Question

If a radioactive element with a half-life of 30 min undergoes beta decay. The fraction of the radioactive element that remains undecayed after 90 min is:

• A. \(\frac{1}{8}\)
• B. \(\frac{1}{4}\)
• C. \(\frac{1}{16}\)
• D. \(\frac{1}{2}\)

Hint:

For radioactive decay:

• Use the formula \[\frac{N}{N_0} = \left(\frac{1}{2}\right)^n\], where n is the number of half-lives.
• Calculate n by dividing the total time by the half-life.

Answer 1

The Correct Option is A

To determine the fraction of the radioactive element that remains undecayed after a certain period, we use the concept of half-life. The half-life (\(t_{1/2}\)) of a radioactive element is the time required for half of the radioactive nuclei in a sample to decay.

Given: The half-life of the radioactive element is 30 minutes.

We need to find out how much of the element remains undecayed after 90 minutes.

First, we calculate how many half-lives have elapsed in 90 minutes:

\[ \text{Number of half-lives} = \frac{\text{Total time elapsed}}{\text{Half-life}} = \frac{90 \, \text{minutes}}{30 \, \text{minutes}} = 3 \]

After each half-life, the fraction of the remaining undecayed radioactive element reduces to half. Therefore, the fraction remaining after 3 half-lives is:

\[ \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]

This calculation shows that after 90 minutes, \(\frac{1}{8}\) of the original radioactive element remains undecayed.

Conclusion: The correct answer is \(\frac{1}{8}\).