Half-Life Calculator
Model exponential decay. Solve for whichever of N₀, Nₜ, t, or t½ you don't know — then convert freely between half-life, mean lifetime, and decay constant.
Half-Life Solver
Fill any three fields and leave the fourth blank — the calculator solves for the missing one.
Half-Life ↔ Mean Lifetime ↔ Decay Constant
Enter one of the three constants and get the other two instantly.
Definition and intuition
Half-life is the amount of time required for a quantity that decays exponentially to fall to exactly half of its initial value. The term is most commonly associated with radioactive atoms — nuclei that spontaneously emit particles and transform into other elements — but the same idea describes any decay process in which the rate of change is proportional to the amount currently present. That includes drug elimination from the bloodstream, the discharge of a capacitor through a resistor, the cooling of a hot object toward its surroundings, and the intensity of light passing through an absorbing medium.
The key intuition is that half-life is constant for a given substance or process: it does not depend on how much material you start with. Begin with 1,000 atoms of an isotope whose half-life is one hour and you expect about 500 to remain after one hour, 250 after two, 125 after three, and so on. Start with a kilogram of the same isotope and after one hour you still have half a kilogram — the fraction is what stays fixed, not the absolute amount lost.
The most famous application is radiocarbon dating. Living organisms continually exchange carbon with the atmosphere, so their ratio of carbon-14 to carbon-12 stays close to the atmospheric value. Once an organism dies that exchange stops, and the carbon-14 already inside it begins to decay with a half-life of about 5,730 years. Measuring the remaining C-14 fraction in a sample and inverting the decay formula gives its age. The technique is reliable to roughly 50,000 years — after about ten half-lives there is too little C-14 left to measure precisely.
Half-life calculator, piece by piece
Each card explains one thing the tool does with your input — the three interchangeable decay formulas it uses, how it relates half-life to mean lifetime and decay constant, and how it solves for whichever quantity you left blank.
The decay curve — what the calculator plots and reports
The tool models a quantity that shrinks by the same fractional amount in every fixed time step. After one half-life half is left, after two half-lives a quarter is left, after three an eighth. The result panel shows N(t) at the time you entered; the curve behind the scenes is a smooth exponential, not a staircase — the halving pattern only shows up if you sample at whole multiples of the half-life.
t½, τ, λ — three names for the same rate
The calculator accepts (and outputs) three different decay constants because different fields use different conventions. The half-life t½ is popular in nuclear physics and pharmacology, the mean lifetime τ shows up in particle physics and RC circuits, and the decay constant λ is what appears in the differential equation dN/dt = −λN. All three are locked together by ln 2, so entering any one fills in the other two.
Solving for whichever field you left blank
The same equation N(t) = N₀·(½)^(t/t½) can be rearranged four ways. Leave N(t) blank and the tool substitutes directly. Leave t blank and it takes logs to get t = t½·log₂(N₀/N). Leave N₀ blank and it multiplies through by 2^(t/t½). Leave t½ blank and it solves t½ = t·ln 2 / ln(N₀/N). The solution steps under the answer show which branch was taken.
Common half-lives you'll meet
| Isotope | Half-life | Typical use |
|---|---|---|
| Carbon-14 (¹⁴C) | 5,730 years | Archaeological dating of organic material |
| Uranium-238 (²³⁸U) | 4.468 billion years | Dating rocks and the age of the Earth |
| Potassium-40 (⁴⁰K) | 1.25 billion years | Dating volcanic rock and early hominid fossils |
| Iodine-131 (¹³¹I) | 8.02 days | Medical imaging and thyroid cancer treatment |
| Technetium-99m (⁹⁹ᵐTc) | 6.01 hours | Most common medical diagnostic tracer |
| Cesium-137 (¹³⁷Cs) | 30.17 years | Radiation therapy, fallout tracing |
| Radon-222 (²²²Rn) | 3.82 days | Indoor air-quality risk assessment |
| Plutonium-239 (²³⁹Pu) | 24,110 years | Reactor fuel and nuclear weapons |
Half-lives span an enormous range — from fractions of a second for some artificial nuclei to billions of years for the isotopes used to date the Earth. The mathematics is identical; only the timescale changes.
Real-world applications
- Radiocarbon dating of bones, wood, and cloth up to ~50,000 years old (C-14, t½ ≈ 5,730 years).
- Geochronology: potassium-argon and uranium-lead dating of rocks over millions to billions of years.
- Nuclear medicine: choosing a tracer with a half-life long enough to image but short enough to clear from the body quickly (Tc-99m at ~6 hours is the classic example).
- Pharmacokinetics: the plasma half-life of a drug determines dosing intervals — after roughly five half-lives only about 3% of a dose remains.
- RC circuits: a capacitor discharging through a resistor loses charge exponentially with τ = R · C, giving t½ = R · C · ln(2).
- Environmental science: modeling how long pollutants such as PCBs or persistent pesticides remain in soil and water.
Common mistakes to avoid
- Mixing time units. If t½ is in years, t must also be in years. The calculator uses whatever unit you supply — convert first if your inputs disagree.
- Confusing τ with t½. Mean lifetime is longer than half-life by a factor of 1/ln(2) ≈ 1.443. Plugging τ into a half-life formula (or vice versa) will be off by ~44%.
- Using base-10 log instead of natural log. The mean-lifetime and decay-constant forms use e, so the inverse is ln (natural log), not log₁₀.
- Expecting exact halving with small samples.Radioactive decay is statistical. With billions of atoms the formula is essentially exact; with only a few dozen, random fluctuations dominate.
Features of this calculator
- Solve for the initial amount, remaining amount, elapsed time, or half-life given any three
- Built-in half-life ↔ mean lifetime ↔ decay constant converter
- Shows the substituted formula and the algebraic rearrangement for every result
- Includes a reference table of common isotopes and their half-lives
- Works for radioactive decay, drug dosing, and any first-order decay process
Frequently asked questions
+What is half-life used for?
Half-life measures how quickly a quantity that decays exponentially shrinks. It is most famous in nuclear physics for radioactive isotopes, but the same idea models drug elimination from the body, capacitor discharge in an RC circuit, the cooling of a hot object toward room temperature, and the removal of pollutants from an ecosystem.
+How is carbon dating calculated?
Living organisms take in carbon-14 at a roughly steady rate; once they die, the C-14 decays with a half-life of about 5,730 years. Measuring the remaining fraction of C-14 in a sample and solving N(t) = N₀·(1/2)^(t/5730) for t gives the sample's age. The method is reliable up to roughly 50,000 years, after which too little C-14 remains to measure precisely.
+What is the difference between half-life and mean lifetime?
Half-life (t½) is the time for a quantity to fall to half its starting value. Mean lifetime (τ) is the average time an individual particle survives before decaying. They are related by t½ = τ · ln(2), so τ is always about 1.44× longer than the half-life for the same process.
+How do I get the decay constant from a half-life?
The decay constant λ is ln(2) divided by the half-life: λ = ln(2) / t½. It represents the fraction of the remaining quantity that decays per unit time — for example, λ = 0.1/year means about 10% of what's left decays each year.
+Does half-life depend on how much material you start with?
No. For a truly exponential process, the half-life is a constant of the substance itself — it doesn't matter whether you start with a gram or a kilogram, half of it will be gone after one t½.
+Can half-life be used for non-radioactive processes?
Yes. Any process where the rate of change is proportional to the current amount is exponential and has a half-life. Common examples include first-order chemical reactions, drug clearance in the bloodstream, and the intensity of light passing through an absorbing medium.