JEE Main · 2022 · 29 Jul 2022 · Shift 2 · Modern Physics

Two radioactive materials A and B have decay constants

25 λ

and

16 λ

respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of B to that of A will be "e" after a time

\frac{1}{aλ}

. The value of a is ______.

Answer: 9

Solution

The number of nuclei at time

t

is given by

N = N_{0} e^{- λ t}

.

For materials A and B with initial equal nuclei:

\frac{N _{B}}{N _{A}} = \frac{N _{0} e ^{- λ_{B} t}}{N _{0} e ^{- λ_{A} t}} = e^{(λ_{A} - λ_{B}) t}

Given that this ratio equals

e

e^{(λ_{A} - λ_{B}) t} = e^{1}

Therefore:

(λ_{A} - λ_{B}) t = 1

t = \frac{1}{λ _{A} - λ _{B}} = \frac{1}{25 λ - 16 λ} = \frac{1}{9 λ}

Comparing with

\frac{1}{aλ}

, we get

a = 9

JEE Main · 2022 · 29 Jul 2022 · Shift 2 · Integer answer

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