Compute remaining quantity, elapsed time, or half-life for any first-order decay — radioactive, pharmacological, or chemical — with step-by-step working and a live decay curve.
Pick what you want to solve for — results update live.
Remaining quantity
% Remaining
25%
fraction of N₀
Decayed
75
N₀ − N
Half-lives elapsed
2
t / t½
Decay constant λ
0.1386
ln(2) / t½
| Isotope | Half-life | Use |
|---|---|---|
| Carbon-14 (¹⁴C) | 5,730 years | radiocarbon dating |
| Iodine-131 (¹³¹I) | 8.02 days | thyroid imaging / therapy |
| Technetium-99m | 6.01 hours | medical imaging tracer |
| Caesium-137 (¹³⁷Cs) | 30.17 years | radiation source, fallout |
| Uranium-238 (²³⁸U) | 4.47 × 10⁹ years | geological dating |
| Plutonium-239 (²³⁹Pu) | 24,110 years | nuclear fuel |
The Method
Half-life describes any process where the rate of decrease is proportional to what remains — so-called first-order kinetics. The amount drops by half in a fixed interval, again by half in the next, and so on. The corresponding differential equation dN/dt = −λN has the exponential solution N(t) = N₀·e^(−λt), where λ = ln(2) / t½. Rewriting in base 2 gives the friendlier form N(t) = N₀·(½)^(t/t½), used directly by this calculator.
Working for current values
About This Tool
A half-life calculator solves the exponential-decay equation N(t) = N₀ × (½)^(t/t½) for whichever of the four variables you don't know. Enter any three of initial quantity N₀, remaining quantity N, elapsed time t, and half-life t½, and the calculator returns the fourth, along with the decay constant λ, the fraction remaining, and the number of half-lives elapsed.
Half-life crops up far beyond physics. Pharmacokinetics uses it to model drug elimination; geology uses it to date rocks via uranium-lead and potassium-argon methods; archaeology uses it for carbon-14 dating; and electronics uses the same maths for RC-circuit discharge and thermistor cooling. Wherever the rate of change scales with what's left, half-life is the natural way to describe it.
The underlying maths is exponential. Each half-life multiplies what remains by ½: after one half-life 50% is left, after two 25%, after three 12.5%, and so on. By 10 half-lives less than 0.1% remains, which is why most safety guidance treats a sample as effectively gone after seven to ten half-lives.
This free half-life calculator runs entirely in your browser — no sign-up, no tracking. Inputs accept any positive units; pick years for geology, hours for medical isotopes, or seconds for short-lived radionuclides — the formula doesn't care about the unit choice as long as t and t½ use the same one.
Solve Any Variable
Find N, t, or t½ from the other three known values — three modes in one tool.
Live Decay Curve
Interactive SVG curve highlights your current point on N(t) = N₀·(½)^(t/t½).
Decay Constant λ
Also reports λ = ln(2)/t½ for use in continuous models like N = N₀·e^(−λt).
Step-by-Step Working
Every substitution shown — useful for chemistry, physics or pharmacology classes.
100% Free & Private
No account, no tracking — every calculation runs locally in your browser.
Any Time Unit
Seconds to billions of years — choose whatever fits your problem.
Find any decay variable in under a minute.
Use the tabs to choose what you want to solve for — Remaining (N), Time (t), or Half-Life (t½).
Fill in the three known fields. Use whichever time unit you like — just be consistent across t and t½.
The headline value is the variable you chose. Stat tiles show % remaining, decayed amount, half-lives elapsed, and λ.
The chart plots N(t) over 5 half-lives and marks your current point — useful for visualising how fast 99% of the sample is gone.
The reference table lists half-lives for C-14, I-131, U-238 and others — useful for sanity-checking real-world calculations.
The formula card shows the exact substitution — handy for homework write-ups and verifying hand calculation.
Everything you need to know about half-life and exponential decay.
Half-life (t½) is the time required for a quantity following first-order decay to fall to half its initial value. It is constant for a given substance and independent of the starting amount — every half-life reduces what's left by exactly 50%.
Remaining quantity: N(t) = N₀ × (½)^(t/t½). Equivalently N(t) = N₀ × e^(−λt), where the decay constant λ = ln(2) / t½ ≈ 0.6931 / t½.
No. Any process with first-order kinetics follows the same maths: drug elimination in pharmacology, RC-circuit discharge in electronics, Newton's law of cooling, and many chemical reactions all behave exponentially and have a meaningful half-life.
Carbon-14 has a half-life of approximately 5,730 years, which is the basis of radiocarbon dating of organic remains up to about 50,000 years old. Beyond that, less than 0.2% of the original ¹⁴C remains and measurement becomes unreliable.
After 10 half-lives less than 0.1% remains. Most radiation safety guidance treats a sample as effectively decayed after 7-10 half-lives. For example, iodine-131 (t½ = 8 days) is considered gone after about two to three months.
The decay constant λ is the probability per unit time that any one nucleus (or molecule) decays. It is the natural rate parameter of the continuous form N = N₀·e^(−λt), and is related to half-life by λ = ln(2) / t½.
For radioactive decay, no — it is set by the nucleus and unaffected by ordinary chemistry or environment. For chemical first-order reactions, the apparent half-life does depend on temperature via the rate constant (Arrhenius equation), so context matters.
Any consistent pair will do. If t is in days, t½ must also be in days. The calculator does not assume a unit, so pick whatever makes your problem readable — seconds for short-lived isotopes, hours for medical tracers, years for geology.
The mean lifetime τ is the average time a nucleus survives before decaying: τ = 1/λ = t½ / ln(2). So τ is slightly longer than t½ — for carbon-14, τ ≈ 8,267 years versus t½ = 5,730 years.