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# The half life of $\;^{215}At\;$ is $\;200 \mu s\;$ . The time taken for radioactivity of a sample of $\;^{215}At\;$ to decay to $\;{\large\frac{1}{4}}^{th}\;$ of its initial value is

$(a)\;100\;\mu s\qquad(b)\;300\;\mu s\qquad(c)\;200\;\mu s\qquad(d)\;400\;\mu s$

Answer: $\;400\;\mu s$
If A is activity of radioactive substance after n half-lives and $\;R_{0}\;$ is initial activity, then $A=R_{0}\;(\large\frac{1}{2})^{n}$
Given $A=\large\frac{R_{0}}{4}$$, \rightarrow n = 2$
$\Rightarrow \;t=nt_{\frac{1}{2}}=2 \times 200 \mu s =400\; \mu s\;.$
edited Aug 12, 2014