3/4 th of first order reaction was completed in 32 min, 15/16 the part will be completed in

Options

A. 24 min
B. 64 min
C. 16 min
D. 32 min

Explanation

To solve the problem of how long it will take for \( \frac{15}{16} \) of a first-order reaction to be completed, given that \( \frac{3}{4} \) of the reaction was completed in 32 minutes, we can follow these steps:

### Step 1: Understand the first-order reaction kinetics
For a first-order reaction, the rate constant \( k \) can be determined using the formula:
\[ k = \frac{2.303}{t} \log \left( \frac{A_0}{A_0 - x} \right) \]
where:
- \( A_0 \) is the initial concentration,
- \( x \) is the amount reacted,
- \( t \) is the time taken.

### Step 2: Calculate the value of \( k \)
Given that \( \frac{3}{4} \) of the reaction is completed in 32 minutes, we can set:
- \( A_0 = 1 \) (initial concentration),
- \( x = \frac{3}{4} \).

Thus, the concentration remaining \( A = A_0 - x = 1 - \frac{3}{4} = \frac{1}{4} \).

Now substituting into the equation:
\[ k = \frac{2.303}{32} \log \left( \frac{1}{1/4} \right) = \frac{2.303}{32} \log(4) \]
Since \( \log(4) = \log(2^2) = 2 \log(2) \) and \( \log(2) \approx 0.301 \):
\[ \log(4) = 2 \times 0.301 = 0.602 \]
Now substituting this back into the equation for \( k \):
\[ k = \frac{2.303}{32} \times 0.602 \approx 0.043 \text{ min}^{-1} \]

### Step 3: Calculate the time for \( \frac{15}{16} \) of the reaction to be completed
Now we need to find the time taken for \( \frac{15}{16} \) of the reaction to be completed. Here:
- \( A_0 = 1 \),
- \( x = \frac{15}{16} \),
- Remaining concentration \( A = A_0 - x = 1 - \frac{15}{16} = \frac{1}{16} \).

Using the first-order reaction formula again:
\[ t = \frac{2.303}{k} \log \left( \frac{A_0}{A_0 - x} \right) = \frac{2.303}{0.043} \log \left( \frac{1}{1/16} \right) \]
Calculating \( \log(16) \):
\[ \log(16) = \log(2^4) = 4 \log(2) \approx 4 \times 0.301 = 1.204 \]
Now substituting this back into the time equation:
\[ t = \frac{2.303}{0.043} \times 1.204 \approx 64 \text{ minutes} \]

### Final Answer
Thus, the time taken for \( \frac{15}{16} \) of the reaction to be completed is **64 minutes**.

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