Two radioactive materials and have decay constants and , respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of to that of will be after a time [2000]

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, then the ratio of the number of nuclei of $X_{1}$ to that of $X_{2}$ will be $\frac{1}{e}$ after a time [2000]

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(4)

$Number of nuclei of X_{1},$ $10 λ$ $and number of nuclei of X_{2}$ $N_{2} = N_{0} e^{- λ t} .$

$∴$ $\frac{N_{1}}{N_{2}} = \frac{e^{- 10 λ t}}{e^{- λ t}} = \frac{1}{e^{9 λ t}}$

$Given,$ $\frac{N_{1}}{N_{2}} = \frac{1}{e};$

$∴$ $\frac{1}{e^{9 λ t}} = \frac{1}{e}$ $(∵ \frac{N_{1}}{N_{2}} = \frac{1}{e} is given)$

$\Rightarrow 9 λ t = 1$ $∴$ $t = (\frac{1}{9 λ})$

$i.e., after time$ $t = \frac{1}{9 λ},$ $the ratio of the number of nuclei of$ $X_{1}$ $to that of$ $X_{2}$ $will be$ $\frac{1}{e} .$

Solution

Q.
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, then the ratio of the number of nuclei of $X_{1}$ to that of $X_{2}$ will be $\frac{1}{e}$ after a time [2000]

1 $\frac{1}{10 λ}$

2 $\frac{1}{11 λ}$

3 $\frac{11}{10 λ}$

4 $\frac{1}{9 λ}$

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Solution

Q. Two radioactive materials X1 and X2 have decay constants 10λ and λ, respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of X1 to that of X2 will be 1e after a time [2000]

1 110λ

2 111λ

3 1110λ

4 19λ