⚛️Half-Life Calculator
Calculate any variable in the half-life formula: quantity remaining (Nₜ), initial quantity (N₀), time elapsed (t), or half-life (t½). Instantly converts between half-life, decay constant (λ), and mean lifetime (τ). Applies to radioactive decay, pharmacokinetics, carbon dating, and any exponential decay process.
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Half-Life Calculator: Radioactive Decay, Carbon Dating, and Pharmacokinetics
The half-life (t½) is the time required for exactly half of a substance to decay or transform. It is constant for a given process regardless of the initial amount. The formula N(t) = N₀ × (½)^(t/t½) connects four variables: current quantity Nₜ, initial quantity N₀, time t, and half-life t½. Enter any three to solve for the fourth.
Formula: N(t) = N₀ × (½)^(t/t½) — equivalently N(t) = N₀ × e^(−λt) where λ = ln(2)/t½
| Example | Half-Life | After 10 half-lives |
|---|---|---|
| Carbon-14 (¹⁴C) | 5,730 years | 0.098% remains |
| Ibuprofen (plasma) | ~2 hours | 0.098% remains |
| Uranium-238 (²³⁸U) | 4.47 billion years | 0.098% remains |
The half-life concept is central to nuclear physics (radioactive decay), pharmacokinetics (drug elimination), archaeology (radiocarbon dating), and any exponential decay process. Regardless of the initial quantity, after n half-lives exactly (½)ⁿ of the original amount remains — after 10 half-lives, only 1/1024 ≈ 0.098% is left.
Half-Life, Decay Constant, and Mean Lifetime
Three equivalent ways to characterize exponential decay are mathematically related:
Half-life t½: time for 50% reduction. Most intuitive — used in chemistry and nuclear physics.
Decay constant λ = ln(2)/t½ ≈ 0.6931/t½: rate of decay per unit time. Used in the continuous form N(t) = N₀e^(−λt). Units are inverse time (s⁻¹, yr⁻¹, etc.).
Mean lifetime τ = 1/λ = t½/ln(2) ≈ 1.4427 × t½: average time before a single particle decays. Longer than t½ because surviving particles can live much longer than average. The mean lifetime is used in quantum mechanics and particle physics.
Example for Carbon-14 (t½ = 5,730 yr): λ = 0.6931/5730 = 1.210 × 10⁻⁴ yr⁻¹; τ = 8,267 years (the average ¹⁴C atom lives 8,267 years before decaying).
Radiocarbon Dating
Carbon-14 is continuously produced in the atmosphere and maintains a roughly constant ratio to Carbon-12 in living organisms (which constantly exchange carbon). When an organism dies, ¹⁴C is no longer replenished and begins decaying with t½ = 5,730 years. Measuring the ¹⁴C/¹²C ratio in a sample and comparing to the atmospheric ratio gives the time since death: t = 5730 × log₂(N₀/Nₜ). This method is accurate to about ±40 years for samples up to 50,000 years old. Beyond that, too little ¹⁴C remains to measure accurately.
Drug Half-Life in Pharmacokinetics
In medicine, a drug's plasma half-life determines dosing frequency. A drug with t½ = 4 hours: after one dose, 50% remains at 4h, 25% at 8h, 12.5% at 12h. Steady state (where intake equals elimination) is reached after approximately 4–5 half-lives regardless of dosing frequency. With repeated dosing, drug accumulates until equilibrium. Drugs with very long half-lives (like amiodarone, t½ ≈ 40–55 days) require loading doses to reach therapeutic levels faster and take weeks to clear after stopping.
Frequently Asked Questions
What is the half-life formula?
The half-life formula is N(t) = N₀ × (½)^(t/t½), where N(t) is the quantity remaining at time t, N₀ is the initial quantity, and t½ is the half-life. This can be rearranged to solve for any variable: t½ = t × ln(2)/ln(N₀/N(t)); t = t½ × log₂(N₀/N(t)); N₀ = N(t) × 2^(t/t½). The equivalent continuous form is N(t) = N₀ × e^(−λt), where the decay constant λ = ln(2)/t½ ≈ 0.6931/t½.
What is the difference between half-life and mean lifetime?
Half-life (t½) is the time for exactly 50% of the substance to decay. Mean lifetime (τ) is the average time a single particle exists before decaying. They are related by τ = t½/ln(2) ≈ 1.4427 × t½ — the mean lifetime is always about 44% longer than the half-life. For Carbon-14: t½ = 5,730 years, τ ≈ 8,267 years. The mean lifetime is longer because even though most atoms decay within a few half-lives, a small fraction persists much longer, pulling the average up.
How many half-lives until a substance is gone?
A substance never completely reaches zero in pure exponential decay — it asymptotically approaches zero. After 10 half-lives, 0.098% remains; after 20 half-lives, 0.000095% remains. In practice, for radioactive materials, once the activity drops below background radiation levels (typically after 10–20 half-lives) it is considered negligible. For drugs, the medical convention is that a drug is "essentially eliminated" after 4–5 half-lives, when 3.1–6.25% remains.
Why is the half-life constant regardless of initial quantity?
The constancy of half-life follows directly from the exponential decay law N(t) = N₀e^(−λt). At any time T, the fraction remaining after one more half-life is N(T+t½)/N(T) = [N₀e^(−λ(T+t½))]/[N₀e^(−λT)] = e^(−λt½) = e^(−ln2) = ½. This fraction (½) is independent of T and N₀ — it depends only on λ and t½. The physical reason: each atom has the same probability λ of decaying per unit time regardless of how old it is (memoryless property), so the fraction surviving per unit time is always the same.
How does this calculator work for pharmacokinetics (drug half-life)?
Use the same formula with time in hours and quantities in any unit (mg, ng/mL plasma concentration, etc.). Example: a drug has t½ = 6 hours. After a 500 mg dose, how much remains after 24 hours? Set N₀ = 500, t½ = 6, t = 24, solve for Nₜ: N(24) = 500 × (½)^(24/6) = 500 × (½)^4 = 500 × 0.0625 = 31.25 mg. Time to clear 95%: solve for t with Nₜ = 25 (5% of 500): t = 6 × log₂(500/25) = 6 × log₂(20) ≈ 6 × 4.322 ≈ 25.9 hours.