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, $t=\frac{1}{2 \lambda }$