A radioactive isotope decays at such a rate that after 96 min, only `1//8th` of the original amount remains.
The value of `t_(1//2)` of this nuclide is
a. 12 min b. 32 min c. 24 min d. 48 min

Options

A. a. 12 min
B. b. 32 min
C. c. 24 min
D. d. 48 min

Explanation

To solve the problem, we need to determine the half-life (t₁/₂) of a radioactive isotope given that after 96 minutes, only 1/8th of the original amount remains.

### Step-by-Step Solution:

1. **Understanding the Decay**: We know that after a certain time, the amount of the radioactive substance remaining can be expressed in terms of its half-lives. The formula for the remaining amount after n half-lives is:
\[ \text{Remaining amount} = \frac{1}{2^n} \times \text{Original amount} \]
Here, we are given that the remaining amount is 1/8th of the original amount.

2. **Expressing 1/8 in terms of powers of 2**: We can express 1/8 as:
\[ \frac{1}{8} = \frac{1}{2^3} \]
This indicates that after 3 half-lives (n = 3), the amount remaining is 1/8th of the original amount.

3. **Relating time to half-lives**: We know that the total time elapsed (96 minutes) corresponds to 3 half-lives. Therefore, we can write:
\[ \text{Total time} = n \times t_{1/2} \]
Substituting the values we have:
\[ 96 \text{ min} = 3 \times t_{1/2} \]

4. **Calculating the half-life**: To find the half-life (t₁/₂), we can rearrange the equation:
\[ t_{1/2} = \frac{96 \text{ min}}{3} \]
Performing the division:
\[ t_{1/2} = 32 \text{ min} \]

5. **Conclusion**: The half-life of the radioactive nuclide is 32 minutes. Therefore, the correct answer is option (b) 32 min.