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A parent nucleus $X$ is decaying into daughter nucleus $Y$ which in turn decays to $Z$. The half lives of $X$ and $Y$ are $40000 \,yr$ and $20 \,yr$, respectively. In a certain sample, it is found that the number of $Y$ nuclei hardly changes with time. If the number of $X$ nuclei in the sample is $4 \times 10^{20}$, the number of $Y$ nuclei present in it is
$2 \times 10^{17}$
$2 \times 10^{20}$
$4 \times 10^{23}$
$4 \times 10^{20}$
Solution
(a)
Decay occurs as
$X \stackrel{40000 yr }{\longrightarrow} Y \stackrel{20 yr }{\longrightarrow} Z$
As number of $Y$ nuclei does not changes with time, this means decay rate of $X=$ decay rate of $Y$.
$\Rightarrow \lambda_X N_X =\lambda_Y N_Y$
$\Rightarrow \frac{N_X}{T_X} =\frac{N_Y}{T_Y}$
$\Rightarrow N_Y =\frac{T_Y}{T_X} N_X$
$=\frac{20}{40000} \times 4 \times 10^{20}$
$=2 \times 10^{17}$ nuclei
