Question

$Uranium-238$ decays to $Thorium-234$ with half-life $5\times 10^{9} \, yr$ . The resulting nucleus is in the excited state and hence further emits $\gamma $ -rays to come to the ground state. It emits $20 \, \gamma $ -rays per second. The emission rate will drop to $5 \, \gamma $ -rays per second in

• A. $1.25\times 10^{9} \, yr$
• B. $10^{10} \, yr$
• C. $10^{- 8} \, yr$
• D. $1.25\times 10^{- 9} \, s$

Answer 1

Answer: (B)

Number of emitted $\gamma $ -rays per second is proportional to no of uranium particles left undecayed
fraction of particles undecayed, $\frac{N}{N_{0}} = \left(\right. \frac{1}{2} \left.\right)^{n}$
Where $n$ is the number of half-lives
$\Rightarrow \frac{5}{20} = \left(\right. \frac{1}{2} \left.\right)^{n}$
$n = 2$
Total time $= 2 \times 5 \times 10^{9}$ years
$= 10^{10} yr$