Physics Nuclear Physics MDCAT 2016
PMDC Verified Question 67 of 98
The relation between decay constant '\(\lambda\)' and half-life '\(T_{1/2}\)' of radioactive substance is:
A
\( \lambda = \frac{1}{T_{1/2}} \)
B
\( \lambda = T_{1/2} \)
C
\( \lambda = 0.693 \, T_{1/2} \)
D
\( \lambda = \frac{0.693}{T_{1/2}} \)
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Propolis Cognitive Error Autopsy

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


Concept:

The half-life of a radioactive isotope is mathematically derived from the radioactive decay law \(N = N_0 e^{-\lambda t}\).

Formula:

$$ T_{1/2} = \frac{\ln(2)}{\lambda} $$

Solution:

  • The natural logarithm of 2 (\(\ln 2\)) is approximately 0.693.


  • Substitute this into the formula: $$ T_{1/2} = \frac{0.693}{\lambda} $$


  • Rearrange algebraically to solve for \(\lambda\): $$ \lambda = \frac{0.693}{T_{1/2}} $$


Why other options are incorrect:

Option A lacks the \(\ln 2\) constant. Option B incorrectly implies they are equal. Option C proposes a direct proportionality, but they are inversely proportional.

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