Half-Life (\(t_{1/2}\))

The half-life of a radioactive isotope is the time required for half of the nuclei in a sample to decay.

• It does not depend on:
  
  • Amount of material.
  • Physical conditions (temperature, pressure, etc.).
• The number of undecayed nuclei decreases exponentially with time.

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

Where: 

• \(N(t)\) = number of nuclei at time \(t\) 
• \(N_0\) = initial number of nuclei
• \(\lambda\) = decay constant (probability of decay per unit time)

Relationship between half-life and decay constant: $ t_{1/2} = \dfrac{\ln 2}{\lambda} $

Points about Half-Life:

• After one half-life → \( \dfrac{1}{2} \) of the nuclei remain.
• After two half-lives → \( \dfrac{1}{4} \) remain.
• After three half-lives → \( \dfrac{1}{8} \) remain, and so on.
• This pattern is exponential decay, not linear.

Applications of Half-Life:

• Carbon dating: Using \(^{14}\text{C}\) to estimate the age of fossils and artifacts (useful up to ~50,000 years).
• Medical tracers: Using isotopes like \(^{99m}\text{Tc}\) (technetium-99m) with short half-lives for imaging, to minimize patient exposure.
• Nuclear waste management: Knowing the half-life helps determine safe storage times.