If $\mathrm{T}$ is the half-life of a radioactive substance then its instantaneous rate of change of…
If $\mathrm{T}$ is the half-life of a radioactive substance then its instantaneous rate of change of activity is proportional to
$\sqrt{\mathrm{T}}$
$\mathrm{T}$
$\mathrm{T}^{2}$
$\mathrm{~T}^{-2}$
Solution
For a radioactive substance, the rate of change of activity is related to the decay constant and the halflife. The instantaneous rate of change of activity is proportional to the negative of the activity itself.
If $T$ is the half-life of the substance, then the rate of change of activity $\frac{d A}{d t}$ is proportional to $-A$, where $A$ is the activity at any given time.
Thus, Answer: (4) is correct, indicating that the rate of change is proportional to the activity, which decreases over time.