Two radioactive substances A and B have decay constants 5λ and λ...

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(\dfrac{1}{e}\right)}^{2} will be

Options

A $\dfrac{1}{\lambda}$
B $\dfrac{1}{2\lambda}$
C $\dfrac{2}{\lambda}$
D $\dfrac{1}{4\lambda}$

Correct answer

B. $\dfrac{1}{2\lambda}$

Step-by-step solution

N = {N}_{0}e^{- \lambdat}

{N}_{1} = {N}_{0}e^{- 5\lambdat}

...........(i)

{N}_{2} = {N}_{0}e^{- \lambdat}

............(ii)
From equation (i) and (ii)

\dfrac{{N}_{1}}{{N}_{2}} = \dfrac{e^{- 5\lambdat}}{e^{- \lambdat}}

{\left(\dfrac{1}{e}\right)}^{2} = e^{- 4\lambdat}

4\lambdat = 2

t = \dfrac{1}{2\lambda}

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