Two radioactive substances A and B have decay constants 5 λ and λ respectively. At t = 0 they have the same number of nuclei. The ratio of number of nuclei of A to those of B will be ${(\frac{1}{e})}^{2}$ after a time interval

$\frac{1}{4 λ}$

4λ

2λ

$\frac{1}{2 λ}$

Solution

$\frac{1}{2 λ}$

Number of nuclei remained after time t can be written as
N = N_o e^-^λt
where N_o is initial number of nuclei of both the substances.

N₁ = N_o e^-5λt ....... (i)

and N₂ = N_o e^-λt ...... (ii)

Dividing Eqⁿ (i) by Eqⁿ (ii), we obtain

$\frac{N_{1}}{N_{2}} = e^{(- 5 λ + λ) t} = e^{- 4 λt} = \frac{1}{e^{4 λt}}$

But we have given

$\frac{N_{1}}{N_{2}} = {(\frac{1}{e})}^{2} = \frac{1}{e^{2}} \frac{1}{e^{2}} = \frac{1}{e^{4 λt}}$

Comparing the powers we get

2 = 4λt

$t = \frac{2}{4 λ} t = \frac{1}{2 λ}$

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