Question

The activity of two radioactive nuclei Q and R is plotted on a semi-log graph. The half-life and decay constant of Q and R are (${\tau }_Q,{\lambda }_Q$) and (${\tau }_R,{\lambda }_R$) respectively. The plot implies that (the symbol "$\approx$" means "approximately equal to")

• A. ${\tau }_Q\approx {\tau }_R$ and ${\lambda }_Q>{\lambda }_R$
• B. ${\tau }_Q>{\tau }_R$ and ${\lambda }_Q<{\lambda }_R$
• C. ${\tau }_Q\approx {\tau }_R$ and ${\lambda }_Q\approx {\lambda }_R$
• D. ${\tau }_Q<{\tau }_R$ and ${\lambda }_Q>{\lambda }_R$

Answer 1

The correct option is D ${\tau }_Q<{\tau }_R$ and ${\lambda }_Q>{\lambda }_R$

We know that:-

$A=A_o{e}^{-\lambda t}$

$⟹\frac{dA}{dt}=-\lambda A_o{e}^{-\lambda t}$

At $t=0,\frac{dA}{dt}=-\lambda A_o$

Since,magnitude of slope of graph for $Q$ is greater than $R$ at $t=0$, hence

${\lambda }_Q>{\lambda }_R$

Again, half-life $T\propto \frac{1}{\lambda }$

Hence, $T_R>T_Q$

Hence, answer is option-(D).