Question

A radioactive substance $X$ decays into another radioactive substance $Y$. Initially only $X$ was present, $\lambda x$ and $\lambda y$ are the disintegration constants of $X$ and $Y$. $N_x$ and $N_y$ are the number of nuclei of $X$ and $Y$ at any time $t$. Number of nuclei $N_y$ will be maximum when

• A. $\frac{N_y}{N_x-N_y}=\frac{{\lambda }_y}{{\lambda }_x-{\lambda }_y}$
• B. $\frac{N_x}{N_x-N_y}=\frac{{\lambda }_x}{{\lambda }_x-{\lambda }_y}$
• C. $\frac{{\lambda }_y}{N_y}=\frac{{\lambda }_x}{N_x}$
• D. ${\lambda }_y{N}_y={\lambda }_x{N}_x$

Answer 1

Answer: (D)

${\lambda }_y{N}_y={\lambda }_x{N}_x$

Net rate of formation of $y$ at any time $t$ is given by:

$\frac{d{N}_y}{dt}={\lambda }_x{N}_x-{\lambda }_y{N}_y$

$N_y$ is maximum when $\frac{d{N}_y}{dt}=0$

or, ${\lambda }_x{N}_x={\lambda }_y{N}_y$