Question

Two radioactive substances $A$ and $B$ have decay constants $5\lambda$ and $\lambda$ respectively. At $t=0$ , a sample has the same number of the two nuclei. The time taken for the ratio of the number of nuclei to become ${\left(\frac{1}{e}\right)}^2$ will be

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

Answer 1

Answer: (B)

Using the decay equation, $N=N_0{e}^{-\lambda t}$
For radioactive substance $A$ , $N_1=N_0{e}^{-5\lambda t}$ ...........(i)
For radioactive substance $B$ , $N_2=N_0{e}^{-\lambda t}$ ............(ii)
From equation (i) and (ii)
$\frac{N_1}{N_2}=\frac{e^{-5\lambda t}}{e^{-\lambda t}}$
${\left(\frac{1}{e}\right)}^2=e^{-4\lambda t}$
$4\lambda t=2$
Time taken, $t=\frac{1}{2\lambda }$