For the first order reaction, the taken to reduce the initial concentration to a factor of `1/4` is 10 minutes if the reduction in concentration is carried out to a factor of `1/16`, then time required will be

Options

A. 10 minutes
B. 20 minutes
C. 40 minutes
D. 60 minutes

Answer

To solve the problem, we will use the first-order reaction kinetics formula. The formula for a first-order reaction is given by:

\[ k = \frac{2.303}{t} \log \left( \frac{a}{a - x} \right) \]

Where:
- \( k \) is the rate constant,
- \( t \) is the time,
- \( a \) is the initial concentration,
- \( x \) is the change in concentration.

### Step 1: Determine the rate constant \( k \)

Given that the time taken to reduce the concentration from \( a \) to \( \frac{a}{4} \) is 10 minutes, we can set up the equation:

- Initial concentration \( a \)
- Final concentration \( a - x = \frac{a}{4} \)

Thus, \( x = a - \frac{a}{4} = \frac{3a}{4} \).

Substituting into the formula:

\[ k = \frac{2.303}{10} \log \left( \frac{a}{\frac{a}{4}} \right) \]

This simplifies to:

\[ k = \frac{2.303}{10} \log(4) \]

Calculating \( \log(4) \):

\[ \log(4) = 2 \log(2) \approx 2 \times 0.301 = 0.602 \]

So,

\[ k = \frac{2.303}{10} \times 0.602 \approx 0.138 \, \text{min}^{-1} \]

### Step 2: Calculate the time for reduction to \( \frac{a}{16} \)

Now we need to find the time taken to reduce the concentration from \( a \) to \( \frac{a}{16} \).

Here, \( a - x = \frac{a}{16} \), thus \( x = a - \frac{a}{16} = \frac{15a}{16} \).

Using the first-order kinetics formula again:

\[ t = \frac{2.303}{k} \log \left( \frac{a}{\frac{a}{16}} \right) \]

This simplifies to:

\[ t = \frac{2.303}{k} \log(16) \]

Calculating \( \log(16) \):

\[ \log(16) = 4 \log(2) \approx 4 \times 0.301 = 1.204 \]

Now substituting \( k \):

\[ t = \frac{2.303}{0.138} \times 1.204 \]

Calculating \( t \):

\[ t \approx \frac{2.303 \times 1.204}{0.138} \approx \frac{2.775}{0.138} \approx 20.1 \, \text{minutes} \]

### Final Answer

The time required to reduce the concentration to a factor of \( \frac{1}{16} \) is approximately **20.1 minutes**.

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