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Ther percentage of ${ }^{235} U$ presently on earth is $0.72$ and the rest $(99.28 \%)$ may be taken to be ${ }^{233} U$. Assume that all uranium on earth was produced in a supernova explosion long ago with the initial ratio ${ }^{235} U /^{335} U =2.0$. How long ago did the supernova event occur? (Take the half-lives of ${ }^{235} U$ and ${ }^{238} U$ to be $7.1 \times 10^5$ years and $4.5 \times 10^{9}$ years respectively)
$4 \times 10^9$ years
$5 \times 10^9$ years
$6 \times 10^9$ years
$7 \times 10^9$ years
Solution
(D)
$N _{01}$ for ${ }^{235} U$
$N _{02}$ for $^{235} U$
$\frac{N_1}{N_2}=\frac{N_{01}}{N_{02}} \frac{e^{-\lambda_1 t}}{e^{-\lambda_2 t}}$
$\frac{0.72}{99.28}=2 e ^{-\left(\lambda_1-\lambda_2\right) t}$
$\Rightarrow \frac{2 \times 99.28}{0.72}= e ^{\left(\lambda_1-\lambda_2\right) t }$
$\Rightarrow\left(\lambda_1-\lambda_2\right) t=\ln \left(\frac{2 \times 99.28}{0.72}\right)=5.62$
$\Rightarrow\left(\frac{1}{7.1}-\frac{1}{45}\right) t =\frac{5.62}{\ln 2} \times 10^8$
$\Rightarrow t =\frac{8.109}{0.1186} \times 10^8 \approx 7 \times 10^9$ years
