Question

$ _{87}^{221} $ undergoes radioactive decay with a half-life of $4$ days. The probability that a $Ra$ nucleus will disintegrate in $8$ days is

• A. $ \frac{1}{2} $
• B. $ \frac{3}{8} $
• C. $ \frac{1}{4} $
• D. $ \frac{3}{4} $

Answer 1

Answer: (D)

Number of half-lives $ n=\frac{8}{4}=2 $
As, $ N=\frac{{{N}_{0}}}{{{2}^{n}}}=\frac{{{N}_{0}}}{{{2}^{2}}}=\frac{{{N}_{0}}}{4} $
Thus, remaining atom $ N=\frac{{{N}_{0}}}{4} $
Decayed atoms $ ={{N}_{0}}-N={{N}_{0}}-\frac{{{N}_{0}}}{4}=\frac{3}{4}{{N}_{0}} $
$ \therefore $ Probability of decay
$ =\frac{{{N}_{0}}-N}{{{N}_{0}}}=\frac{\frac{3}{4}{{N}_{0}}}{{{N}_{0}}}=\frac{3}{4} $