What is Half-Life?
The half-life of a substance is the time required for half of it to decay (radioactive isotopes), be eliminated (drugs in the body), or otherwise reduce to 50% of its original quantity. Halflife governs first-order decay processes and underlies carbon dating, drug dosing intervals, and reactor calculations.
The Formula
N(t) = N₀ × (1/2)t/T
where N₀ is the starting quantity, N(t) is the remaining quantity after time t, and T is the half-life. The relationship is symmetric — given any three of the four variables, you can solve for the fourth:
- Remaining: N(t) = N₀ × 0.5^(t/T)
- Initial: N₀ = N(t) × 2^(t/T)
- Time: t = T × log(N(t)/N₀) / log(0.5)
- Half-life: T = t × log(0.5) / log(N(t)/N₀)
Worked Examples
Carbon-14: Half-life T = 5,730 years. After 11,460 years (2 half-lives), only 25% of the original C-14 remains. After 17,190 years (3 half-lives), 12.5% remains.
Drug elimination: A drug with a 6-hour half-life and a 100 mg starting dose decays to 50 mg at 6h, 25 mg at 12h, 12.5 mg at 18h, etc.
Iodine-131 (medical imaging): T = 8.02 days. To estimate when a patient's body has reduced a tracer by 99%, solve t = 8.02 × log(0.01)/log(0.5) ≈ 53.3 days.
Half-life vs Mean Lifetime vs Decay Constant
Three related but distinct quantities describe the same exponential decay:
- Half-life T — time for half to decay. The "everyday" parameter.
- Mean lifetime τ — average lifetime of one particle/molecule before decay. τ = T / ln(2) ≈ T × 1.4427.
- Decay constant λ — probability per unit time. λ = ln(2) / T ≈ 0.693 / T.
Use whichever fits your context — physicists tend to use λ, doctors quote half-life, and biologists use mean lifetime.
Multi-step Decay & Mixed Substances
Real samples often contain multiple isotopes or mixed compounds, each with its own half-life. The total quantity at time t is the sum of N₀ᵢ × 0.5^(t/Tᵢ) over each component i. This calculator models a single half-life — to handle mixtures, run the calc separately for each component and add the results.
Caveats
- First-order assumption. The formula assumes constant fractional decay rate. Drug elimination follows it for most medications, but enzyme-saturated kinetics (alcohol, some drugs at high doses) is zero-order — different formula.
- No decay product tracking. The calculator returns remaining parent material, not daughter products. For radioactive chains (U-238 → Ra-226 → Rn-222), each step needs its own calc.
- Time units must match. If half-life is in days, time t must also be in days — the calculator uses dimensionless ratio t/T.