At a given instant, there are 25% undecayed radioactive nuclei in a sample. After 10 seconds the number of undecayed nuclei reduces to 12.5%, the mean life of the nuclei is

Options

A. 10.21 s
B. 14.43 s
C. 5.31 s
D. 7.43 s

Explanation

To find the mean life (or average life) of the radioactive nuclei given the information in the question, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- At time \( t = 0 \), let the number of undecayed nuclei be \( N_0 \).
- At a given instant, 25% of the nuclei are undecayed, which means:
\[ N(t) = \frac{N_0}{4} \]
- After 10 seconds, the number of undecayed nuclei reduces to 12.5%, which means:
\[ N(t + 10) = \frac{N_0}{8} \]

2. **Relating the Time to Half-Life**:
- The decay of radioactive nuclei follows the exponential decay law. The fraction of undecayed nuclei after time \( t \) can be expressed in terms of half-life \( t_{1/2} \):
\[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \]
- For \( t = 0 \) to \( t \) (where \( N(t) = \frac{N_0}{4} \)):
\[ \frac{N_0}{4} = N_0 \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \]
Simplifying gives:
\[ \frac{1}{4} = \left( \frac{1}{2} \right)^{\frac{t}{t_{1/2}}} \]
- Recognizing that \( \frac{1}{4} = \left( \frac{1}{2} \right)^2 \), we find:
\[ \frac{t}{t_{1/2}} = 2 \implies t = 2 t_{1/2} \]

3. **Finding the Next Time Interval**:
- After 10 seconds, the number of undecayed nuclei becomes \( \frac{N_0}{8} \):
\[ N(t + 10) = N_0 \left( \frac{1}{2} \right)^{\frac{t + 10}{t_{1/2}}} \]
- Setting this equal to \( \frac{N_0}{8} \):
\[ \frac{N_0}{8} = N_0 \left( \frac{1}{2} \right)^{\frac{t + 10}{t_{1/2}}} \]
Simplifying gives:
\[ \frac{1}{8} = \left( \frac{1}{2} \right)^{\frac{t + 10}{t_{1/2}}} \]
- Recognizing that \( \frac{1}{8} = \left( \frac{1}{2} \right)^3 \), we find:
\[ \frac{t + 10}{t_{1/2}} = 3 \]

4. **Setting Up the Equations**:
- We have two equations:
1. \( t = 2 t_{1/2} \)
2. \( t + 10 = 3 t_{1/2} \)

5. **Substituting and Solving**:
- Substitute \( t = 2 t_{1/2} \) into the second equation:
\[ 2 t_{1/2} + 10 = 3 t_{1/2} \]
- Rearranging gives:
\[ 10 = 3 t_{1/2} - 2 t_{1/2} \implies t_{1/2} = 10 \text{ seconds} \]

6. **Finding the Mean Life**:
- The mean life \( \tau \) is related to the half-life \( t_{1/2} \) by:
\[ \tau = \frac{t_{1/2}}{\ln(2)} \]
- Substituting \( t_{1/2} = 10 \):
\[ \tau = \frac{10}{\ln(2)} \approx \frac{10}{0.693} \approx 14.43 \text{ seconds} \]

### Final Answer:
The mean life of the nuclei is approximately \( 14.43 \) seconds.