Half-Life
Stacked halving boxes — 4 isotopes (C-14, I-131, U-238, P-32).
Key Notes
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.
Formulas
Half-life decay
After n half-lives, 2⁻ⁿ remain.
Half-life vs decay constant
Direct relation.
Mean lifetime
Average time before any single nucleus decays.
Carbon-14 dating
Determines age from current vs original ¹⁴C activity.
Important Points
T_½ refers to STATISTICAL behaviour — a SINGLE nucleus has no 'half-life'. After T_½, each nucleus has 50% chance of having decayed.
Activity halves at the same rate as N — both follow the same exponential.
Memorise (1/2)^n: 1, ½, ¼, ⅛, 1/16, 1/32, … Useful for 'after 5 half-lives' questions.
After 10 half-lives, only ~0.1% (1/1024) of the initial amount remains.
Carbon-14 dating works up to ~50,000 years (after that, activity is too low to measure reliably).
Half-life is NOT the same as mean lifetime — τ ≈ 1.44 T_½.
Half-Life notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 12 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.