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

1. Number of Half-Lives: - Time elapsed: t = 90 min. - Half-life: T_1/2 = 30 min. - Number of half-lives:

\[n = \frac{t}{T_{1/2}} = \frac{90}{30} = 3.\]

2. Remaining Fraction: - Fraction remaining after n half-lives:

\[\frac{N}{N_0} = \left(\frac{1}{2}\right)^n = \left(\frac{1}{2}\right)^3 = \frac{1}{8}.\]

Final Answer:

\(\boxed{\frac{1}{8}}\)