for a first order reaction , the time taken to reduce the initial concentration by a factor of `(1)/(4) ` is 20 minutes . The time required to reduce intitial concentration by a factor of `(1)/(16)` is

Explanation

To solve the problem, we need to determine the time required to reduce the initial concentration of a substance by a factor of \( \frac{1}{16} \) for a first-order reaction, given that the time taken to reduce the concentration by a factor of \( \frac{1}{4} \) is 20 minutes.

### Step-by-Step Solution:

1. **Understanding the First-Order Reaction:**
For a first-order reaction, the rate constant \( k \) can be expressed using the formula:
\[
k = \frac{2.303}{t} \log \left( \frac{A}{A - X} \right)
...