Question

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

• A. $\sqrt{T}$
• B. $T$
• C. $T^2$
• D. $T^{-2}$

Hint:

Shorter half-life means faster decay and a larger rate of change of activity.

Answer 1

The Correct Option is D

Step 1: Relation between decay constant and half-life.
\[ T = \frac{\ln 2}{\lambda} \Rightarrow \lambda \propto \frac{1}{T} \]

Step 2: Expression for activity.
Activity $A$ is given by:
\[ A = \lambda N \]

Step 3: Instantaneous rate of change of activity.
\[ \frac{dA}{dt} \propto \lambda^2 \]

Step 4: Substitute relation with half-life.
\[ \frac{dA}{dt} \propto \left(\frac{1}{T}\right)^2 = T^{-2} \]

Step 5: Conclusion.
The instantaneous rate of change of activity is proportional to $T^{-2}$.