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 $4\, N_0$ 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. $2\,N_0$
• B. $3\,N_0$
• C. $\frac{9\,N_0}{2}$
• D. $\frac{5\,N_0}{2}$

Answer 1

Answer: (C)

$\frac{9\,N_0}{2}$

Answer 2

$T _{1 / 2}$
$N _{ P }= N _{ Q }$
$\frac{4 N _{0}}{2^{ t / 1}}=\frac{ N _{0}}{2^{ t / 2}}$
$4=2^{ t / 2}$
$2^{2}=2^{ t / 2}$
$\frac{ t }{2}=2$
$\Rightarrow t =4 \,min$
Disactive nucleus or Nuclei of $R$
$=\left(4 N _{0}-\frac{4 N _{0}}{2^{4}}\right)+\left( N _{0}-\frac{ N _{0}}{2^{2}}\right)$
$=4 N _{0}-\frac{ N _{0}}{4}+ N _{0}-\frac{ N _{0}}{4}=5 N _{0}-\frac{ N _{0}}{2}$
$=\frac{9}{2} N _{0}$