Question

Two radioactive nuclei $A$ and $B$ have their disintegration constant $\lambda_{A}$ and $\lambda_{B}$ respectively initially $N_{A}$ and $N_{B}$ number of nuclei are taken then the time after which their undisintegrated nucleus are same is:

• A. $\frac{1}{\lambda_{ A }+\lambda_{ B }} \ln \frac{ N _{ B }}{ N _{ A }}$
• B. $\frac{\lambda_{A} \lambda_{B}}{\lambda_{A}-\lambda_{B}} \ln \frac{N_{B}}{N_{A}}$
• C. $\frac{1}{\lambda_{ A }-\lambda_{ B }} \ln \frac{ N _{ A }}{ N _{ B }}$
• D. $\frac{1}{\lambda_{A}-\lambda_{B}} \ln \frac{N_{B}}{N_{A}}$

Answer 1

Answer: (C)

$N _{ A } e ^{-\lambda_{A}{ }{t}}= N _{ B } e ^{-\lambda_{ B }t}$
$e ^{\left(\lambda_{ B }-\lambda_{A}\right) t}=\frac{ N _{ B }}{ N _{ A }}$
$\left(\lambda_{B}-\lambda_{A}\right) t=\ln \frac{N_{B}}{N_{A}}$
$t =\frac{1}{\lambda_{ B }-\lambda_{ A }} \ln \frac{ N _{ B }}{ N _{ A }}$