Physics - Modern Physics Question with Solution | TestHub

The activity of a radioactive substance is given by $A=\lambda N$.

The instantaneous rate of change of activity is $\frac{dA}{dt}=\frac{d}{dt}(\lambda N)=\lambda \frac{dN}{dt}$.

Since the rate of decay is $\frac{dN}{dt}=-\lambda N$, we get $\frac{dA}{dt}=-{\lambda }^{2}N$.

The decay constant $\lambda$ is related to the half-life $T$ by $\lambda =\frac{\ln 2}{T}$.

Substituting the value of $\lambda$, we get $\frac{dA}{dt}=-{\left(\frac{\ln 2}{T}\right)}^{2}N$.

Thus, for a given number of nuclei $N$, the instantaneous rate of change of activity is proportional to $T^{-2}$.

Answer: ${\mathrm{T}}^{-2}$