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: Identify the reaction order. The presence of completion percentages and time indicates a first-order reaction. The equation governing the time for a specific completion percentage in a first-order reaction is: \[ t = \frac{2.303}{k} \log \frac{[A]_0}{[A]}, \] where: - \( t \) represents time, - \( k \) is the rate constant, - \( [A]_0 \) is the initial concentration, - \( [A] \) is the concentration at time \( t \).

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

Step 3: Calculate the time for 90% completion. For 90% completion, the ratio \( [A]_0/[A] = 10 \). 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 Result. The duration required for 90% reaction completion is: \[ \boxed{49.44 \, \text{minutes}}. \]