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Question

Physics Question on deccay rate

Two radioactive materials X1X_1 and X2X_2 have decay constants 10λ10\lambda and λ\lambda respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of X1X_1 to that of X2X_2 will be 1/e after a time

A

1/10λ1/10\lambda

B

1/11λ1/11\lambda

C

11/10λ11/10\lambda

D

1/9λ1/9\lambda

Answer

1/9λ1/9\lambda

Explanation

Solution

Nx1(t)Nx2(t)Ie\frac{N_{x1}(t)}{N_{x2}(t)}\frac{I}{e}
N0e10λtN0eλt=Ie\frac{N_0 e^{-10\lambda t}}{N_0e^{-\lambda t}}=\frac{I}{e}
(Initially, both have same number of nuclei say N0N_0)
or e=eλt/e10λte=e^{\lambda t/e-10\lambda t} or e=e9λte=e^{9\lambda t}
or $$9\lambda t=1 or t=19λt=\frac{1}{9\lambda}