`7/8th` of the active nuclei present in a radioactive sample has decayed in 8 s. The half-life of the sample is

Options

A. (a) 2s
B. (b) 1s
C. (c) 7s
D. (d) 8/3 s

Explanation

To solve the problem, we need to find the half-life of a radioactive sample given that \( \frac{7}{8} \) of the active nuclei have decayed in 8 seconds.

### Step-by-Step Solution:

1. **Understand the Decay**: If \( \frac{7}{8} \) of the active nuclei have decayed, then the remaining active nuclei is:
\[ N = N_0 - \frac{7}{8}N_0 = \frac{1}{8}N_0 \]
where \( N_0 \) is the initial number of nuclei.

2. **Use the Decay Formula**: The number of nuclei remaining after time \( t \) can also be expressed as:
\[ N = N_0 \left( \frac{1}{2} \right)^n \]
where \( n \) is the number of half-lives that have passed.

3. **Set Up the Equation**: From the previous step, we can equate the two expressions for \( N \):
\[ \frac{1}{8}N_0 = N_0 \left( \frac{1}{2} \right)^n \]

4. **Simplify the Equation**: Dividing both sides by \( N_0 \) (assuming \( N_0 \neq 0 \)):
\[ \frac{1}{8} = \left( \frac{1}{2} \right)^n \]

5. **Express \( \frac{1}{8} \) as a Power of \( \frac{1}{2} \)**: We know that:
\[ \frac{1}{8} = \left( \frac{1}{2} \right)^3 \]
Therefore, we can equate the exponents:
\[ n = 3 \]

6. **Relate Time to Half-Lives**: Given that 3 half-lives have passed in 8 seconds, we can find the duration of one half-life:
\[ \text{Time for one half-life} = \frac{8 \text{ s}}{3} = \frac{8}{3} \text{ s} \]

7. **Final Answer**: Thus, the half-life of the sample is:
\[ \frac{8}{3} \text{ seconds} \]

### Summary:
The half-life of the radioactive sample is \( \frac{8}{3} \) seconds.