Question

Two radioactive nuclei $P$ and $Q$, in a given sample decay into a stable nucleus $R$. At time $t=0$, number of $P$ species are $4N_0$ and that of $Q$ are $N_0$. Half-life of $P$ (for conversion to $R$) is $1$ minute where as that of $Q$ is $2$ minutes. Initially there are no nuclei of $R$ present in the sample. When number of nuclei of $P$ and $Q$ are equal, the number of nuclei of $R$ present in the sample would be

• A. $2N_0$
• B. $3N_0$
• C. $\frac{9N_0}{2}$
• D. $\frac{5N_0}{2}$

Answer 1

Answer: (C)

$T_{1/2}$
$N_P=N_Q$
$\frac{4N_0}{2^{t/1}}=\frac{N_0}{2^{t/2}}$
$4=2^{t/2}$
$2^2=2^{t/2}$
$\frac{t}{2}=2$
$\Rightarrow t=4min$
Disactive nucleus or Nuclei of $R$
$=\left(4N_0-\frac{4N_0}{2^4}\right)+\left(N_0-\frac{N_0}{2^2}\right)$
$=4N_0-\frac{N_0}{4}+N_0-\frac{N_0}{4}=5N_0-\frac{N_0}{2}$
$=\frac{9}{2}{N}_0$