Half-Life Calculator
Radioactive decay halves a sample every half-life, following N = N₀ × (1/2)^(t/T). Choose what to solve for, pick your time unit, and get the result alongside the decay constant, mean lifetime, and a full decay schedule.
Formula
Worked example
Carbon-14 dating: Start with 100% (N₀ = 100), half-life T = 5,730 years, and find how old a sample is when 25% remains. n = ln(100/25) / ln 2 = ln(4) / 0.693 = 2 half-lives. t = 2 × 5,730 = 11,460 years. Decay constant λ = 0.693 / 5,730 = 0.0001210 per year. Mean lifetime τ = 5,730 / 0.693 = 8,267 years.
How the half-life formula works
Radioactive decay is a random process at the level of individual atoms, but across a large sample it follows a precise exponential law. The half-life T is the time for exactly half of the radioactive atoms to decay. After one half-life, half the sample remains; after two half-lives, a quarter; after three, just one-eighth. The relationship is N = N₀ × (1/2)^(t/T), where N₀ is the initial amount and t is the elapsed time. The exponent t/T is simply the count of half-lives that have passed, so the formula works with any consistent units for amount (atoms, grams, moles, or activity in Becquerels) as long as t and T share the same time unit.
Decay constant and mean lifetime
The half-life T, decay constant λ, and mean lifetime τ are three equivalent ways to describe the same decay rate. The decay constant λ = ln(2) / T is the probability per unit time that a given nucleus decays; it lets you write the decay law as N = N₀ × e^(-λt). The mean lifetime τ = 1/λ = T/ln(2) ≈ 1.4427 T is the average time an individual atom survives before it decays. Knowing any one of the three gives the others instantly: this calculator shows all three for every calculation.
Solving for time or half-life (reverse mode)
The decay equation rearranges to find any unknown. To find elapsed time, take logarithms: t = T × ln(N₀/N) / ln 2. This is exactly how radiocarbon dating works: you measure the fraction of Carbon-14 remaining and recover the age. To find the half-life itself, rearrange to T = t × ln 2 / ln(N₀/N), which is useful when you have tracked how much has decayed over a known interval. The decay schedule table below shows the remaining amount at every half-life from 0 through 10, giving you a quick sanity-check for any result.
Real-world half-lives and applications
Half-lives span an enormous range. Carbon-14 (5,730 years) anchors archaeological dating up to about 50,000 years. Iodine-131 (8.02 days) governs thyroid treatment dosing schedules. Uranium-238 (4.47 billion years) is used to date rocks billions of years old. Polonium-214 decays in 164 microseconds, and some synthetic superheavy elements last milliseconds. The same exponential model applies to drug clearance in pharmacology, capacitor discharge in electronics, and population decline in ecology. Use the time-unit selector to match whichever scale fits your application.
Common isotope half-lives
| Isotope | Half-life | Application |
|---|---|---|
| Polonium-214 | 164 μs | Alpha-decay chain of Uranium-238 |
| Fluorine-18 | 109.8 min | PET scan tracer |
| Iodine-131 | 8.02 days | Thyroid cancer therapy |
| Cobalt-60 | 5.27 years | Radiation therapy, food irradiation |
| Carbon-14 | 5,730 years | Radiocarbon (archaeological) dating |
| Chlorine-36 | 301,000 years | Groundwater age dating |
| Uranium-235 | 703.8 million years | Nuclear fuel, geological dating |
| Uranium-238 | 4.47 billion years | Dating Earth and meteorites |
Representative half-lives spanning the full range from microseconds to billions of years.
Frequently asked questions
What is the half-life formula?
The amount remaining after time t is N = N₀ × (1/2)^(t/T), where N₀ is the initial amount and T is the half-life. Equivalently, N = N₀ × e^(-λt), where λ = ln(2)/T is the decay constant. The exponent t/T counts the half-lives elapsed, so each whole half-life cuts the remaining amount in half.
How do I find elapsed time from the remaining fraction?
Rearrange the decay law: t = T × ln(N₀ / N) / ln 2. Divide the initial amount by the remaining amount, take the natural logarithm, divide by ln 2 (about 0.6931), and multiply by the half-life. For example, if 25% remains, ln(100/25) / 0.6931 = 2 half-lives have elapsed. This is the principle behind radiocarbon dating.
What is the decay constant and how does it relate to the half-life?
The decay constant λ is the probability per unit time that a nucleus decays. It relates to the half-life by λ = ln(2) / T ≈ 0.6931 / T. The mean lifetime τ = 1/λ = T / 0.6931. All three constants fully describe the same decay process; knowing any one gives the others. This calculator shows all three outputs automatically.
Does temperature or pressure change an isotope's half-life?
No. For ordinary nuclear radioactive decay the half-life is an intrinsic property of the nucleus and is unaffected by temperature, pressure, or chemical state. The same isotope decays at the same rate whether it is hot, cold, solid, or dissolved. (Electron capture is an exception where extreme pressures can have a tiny measurable effect, but this is negligible in practice.)
What time unit should I choose?
Choose the unit that matches the half-life of your isotope. Carbon-14 dating uses years (half-life 5,730 years). Medical isotopes like Iodine-131 (8 days) work naturally in days. Short-lived tracers like Fluorine-18 (110 minutes) are best in minutes. The calculator applies the same unit to both the half-life and the elapsed time, so the ratio t/T is dimensionless and always correct.