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Half-Life Calculator

Solve for the remaining amount, elapsed time, half-life, or initial amount in exponential decay — for radioactive decay, drug elimination or any half-life process.

How to use this calculator

  1. 1Choose which value you want to solve for.
  2. 2Fill in the other three known values — units just need to be consistent between the half-life and elapsed time.

How the calculation works

N(t) = N₀ × (1/2)^(t / half-life)
N(t)
Amount remaining after time t
N₀
Initial amount
t
Elapsed time, in the same unit as the half-life

This is equivalent to the exponential decay form N(t) = N₀e^(−λt), with the decay constant λ = ln(2) / half-life — both describe the same curve, this one is just phrased in terms of the half-life directly.

Worked example

100g of Cobalt-60 (half-life 5.27 years), after 10.54 years

  1. 1.10.54 years ÷ 5.27-year half-life = exactly 2 half-lives.
  2. 2.Remaining = 100 × 0.5² = 100 × 0.25 = 25 g.

Result: 25 g remaining (2 half-lives elapsed)

What decaying by halves means

Something decays "by half-life" when a fixed fraction of whatever remains disappears in each equal stretch of time — not a fixed amount. After one half-life, half of the original quantity is left. After two, a quarter. After three, an eighth. The amount removed keeps shrinking because it is always half of a smaller and smaller starting point, which is exactly why the decay curve flattens out over time instead of running in a straight line down to zero.

This is different from most everyday ideas of wearing out. A car does not lose exactly half its remaining value every five years, and a battery does not lose exactly half its remaining charge every hour — those processes depend on use, temperature and countless other factors. Half-life describes a narrower, more mathematically precise kind of decline: one where the rate of loss is always proportional to how much is currently there.

A rate that never speeds up, slows down, or resets

The defining feature of half-life decay is that it has no memory. A sample of a radioactive isotope that is a billion years old decays at exactly the same proportional rate as a freshly created sample of the same isotope — age and history make no difference to the physics. That property is what lets a single half-life value describe a substance forever, rather than needing a different figure for every batch or every point in its history.

It also means the quantity mathematically never reaches exactly zero, no matter how long you wait — halving a positive number always leaves a smaller positive number. In practice a decaying sample becomes undetectable long before that point, but the idealised exponential curve this calculator uses keeps approaching zero without ever technically touching it.

Where half-life shows up beyond the physics classroom

Half-life is most associated with radioactivity, but the identical mathematics governs any process where the amount lost is proportional to the amount present.

  • Radiocarbon dating — archaeologists and geologists estimate the age of once-living material by measuring how much carbon-14 remains relative to stable carbon, using its roughly 5,730-year half-life — a technique developed by chemist Willard Libby in the late 1940s.
  • Nuclear medicine — diagnostic isotopes are chosen partly for their half-life — long enough to complete an imaging scan, short enough that the radioactivity clears from the body quickly afterward.
  • Drug dosing — a medication's biological half-life — how long it takes the body to clear half of a dose — determines how often it needs to be taken to keep blood levels within a safe, effective range.
  • Nuclear waste management — the half-lives of different radioactive byproducts, some measured in days and others in tens of thousands of years, determine how long spent material must be safely stored.
  • Caffeine and other stimulants — caffeine has an average half-life of roughly five hours in a healthy adult, which is why a coffee at 4pm can still measurably affect sleep at midnight.

A common mistake: treating half-life as an expiry date

It is tempting to assume that after two half-lives a substance is "gone," but two half-lives only removes three-quarters of the original amount — a quarter is still there. This misunderstanding matters in fields like nuclear safety and pharmacology, where "wait ten half-lives and it is effectively gone" is a genuinely useful rule of thumb (less than 0.1% remains), but "wait one half-life and it is gone" is a dangerous one.

A related mix-up is treating decay as linear — subtracting the same fixed amount every period instead of the same fixed fraction. Linear decay predicts a substance vanishes completely at some specific date; exponential decay predicts it keeps thinning out indefinitely. The two models only agree over a very short stretch near the start.

What this assumes, and where it stops

Assumptions

  • Decay follows the standard exponential model with a constant half-life — true for radioactive decay and a good approximation for first-order elimination processes like many drugs.

Limitations

  • Not every real-world decay process is a clean single half-life — some drug elimination and biological processes follow multi-phase or non-exponential kinetics.

Common questions

What is a half-life?

The time it takes for half of a quantity to decay or be eliminated, regardless of how much you started with — this is what makes it useful: whether you have 100g or 100kg of a radioactive isotope, the same fixed half-life describes how quickly it halves.

Does this apply to things other than radioactivity?

Yes — the same exponential decay maths describes drug elimination from the body (pharmacokinetic half-life), caffeine metabolism, and any process where the rate of decrease is proportional to the amount remaining.

Sources

Formula and content last reviewed on .

Results are estimates for information only, not professional advice.

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