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>> $Two radioactive materials X1 and X2 have$

Question

Two radioactive materials $X_{1}$ and $X_{2}$ have decay constants $10 λ$ and $λ$ , respectively. If initially they have the same number of nuclei, the ratio of the number of nuclei of $X_{1}$ to that of $X_{2}$ be $1 / e$ after a time

A

$\frac{1}{1 0 λ}$

B

$\frac{1}{1 1 λ}$

C

$\frac{1 1}{1 0 λ}$

D

$\frac{1}{9 λ}$

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Updated on : 2022-09-05

Solution

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Correct option is D)

Considering first order decay process, and same no. of initial no. of atoms:
$X_{1} = N_{o} e^{- 10 λ t}$
$X_{2} = N_{o} e^{- λ t}$
$\frac{X _{1}}{X _{2}} = \frac{e ^{- 10 λ t}}{e ^{- λ t}}$
$\frac{X _{1}}{X _{2}} = e^{- 9 λ t}$

Determining the time when, $\frac{X _{1}}{X _{2}} = \frac{1}{2}$
$\frac{1}{e} = e^{- 9 λ t}$
$t = \frac{1}{9 λ}$

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