Question

8. prove that the instantaneous rate of change of activity of a radioactive substance is inversely proportional to the square of its half life.

Answer 1

instantaneous activity, R = $-\frac{dN}{dt}=\lambda N$

now, the instantaneous rate of change of the activity of radioactive substance is...

$\frac{dR}{dt}=\frac{d}{dt}(\lambda N)$

= $\lambda\frac{dN}{dt}$

=$\lambda(-\lambda N)$

=$-\lambda^2N$

we know, in half life reaction, $T_{1/2}=\frac{ln2}{\lambda}$

so, $\frac{dR}{dt}=-\left(\frac{ln2}{T_{1/2}}\right)^2N$

it is clear that, $\frac{dR}{dt}\propto\frac{1}{T^2_{1/2}}$

hence, instantaneous rate of change of activity of a radioactive substance is inversely proportional to the square of its half life.