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In a radioactive sample, ${ }_{10}^a K$ nuclei either decay into stable ${ }_{20}^{* 0} Ca$ nuclei with decay constant $4.5 \times 10^{-10}$ per year or into stable ${ }_{18}^{40}$ Ar muclei with decay constant $0.5 \times 10^{-10}$ per year. Given that in this sample all the stable ${ }_{20}^{\infty 0} Ca$ and ${ }_{15}^{20} Ar$ nuclei are produced by the ${ }_{19}^{* 0} K$ muclei only. In time $t \times 10^{\circ}$ years, if the ratio of the sum of stable ${ }_{30}^{40} Ca$ and ${ }_{15} \operatorname{An}$ nuclei to the radioactive ${ }_{19} K$ muclei is $99$ , the ralue of $t$ will be : [Given $\ln 10=2.3]$
$9.2$
$1.15$
$4.6$
$2.3$
Solution
$\lambda=\lambda_1+\lambda_2=5 \times 10^{-10} \text { per year }$
$N = N _0 e ^{-\lambda . t}$
$N _0- N = N _{\text {sable }}$
$N = N _{\text {rrdiooctire }}$
$\frac{ N _0}{ N }-1=99$
$\frac{ N _0}{ N }=100$
$\frac{ N }{ N _0}= e ^{-\lambda . t }=\frac{1}{100}$
$\Rightarrow \lambda t =2 \text { en10 }$
$=4.6$
$t =9.2 \times 10^9 \text { years }$
