The half-lives of X and Y are 3 min and 27 | vedclass

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(C) The activity $A$ of a radioactive sample is given by $A = \lambda N$,where $\lambda$ is the decay constant and $N$ is the number of nuclei.Given t

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$1 : 9$

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(C) The activity $A$ of a radioactive sample is given by $A = \lambda N$,where $\lambda$ is the decay constant and $N$ is the number of nuclei.Given that the activities are equal,$A_X = A_Y$.Since $\lambda = \frac{\ln 2}{T_{1/2}}$,we have $\frac{\ln 2}{T_X} N_X = \frac{\ln 2}{T_Y} N_Y$.This simplifies to $\frac{N_X}{T_X} = \frac{N_Y}{T_Y}$.Rearranging for the ratio of nuclei,we get $\frac{N_X}{N_Y} = \frac{T_X}{T_Y}$.Given $T_X = 3 \ min$ and $T_Y = 27 \ min$,the ratio is $\frac{N_X}{N_Y} = \frac{3}{27} = \frac{1}{9}$.Thus,the ratio of the number of excited nuclei is $1 : 9$.

The half-lives of $X$ and $Y$ are $3 \ min$ and $27 \ min$ respectively. If they have the same activity at a certain instant,find the ratio of the number of excited nuclei of $X$ and $Y$ at that instant.

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