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(D) The activity $A$ of a radioactive substance at any time $t$ is given by $A = A_0 e^{-\lambda t}$,where $\lambda$ is the decay constant.The decay c

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$T^{-2}$

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(D) The activity $A$ of a radioactive substance at any time $t$ is given by $A = A_0 e^{-\lambda t}$,where $\lambda$ is the decay constant.The decay constant $\lambda$ is related to the half-life $T$ by the relation $\lambda = \frac{\ln 2}{T}$.The rate of change of activity is given by $\frac{dA}{dt} = \frac{d}{dt}(A_0 e^{-\lambda t}) = -\lambda A_0 e^{-\lambda t} = -\lambda A$.The magnitude of the rate of change of activity is $|\frac{dA}{dt}| = \lambda A$.Since $\lambda = \frac{\ln 2}{T}$,we have $|\frac{dA}{dt}| = \frac{\ln 2}{T} A$.Thus,the rate of change of activity is proportional to $T^{-1}$. However,looking at the instantaneous rate of change of activity $\frac{dA}{dt} = -\lambda^2 N_0 e^{-\lambda t}$,where $N = N_0 e^{-\lambda t}$.Since $A = \lambda N$,then $\frac{dA}{dt} = -\lambda A = -\lambda (\lambda N) = -\lambda^2 N$.Substituting $\lambda = \frac{\ln 2}{T}$,we get $\frac{dA}{dt} \propto \lambda^2 \propto (\frac{1}{T})^2 = T^{-2}$.Therefore,the rate of change of activity is proportional to $T^{-2}$.

If $T$ is the half-life of a radioactive substance,then its instantaneous rate of change of activity is proportional to

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