Question

An unstable radio nuclei $X$ can decay into two stable nuclei $Y$ and $Z$. A sample containing only $X$ is taken at $t=0$, three graphs $lo{g}_e\left(N_x\right)$ vs $t$, $N_y$ vs $t$, $N_z$ vs $t$ are drawn as shown below, here $N_x$, $N_y$ and $N_z$represents number of nuclei of $X$, $Y$ and $Z$ respectively any instant $t$.
Choose the correct statement(s) from the following :

• A. Decay constant for decay of $X$ into $Y$ is $\frac{btan\theta }{e^a}$
• B. Decay constant for decay of $X$ into $Y$ is $\frac{ctan\theta }{e^a}$
• C. Number of nuclei of $X$ at $t=0$ is $e^a$.
• D. None of these.

Answer 1

The correct option is C Number of nuclei of $X$ at $t=0$ is $e^a$.

$\begin{array}{c}\text{From Graph}1,\text{we conclude that}\\ \begin{array}{c}\ln \left(N_X\right)=-t\cdot \tan \theta +a\\ N_X=e^{a(-t)\cdot \tan \theta }\\ N_X=e^a\cdot e^{-t\tan \theta }\\ \text{at}t=0,\text{we have}{N}_X=e^a\end{array}\\ \Rightarrow \text{Hence, option}\left(c\right)\text{is correct.}\end{array}$