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, $\frac{N}{N_0}=(\frac{1}{2}{)}^n$
Where $n$ is the number of half-lives
$\Rightarrow \frac{5}{20}=(\frac{1}{2}{)}^n$
$n=2$
Total time $=2\times 5\times 10^9$ years
$=10^{10}yr$