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The half-life of $^{238} _{92} U$ undergoing $\alpha$ -decay is $4.5 \times 10^{9}$ $years$. What is the activity of $1\; g$ sample of $^{238} _{92} U$?
$3.46 \times 10^{5}\; Bq$
$8.27 \times 10^{3}\; Bq$
$5.96 \times 10^{4}\; Bq$
$1.23 \times 10^{4}\; Bq$
Solution
$T_{1 / 2}=4.5 \times 10^{9}\, y$
$=4.5 \times 10^{9} y \times 3.16 \times 10^{7} \,s / y$
$=1.42 \times 10^{17}\, s$
One $k$ mol of any isotope contains Avogadro's number of atoms, and so lg of $^{238}_{92} U$ contains
$\frac{1}{238 \times 10^{-3}}\, kmol \times 6.025 \times 10^{26} \text { atoms } / kmol$
$=25.3 \times 10^{20}$ atoms.
The decay rate $R$ is $R=\lambda N$
$=\frac{0.693}{T_{1 / 2}} \,N=\frac{0.693 \times 25.3 \times 10^{20}}{1.42 \times 10^{17}} \,s ^{-1}$
$=1.23 \times 10^{4}\, s ^{-1}$
$=1.23 \times 10^{4}\; Bq$
