Question

Two radioactive nuclei $P$ and $Q$, in a given sample decay into a stable nucleous $R$. 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. $3N_0$
• B. $\frac{9N_0}{2}$
• C. $\frac{5N_0}{2}$
• D. $2N_0$

Answer 1

The correct option is B $\frac{9N_0}{2}$

Initially $P\to 4N_0$
$Q\to N_0$
Half life, $T_P=1min$
$T_Q=2min$
Let after time number of nuclei of $P$ and $Q$ are equal that is:
$\frac{4N_0}{2^{t/1}}=\frac{N_0}{2^{t/2}}$

$\frac{4}{2^{t/2}}=1$ or $t=4min$

Ao at $t=4min$:
$N_P=\frac{\left(4N_0\right)}{2^{4/1}}=\frac{N_0}{4}$

At $t=4min$

$N_Q=\frac{N_0}{2^{4/2}}=\frac{N_0}{4}$

Population of $R$:
$=(4N_0-\frac{N_0}{4})+(N_0-\frac{N_0}{4})$$=\frac{9N_0}{2}$