Question

The half-life of $²15$At is 100 $μ$s. The time taken for the radioactivity of a sample of this nucleus to decay to $\frac116$th of its initial value is

• A. 6.3 $μ$s
• B. 40 $μ$s
• C. 300 $μ$s
• D. 400 $μ$s

Hint:

$N = N₀ (1/2)ᵗ/T$ for radioactive decay.

Answer 1

The Correct Option is D

To solve the problem of determining the time taken for the radioactivity of a sample to decay to \(\frac{1}{16}\)th of its initial value, we need to apply the concept of half-life. The half-life of a radioactive substance is the time it takes for half of the substance to decay.

Step-by-step Solution:

1. The half-life of the isotope \(^{215}\text{At}\) is given as 100 \(\mu \text{s}\).
2. We want the radioactivity to decrease to \(\frac{1}{16}\)th of its initial value. A fraction of \(\frac{1}{16}\) corresponds to 4 half-lives because:
   • After 1 half-life: \(\frac{1}{2}\) remains.
   • After 2 half-lives: \(\frac{1}{4}\) remains.
   • After 3 half-lives: \(\frac{1}{8}\) remains.
   • After 4 half-lives: \(\frac{1}{16}\) remains.
3. We calculate the total time for 4 half-lives:

\text{Total time} = 4 \times \text{Half-life} = 4 \times 100 \, \mu \text{s}

1. This gives:

\text{Total time} = 400 \, \mu \text{s}

Hence, the time taken for the radioactivity to decay to \(\frac{1}{16}\)th of its initial value is 400 \(\mu \text{s}\).

Conclusion:

Thus, the correct answer is 400 μs.