The half-life of `""^(215)At` is `100mus` . Find the time taken for the radioactivity of a sample of `""^(215)` At to decay to `1/(16)th` of its initial value:

Options

A. `400mus`
B. `6.3mus`
C. `40mus`
D. `300mus`

Answer

To solve the problem of finding the time taken for the radioactivity of a sample of \(^{215}\text{At}\) to decay to \( \frac{1}{16} \) of its initial value, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Half-Life**:
The half-life of a radioactive substance is the time taken for half of the radioactive nuclei in a sample to decay. For \(^{215}\text{At}\), the half-life is given as \(100 \, \mu s\).

2. **Determine the Initial and Final Values**:
Let the initial amount of the sample be \(A_0\). We need to find the time taken for the amount to decay to \( \frac{1}{16} A_0\).

3. **Calculate the Number of Half-Lives Required**:
To find out how many half-lives it takes for the sample to decay to \( \frac{1}{16} A_0\), we can express this as follows:
- After 1 half-life: \(A_0 \to \frac{1}{2} A_0\)
- After 2 half-lives: \(A_0 \to \frac{1}{4} A_0\)
- After 3 half-lives: \(A_0 \to \frac{1}{8} A_0\)
- After 4 half-lives: \(A_0 \to \frac{1}{16} A_0\)

Thus, it takes 4 half-lives to decay to \( \frac{1}{16} A_0\).

4. **Calculate the Total Time Taken**:
Since each half-life is \(100 \, \mu s\), the total time taken for 4 half-lives is:
\[ \text{Total Time} = 4 \times \text{Half-Life} = 4 \times 100 \, \mu s = 400 \, \mu s \]

5. **Conclusion**:
Therefore, the time taken for the radioactivity of the sample of \(^{215}\text{At}\) to decay to \( \frac{1}{16} \) of its initial value is \(400 \, \mu s\).

### Final Answer:
The time taken is \(400 \, \mu s\).

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