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

JEE Main · 2019 · 10 Apr 2019 · Shift 2 · Modern Physics

Two radioactive substances A and B have decay constants

5 λ

and

λ

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

(\frac{1}{e})^{2}

will be

(A) $1/ λ$
(B) $1/4 λ$
(C) $2/ λ$
(D) $1/2 λ$ ✓ Correct

Answer: (D)

Solution

Each sample thins out exponentially,

N = N_{0} e^{- λ t}

, so the ratio of the two populations is

\frac{N _{A}}{N _{B}} = \frac{e ^{- 5 λ t}}{e ^{- λ t}} = e^{- 4 λ t}

. Setting this equal to

(\frac{1}{e})^{2} = e^{- 2}

forces

4 λ t = 2

, hence

t = \frac{1}{2 λ}

JEE Main · 2019 · 10 Apr 2019 · Shift 2 · Single correct option

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