Half-Life Calculator
Calculate how much of a substance remains after any time using exponential decay, from its half-life — plus the decay constant, mean lifetime, and number of half-lives.
How to use this calculator
- 1Enter the initial amount of the substance.
- 2Enter its half-life and the elapsed time, using the same time unit for both.
- 3Read how much remains, the fraction left, and the number of half-lives that have passed.
- 4Use the decay constant and mean lifetime for the exponential form of the law.
How it works
Exponential decay
N(t) = N₀ × (1/2)^(t / t½) equivalently N(t) = N₀ × e^(−λt) decay constant λ = ln 2 ÷ t½ mean lifetime τ = 1 ÷ λ = t½ ÷ ln 2
The half-life is the time it takes for half of a decaying quantity to disappear, and it is constant regardless of how much is present — a defining feature of exponential decay. After one half-life, half remains; after two, a quarter; after three, an eighth. The amount left after any time is the initial amount multiplied by one-half raised to the number of half-lives elapsed. The same process can be written with the natural exponential and a decay constant λ, equal to ln 2 divided by the half-life, which represents the fraction decaying per unit time. The reciprocal of λ is the mean lifetime — the average time an individual atom or molecule survives, always a bit longer than the half-life because a few survive for a very long time.
Worked example
Iodine-131 has a half-life of about 8 days. Starting with 100 units, after 24 days — three half-lives — an eighth remains: 100 × (1/2)³ = 12.5 units. Its decay constant is ln 2 ÷ 8 ≈ 0.0866 per day, giving a mean lifetime of about 11.5 days.
Half-Life Calculator: the complete guide
Why decay is exponential
Radioactive decay, and many other decay processes, share a peculiar property: the time for half the material to disappear is the same no matter when you start counting. Begin with a kilogram and after one half-life you have 500 grams; begin with 500 grams and after another half-life you have 250. This constant proportional loss is the signature of exponential decay, and it arises because each atom decays independently with a fixed probability per unit time, unaffected by its neighbours or by how many have already decayed.
The consequence is a curve that falls steeply at first and then ever more gently, approaching zero without reaching it. This is why you cannot speak of a substance 'fully decaying' at a specific moment — there is always, in principle, a little left. In practice, after about ten half-lives less than a thousandth remains, which is close enough to gone for most purposes. Understanding this shape is key to reading everything from a radiation safety chart to the concentration of a medicine in the bloodstream.
Half-life, decay constant, and mean lifetime
The same decay can be described by three related numbers, and it helps to see how they connect. The half-life is the most intuitive: the time for half to remain. The decay constant λ is the fraction of the material that decays per unit time, and it equals ln 2 divided by the half-life — a larger λ means faster decay and a shorter half-life. The mean lifetime τ is the average survival time of an individual particle, equal to 1 divided by λ.
A common surprise is that the mean lifetime is longer than the half-life, by a factor of about 1.44 (that is, 1 ÷ ln 2). This is because a minority of particles survive far beyond the half-life and drag the average upward, much as a few very old individuals raise the average lifespan of a population above its median. Physicists often work with λ or τ rather than the half-life because they slot directly into the exponential formula, but all three carry exactly the same information about how fast the substance decays.
Where half-lives show up
The half-life concept reaches far beyond nuclear physics. Radiocarbon dating uses the 5,730-year half-life of carbon-14 to date organic remains: by measuring how much of the original carbon-14 is left, archaeologists calculate how long ago the organism died. Geologists use the much longer half-lives of uranium and potassium isotopes to date rocks and, ultimately, the age of the Earth. Medical imaging relies on short-lived isotopes chosen so they decay away soon after a scan, minimising a patient's exposure.
Pharmacology borrows the same mathematics for a non-radioactive process: the biological half-life of a drug is the time for the body to clear half of it. This governs how often a medicine must be taken to maintain an effective level, and how long it lingers after the last dose — roughly four to five half-lives to be substantially eliminated. Even outside science the pattern recurs, from the discharge of a capacitor to the cooling of a hot object, wherever the rate of change is proportional to the amount present. The half-life is the natural language of all such processes.
Frequently asked questions
How do I calculate how much remains after a time?
Divide the elapsed time by the half-life to get the number of half-lives, then multiply the initial amount by one-half raised to that power. After 24 days with an 8-day half-life, that is 3 half-lives, so (1/2)³ = 1/8 remains — 12.5% of the original.
What is the decay constant?
The decay constant λ is the fraction of a substance that decays per unit time. It equals the natural log of 2 (about 0.693) divided by the half-life. It appears in the exponential form of the decay law, N = N₀e^(−λt), and a larger λ means faster decay.
Why is the mean lifetime longer than the half-life?
Because a minority of particles survive well beyond the half-life and pull the average upward. The mean lifetime equals the half-life divided by ln 2, making it about 1.44 times longer. It is the average survival time of a single particle, not the median.
When is a substance considered fully decayed?
Never exactly — decay is exponential and only approaches zero. As a practical rule, after about 10 half-lives less than 0.1% remains, which is treated as effectively gone. In medicine, a drug is largely cleared after roughly four to five of its biological half-lives.