Question

A sample has two isotopes $A$ and $B$ having masses $50\text{g}$ and $30\text{g}$ respectively. $A$ is radioactive and $B$ is stable. A decays to $A^{\prime }$ by emitting $\alpha$ particles. The half life of $A$ is $2\text{hrs}$. Mass numbers of $A$ and $A^{\prime }$ are $150$ and $146$ respectively. In $4$ hour, if the number of $\alpha$ particles emitted is $\frac{N_A}{n}$ where $N_A=$ Avogadro’s number, then $n=$

Answer 1

As B is stable, its content remains same, while A decays two half lives, so one fourth of A is left.
Hence, after $4\text{hrs}$, sample will contain
$30\text{gm}$ ... B
$12.5\text{gm}$ ... A
$\frac{146}{150}\left(37.5\right)=36.5\text{gm}$....A'
Total mass of the sample after $4\text{hrs}$ is
$30+12.5+36.5=79\text{gm}$
Initial mass of the sample is $80\text{gm}$
Mass of the alpha particle is $80-79=1\text{gm}$
No. of $\alpha$ particles emitted $\frac{m_{\alpha }}{M_{\alpha }}\left(N_A\right)=\frac{1}{4}\left(N_A\right)$
$\therefore n=4$