Question 404: Atoms & Nuclei

Question

If 'T' is the half life of a radioactive substance then its instantaneous rate of change of activity is proportional to

Accepted Answer

The instantaneous rate of change of activity of a radioactive substance is derived based on its half-life, $ T $.
Let's break it down step by step:
Activity Relationship
:
$    R = -\frac{dN}{dt} = \lambda N    $
Here, $ R $ is the activity, $ \lambda $ is the decay constant, and $ N $ is the number of undecayed nuclei.
Rate of Change of Activity
:
$    \frac{dR}{dt} = \frac{d}{dt}(\lambda N) = \lambda \frac{dN}{dt} = \lambda(-\lambda N) = -\lambda^2 N    $
Linking Decay Constant to Half-Life
:
$    \lambda = \frac{\log_e 2}{T_{1/2}}    $
This expression relates the decay constant $ \lambda $ to the half-life $ T_{1/2} $.
Substitute the Decay Constant
:
$    \lambda^2 N = \frac{(\log_e 2)^2 N}{T_{1/2}^2}    $
Proportionality of Rate of Change
:
$    \frac{dR}{dt} \propto \frac{1}{T_{1/2}^2}    $
Thus, the instantaneous rate of change of activity is inversely proportional to the square of the half-life, $ T_{1/2}^2 $.

Answer

T

Answer

T$^{-2}$

Answer

T$^{+2}$

Answer

T$^{-1}$

Atoms & Nuclei

If 'T' is the half life of a radioactive substance then its instantaneous rate of change of activity is proportional to

Choose the correct answer: