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)

$\frac{1}{2 \lambda }$

Answer 2

$N=N_{0}e^{- \lambda t}$ $N_{1}=N_{0}e^{- 5 \lambda t}$ ...........(i) $N_{2}=N_{0}e^{- \lambda t}$ ............(ii) From equation (i) and (ii) $\frac{N_{1}}{N_{2}}=\frac{e^{- 5 \lambda t}}{e^{- \lambda t}}$ $\left(\frac{1}{e}\right)^{2}=e^{- 4 \lambda t}$ $4\lambda t=2$ $t=\frac{1}{2 \lambda }$