 
\( [A] \) is the concentration at time \( t \), 
\( k \) is the rate constant, 
\( t \) is the time. 
Step 5: Calculating the Time for 99.9\% Completion 
For 99.9% completion, \( [A] = 0.001 [A]_0 \). 
Substituting into the first
order reaction formula: \[ \ln \left( \frac{[A]_0}{0.001 [A]_0} \right) = kt. \] \[ \ln (1000) = kt. \] \[ t = \frac{\ln (1000)}{k}. \] 
Substituting \( k = \frac{0.693}{45} \): \[ t = \frac{\ln (1000)}{\frac{0.693}{45}} = \frac{6.908}{0.693} \times 45 \approx 450 \, \text{minutes}. \] 
Converting minutes to hours: \[ t = \frac{450}{60} = 7.5 \, \text{hours}. \] 
Step 6: Matching with the Options 
The calculated time is 7.5 hours, which corresponds to option (B). Final Answer: The time required for 99.9\% of the reaction to be completed is (B) 7.5 hours.