Half-Life & Exponential Decay Calculator
Half-life is the time required for a quantity to fall to half its initial value — fundamental in nuclear physics, pharmacology and radiometric dating. This tool solves the same exponential-decay law for any of the four unknowns: remaining amount, initial amount, elapsed time, or half-life. Pick which one to solve for, fill in the other three, and you also get the percentage remaining, number of half-lives elapsed, decay constant λ, mean lifetime τ, and a live SVG decay curve. The time unit toggle (seconds, minutes, hours, days, years) lets you cover everything from caffeine (~5 h half-life) to carbon-14 (~5,730 y).
Solve for
Fill in the other three inputs (non-negative; half-life and initial amount must be > 0; remaining cannot exceed initial).
Result
50
- Percent remaining
- 50.00 %
- Half-lives elapsed
- 1.000
- Decay constant λ = ln 2 / T½
- —
- Mean lifetime τ = T½ / ln 2
- —
Decay curve (fraction remaining vs time)
Formula
N(t) = N₀ · (½)^(t / T½) = N₀ · e^(−λ · t) λ = ln 2 / T½ τ = T½ / ln 2
t and T½ must share the same time unit — the dropdown applies to both. The "amount" inputs accept any quantity (mass, becquerels, drug concentration, …); the unit shown is just labelled "units".
Formula
N(t) = N₀ · (½)^(t / T½) // base exponential-decay law N(t) = N₀ · e^(−λ · t), λ = ln 2 / T½ // equivalent form T½ = t · ln 2 / ln(N₀ / N) // solve half-life from observation t = T½ · log₂(N₀ / N) // solve elapsed time from remaining τ = T½ / ln 2 // mean lifetime
- · Half-life does not depend on the starting quantity — one kilogram of iodine-131 and one gram of iodine-131 both decay to half after 8.025 days. This scale-invariance is the defining feature of exponential decay: each half-life period reduces the remaining amount by 50 % of whatever was there at the start of that period.
- · You pick the time unit, but t and T½ must share the same unit — the dropdown applies to both. For PET-imaging fluorine-18 (T½ = 109.77 min) choose "minutes"; for archaeology choose "years".
- · Reference half-lives (NIST Atomic Weights & Isotopic Compositions): carbon-14 ≈ 5,730 y (radiocarbon dating); tritium (³H) ≈ 12.32 y; potassium-40 ≈ 1.25 × 10⁹ y; caesium-137 ≈ 30.17 y; iodine-131 ≈ 8.025 d; fluorine-18 ≈ 109.77 min; technetium-99m ≈ 6.01 h.
- · Pharmacological elimination half-lives (terminal t½, Goodman & Gilman 13e): caffeine ≈ 5 h; alcohol ≈ 0.7 h (actually zero-order — this tool is only approximate for ethanol); paracetamol ≈ 2 h; ibuprofen ≈ 2 h; diazepam ≈ 20–80 h; fluoxetine ≈ 1–4 d. Plasma concentration falls below ~5 % after about 4–5 half-lives ("clinically eliminated").
- · Radiocarbon dating: once an organism dies, ¹⁴C uptake stops and the existing ¹⁴C decays. After 11,460 years (two half-lives) ~25 % remains; after 30,000 years ~2.6 % remains; beyond ~50,000 years the ¹⁴C signal is too weak and other methods (K-Ar, U-Pb) take over.
- · Common slip-ups: (1) half-life ≠ mean lifetime τ — τ = T½ / ln 2 ≈ 1.443 · T½; (2) the decay constant λ is in units of 1 / time, not a percentage; (3) "50 % decay" means "50 % remaining" — don't double-count by saying "another half-life lops off another 50 % of the original" (it lops off 50 % of what is currently left).
- · Numerical caveat: standard IEEE-754 doubles lose precision at very small ratios (N / N₀ below ~1e-15). For very long timescales (> 60 half-lives) prefer log-space arithmetic or a symbolic computer-algebra system.
- · Scope: this calculator covers first-order exponential decay only. Branching decay chains (uranium series), zero-order kinetics (high-dose ethanol), or saturable enzymatic clearance need the Bateman equation or Michaelis-Menten kinetics — half-life alone is insufficient.
Frequently asked
After a double espresso (200 mg caffeine), how long until only 25 mg is left in my system?
Caffeine's mean elimination half-life is ~5 h (Goodman & Gilman 13e). Going from 200 mg to 25 mg is an 87.5 % drop (1/8 remains = three half-lives) ≈ 15 hours. Pick "solve for elapsed time", set initial = 200, remaining = 25, half-life = 5 h → the tool returns 15.0 h. So a 3 pm espresso is mostly gone by 6 am the next day — though individual half-lives vary from ~1.5 h to ~9.5 h depending on age, liver function, oral contraceptives and smoking status.
What is the difference between half-life and mean lifetime (τ)?
T½ is the time for the quantity to fall to half its current value; τ is the average lifetime of a single particle (the expectation value of the exponential distribution). They are related by τ = T½ / ln 2 ≈ 1.443 · T½, so τ is always ~44 % longer than T½. For iodine-131, T½ = 8.025 d but τ ≈ 11.58 d. Physicists doing rate calculations often use λ = 1/τ; chemists, pharmacologists and biologists prefer T½ because "how long until half is gone" is more intuitive for back-of-envelope work. This tool reports both so you can cross-check.
Why is radiocarbon dating limited to roughly 50,000 years?
Because carbon-14 has a half-life of only 5,730 years. After 50,000 years N / N₀ ≈ (½)^(50000/5730) ≈ (½)^8.73 ≈ 0.24 % — so close to background that even modern accelerator mass spectrometry (AMS) struggles to separate signal from noise. Pick "solve for remaining" in the tool to reproduce this ratio. For older samples (millions to billions of years), use isotopes with longer half-lives: K-Ar (T½ ≈ 1.25 × 10⁹ y) for volcanic minerals, U-Pb (T½ ≈ 4.47 × 10⁹ y) for zircons.
Why is "90 % decay" not equal to 1.8 half-lives?
It feels like "one half-life = 50 %, so 1.8 ≈ 90 %", but exponential decay is multiplicative, not additive — each half-life multiplies what is left by ½. Reaching 10 % remaining means (½)^n = 0.1, i.e. n = log₂(10) ≈ 3.32 half-lives, not 1.8. Likewise: 1 % remaining needs 6.64 half-lives; 99 % remaining (only 1 % gone) needs 0.0145. The clinical rule of thumb that drugs are "essentially eliminated in 4–5 half-lives" comes from the same logarithm: (½)^5 = 1/32 ≈ 3.1 %.
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