Question

Two radioactive substances $X$ and Y originally have $N _{1}$ and $N _{2}$ nuclei respectively. Half life of $X$ is half of the half life of $Y$. After three half lives of $Y,$ number of nuclei of both are equal. The ratio $\frac{ N _{1}}{ N _{2}}$ will be equal to:

• A. $\frac{1}{8}$
• B. $\frac{3}{1}$
• C. $\frac{8}{1}$
• D. $\frac{1}{3}$

Answer 1

Answer: (C)

$T _{ x }= t ; T _{ y }=2 t$
$3 T _{ y }=6 t$
$N _{1}^{\prime}= N _{2}^{\prime}$
$N _{1} e ^{-\lambda_{1} 6 t }= N _{2} e ^{-\lambda_{2} 6 t }$
$\frac{ N _{1}}{ N _{2}}= e ^{\left(\lambda_{1}-\lambda_{2}\right) 6 t }= e ^{\ln 2\left(\frac{1}{t}-\frac{1}{2 t }\right) \times 6 t }= e ^{( In 2) \times 3}= e ^{ ln 8}=8$
$\frac{ N _{1}}{ N _{2}}=\frac{8}{1}$