Question

Find the time required to complete a reaction 90% if the reaction is completed 50% in 15 minutes.

Hint:

For first-order reactions, the time required for completion depends on the logarithm of the ratio \( [A]_0/[A] \). Memorize the key relationships for easier calculations.

Answer 1

Step 1: Determine the order of reaction.
The problem involves percentages of completion and time, which suggests a first-order reaction. The formula for the time required to achieve a certain completion in a first-order reaction is: \[ t = \frac{2.303}{k} \log \frac{[A]_0}{[A]}, \] where: - \( t \) is the time, - \( k \) is the rate constant, - \( [A]_0 \) is the initial concentration, - \( [A] \) is the concentration at time \( t \).

 Step 2: Calculate the rate constant \( k \).
For 50\% completion, \( [A]_0/[A] = 2 \). Substituting \( t = 15 \, \text{minutes} \): \[ 15 = \frac{2.303}{k} \log 2. \] \[ k = \frac{2.303 \log 2}{15}. \] Using \( \log 2 = 0.3010 \): \[ k = \frac{2.303 \times 0.3010}{15} = 0.04627 \, \text{min}^{-1}. \] 

Step 3: Calculate the time for 90\% completion.
For 90\% completion, \( [A]_0/[A] = 10 \). Substituting into the formula: \[ t = \frac{2.303}{k} \log 10. \] Using \( \log 10 = 1 \): \[ t = \frac{2.303}{0.04627} \times 1. \] \[ t = 49.44 \, \text{minutes}. \] 

Step 4: Final Answer.
The time required to complete 90\% of the reaction is: \[ \boxed{49.44 \, \text{minutes}}. \]