For first order reaction, the time taken to reduce tha initial concentration by a factor of `(1)/(4)` is 20 minutes. The time required to reduce initial concentration by a factor of `1//10` is.

Options

A. 20 min
B. 10 min
C. 80 min
D. 40 min

Explanation

To solve the problem, we will use the first-order reaction kinetics formula. The first-order reaction can be described by the equation:

\[ k = \frac{2.303}{t} \log \left( \frac{[A]_0}{[A]} \right) \]

Where:
- \( k \) = rate constant
- \( t \) = time
- \( [A]_0 \) = initial concentration
- \( [A] \) = concentration at time \( t \)

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

We know that the time taken to reduce the initial concentration by a factor of \( \frac{1}{4} \) is 20 minutes. This means:

\[ [A] = \frac{[A]_0}{4} \]

Substituting this into the first-order rate equation:

\[ k = \frac{2.303}{20} \log \left( \frac{[A]_0}{\frac{[A]_0}{4}} \right) \]

This simplifies to:

\[ k = \frac{2.303}{20} \log(4) \]

Now, we know that \( \log(4) = 2 \log(2) \) and \( \log(2) \approx 0.301 \), thus:

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

Now substituting this value into the equation for \( k \):

\[ k = \frac{2.303}{20} \times 0.602 \]

Calculating \( k \):

\[ k \approx \frac{2.303 \times 0.602}{20} \]
\[ k \approx \frac{1.384}{20} \]
\[ k \approx 0.0692 \, \text{min}^{-1} \]

### Step 2: Calculate the time to reduce concentration by a factor of \( \frac{1}{10} \)

Now we need to find the time \( t \) required to reduce the concentration by a factor of \( \frac{1}{10} \):

\[ [A] = \frac{[A]_0}{10} \]

Using the first-order rate equation again:

\[ k = \frac{2.303}{t} \log \left( \frac{[A]_0}{\frac{[A]_0}{10}} \right) \]

This simplifies to:

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

Since \( \log(10) = 1 \):

\[ k = \frac{2.303}{t} \]

Now we can equate the two expressions for \( k \):

\[ 0.0692 = \frac{2.303}{t} \]

Rearranging for \( t \):

\[ t = \frac{2.303}{0.0692} \]

Calculating \( t \):

\[ t \approx 33.24 \, \text{minutes} \]

### Final Answer

The time required to reduce the initial concentration by a factor of \( \frac{1}{10} \) is approximately **33.24 minutes**.

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