Half-Life

Half-life T_½ is the time for half the nuclei in a sample to decay.

After n half-lives, fraction surviving = (1/2)ⁿ ⇒ N/N₀ = 1/2ⁿ.

T_½ is independent of starting amount or chemistry — depends ONLY on the isotope.

Relation to decay constant: T_½ = (ln 2)/λ ≈ 0.693/λ.

Examples: C-14 T_½ = 5730 yr; I-131 = 8.0 d; Po-214 = 164 μs; U-238 = 4.5 × 10⁹ yr; Tc-99m = 6 hours.

Activity also halves every T_½: A(t) = A₀ × (1/2)^(t/T_½).

Carbon-14 dating: dead organisms stop incorporating fresh C-14; the ratio C-14/C-12 decays exponentially.

Tc-99m (T_½ = 6 hr, γ-emitter) is the most widely used medical imaging isotope — short enough for safety, long enough for procedures.