Question

$A$ 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

• A. $2\times 10^{17}$
• B. $2\times 10^{20}$
• C. $4\times 10^{23}$
• D. $4\times 10^{20}$

Answer 1

Answer: (A)

Decay occurs as
$X\stackrel{40000yr}{\to }Y\stackrel{20yr}{\to }Z$
As number of $Y$ nuclei does not changes with time,
this means decay rate of $X=$ decay rate of $Y.$
${\lambda }_X{N}_X={\lambda }_Y{N}_Y$
$\Rightarrow \frac{N_X}{T_X}=\frac{N_Y}{T_Y}$
$\Rightarrow N_Y=\frac{T_Y}{T_X}{N}_X$
$=\frac{20}{40000}\times 4\times 10^{20}$
$=2\times 10^{20}$
$=2\times 10^{17}$ nuclei