Physics Nuclear Physics DUHS 2022
PMDC Verified Question 43 of 98
Half-life of a radioactive sample is given by:
A
\( T_{1/2} = \frac{(0.693)}{\sqrt{\lambda}} \)
B
\( T_{1/2} = \sqrt{\frac{0.693}{\lambda}} \)
C
\( T_{1/2} = (0.693)\lambda \)
D
\( T_{1/2} = \frac{0.693}{\lambda} \)
Tap any option to test your recall and reveal the step-by-step Propolis autopsy.

Propolis Cognitive Error Autopsy

Official Correct Choice:
Option D: \( T_{1/2} = \frac{0.693}{\lambda} \)


Concept:

The half-life represents the time taken for an exponentially decaying quantity to decrease to half of its initial value.

Formula:

$$ N = N_0 e^{-\lambda t} $$

Solution:

  • Set \(N = \frac{N_0}{2}\) at \(t = T_{1/2}\): $$ \frac{1}{2} = e^{-\lambda T_{1/2}} $$


  • Take the natural logarithm (\(\ln\)) of both sides: $$ \ln\left(\frac{1}{2}\right) = -\lambda T_{1/2} $$


  • $$ -0.693 = -\lambda T_{1/2} \implies T_{1/2} = \frac{0.693}{\lambda} $$


  • The half-life is defined exactly as \(\ln(2)\) divided by the decay constant \(\lambda\).


Why other options are incorrect:

The other options feature mathematically invalid square roots or multiplications that violate the exponential nature of decay.

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