Question

Two radioactive materials $X_1$, and $X_2$ have decay constants 5$\lambda$ and $\lambda$ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of X$_1$ to that X$_2$ will be 1/e after a time

• A. $1/4\lambda$
• B. e/$\lambda$
• C. $\lambda$
• D. $\frac{1}{2}\lambda$

Answer 1

Answer: (A)

$1/4\lambda$

Answer 2

$X_1=N_0e^{-\lambda 1 t} ; X_2=N_0e^{-\lambda 2t}$
$\frac{X_1}{X_2}=e^{-1}=e^{(-\lambda_1 +\lambda_2)t}; e^{-1}=e^{(\lambda_1 - \lambda_2)t}$
$\therefore \, \, \, t=\bigg|\frac{1}{\lambda_1-\lambda_2}\bigg|=\frac{1}{(5\lambda-\lambda)}=\frac{1}{4\lambda}$