question_answer

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

Answer:

                Instantaneous activity, \[R=-\frac{dN}{dt}=\lambda N\] The instantaneous rate of change of the  activity of a radioactive substance, \[\frac{dR}{dt}=\frac{d}{dt}(\lambda N)=\lambda \frac{dN}{dt}=\lambda (-\lambda N)\] \[=-{{\lambda }^{2}}N=-{{\left( \frac{{{\log }_{e}}2}{{{T}_{1/2}}} \right)}^{2}}N\] \[\therefore \]  \[\frac{dR}{dt}\propto \frac{1}{{{({{T}_{1/2}})}^{2}}}\]