Two radioactive materials X _ 1 and X _ 2 have decay constants 10λ...

Two radioactive materials ${X}_{1}and{X}_{2}$ have decay constants $10\lambda and\lambda$ respectively. If initially, they have the same number of nuclei, then the ratio of the number of nuclei of ${X}_{1}$ to that of ${X}_{2}$ will be $\dfrac{1}{e}$ after a time

Options

A $\dfrac{1}{10\lambda}$
B $\dfrac{1}{11\lambda}$
C $\dfrac{11}{10\lambda}$
D $\dfrac{1}{9\lambda}$

Correct answer

D. $\dfrac{1}{9\lambda}$

Step-by-step solution

Here,

\dfrac{{N}_{{x}_{1}} ( t )}{{N}_{{x}_{2}} ( t )} = \dfrac{1}{e}

or\dfrac{{N}_{0}e^{- 10\lambdat}}{{N}_{0}e^{- \lambdat}} = \dfrac{1}{e}

(Because initially, both have the same number of nuclei,

{N}_{0}

).

or

e = \dfrac{e^{- \lambdat}}{e^{- 10\lambdat}} = e^{9\lambdat}

9\lambdat = 1

t = \dfrac{1}{9\lambda}

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