Half-Life Calculator & Radioactive Decay Simulator

What is half-life?

Half-life is the time it takes for half of a radioactive sample to decay. It is written T½, and for any given isotope it is a fixed property — as reliable as a melting point or a density. Carbon-14 has a half-life of 5,730 years whether you have a gram of it or a tonne, whether it is warm or cold, and whether it is chemically bound in a bone or floating free in the atmosphere.

That last point surprises people. Almost every other rate in chemistry can be pushed around: heat a reaction and it speeds up, add a catalyst and it accelerates. Radioactive decay is different because it happens inside the nucleus, insulated from the chemistry going on in the electron shells outside. Nothing you can do in a normal laboratory changes an isotope's half-life in any meaningful way.

The consequence is worth stating plainly: half-life does not mean half the time until it is gone. After one half-life, half remains. After two, a quarter. After three, an eighth. The sample never quite reaches zero by this equation — it keeps halving, and half of something is always something.

What is radioactive decay?

Some atomic nuclei are unstable. Their combination of protons and neutrons sits at an energy that is not the lowest available to them, and sooner or later they rearrange into something more stable, throwing off energy and particles as they go. That rearrangement is radioactive decay.

The crucial feature is that it is random for any individual nucleus. There is no countdown timer inside an atom of uranium-238 ticking towards a scheduled moment. Each nucleus simply has a fixed probability of decaying in the next second, and that probability never changes no matter how long it has already survived. An atom that has sat around for a billion years is exactly as likely to decay in the next minute as a freshly created one.

Randomness at the level of one atom produces near-perfect predictability at the level of a sample, because a realistic sample holds an enormous number of atoms. A single gram of carbon contains on the order of 5 × 10²² atoms. With numbers that large, the statistical average is so overwhelmingly likely that the smooth exponential curve describes the population to many decimal places, even though every individual decay was a coin flip.

How does half-life work?

Because each nucleus has a constant probability of decaying per unit time, a constant fraction of the sample disappears in each equal period — not a constant amount. This is the single idea that makes decay exponential rather than linear, and it is where most mistakes come from.

Picture 1,000 g of an isotope with a 10-year half-life:

• Years 0 to 10: 500 g decays, leaving 500 g.
• Years 10 to 20: 250 g decays, leaving 250 g.
• Years 20 to 30: 125 g decays, leaving 125 g.
• Years 30 to 40: 62.5 g decays, leaving 62.5 g.

Every decade removes half of what was there at the start of that decade. The percentage lost is always 50%, but the mass lost falls each time: 500 g, then 250 g, then 125 g. A sample that started with a great deal of activity becomes quieter and quieter, which is why the curve flattens as it goes right.

The half-life formula

The core equation this calculator uses is:

N(t) = N₀ × (1/2)^{t / T½}

where N₀ is the initial quantity, T½ is the half-life, t is the elapsed time and N(t) is what remains after that time. The exponent t / T½ is just the number of half-lives that have passed — call it n — so the equation can be written even more simply as N = N₀ × (1/2)ⁿ.

The exponential form

You will also meet the same physics written with the base of natural logarithms:

N(t) = N₀ × e^−λt, where λ = ln(2) / T½

These two are not competing models, or approximations of each other. They are algebraically identical. Substituting λ = ln(2)/T½ into the exponential form gives e^{−ln(2)·t/T½}, and since e^ln(2) is exactly 2, that is 2^−t/T½ — the first equation again. Any half-life problem can be solved either way and must give the same answer. This calculator evaluates the power-of-two form because it is numerically better behaved, and reports λ separately for anyone who needs it.

Rearranging for the other unknowns

The same relationship solves for whichever quantity you are missing:

• Time: t = T½ × log₂(N₀ / N)
• Half-life: T½ = t ÷ log₂(N₀ / N)
• Initial quantity: N₀ = N × 2^{t / T½}

All three are available in the calculator above through the mode buttons.

How to calculate radioactive decay

The method is three short steps, and it is worth doing once by hand so the calculator's output means something.

Worked example

You have 1,000 g of an isotope whose half-life is 10 years. How much is left after 30 years?

Step 1 — make the units match. The half-life is in years and the elapsed time is in years, so there is nothing to convert. If the half-life had been given in days you would convert one to the other first; this is the single most common source of wrong answers.

Step 2 — count the half-lives. n = t ÷ T½ = 30 ÷ 10 = 3 half-lives.

Step 3 — apply the equation. N = 1000 × (1/2)³ = 1000 × 0.125 = 125 g.

So 125 g remains, which is 12.5% of the original. The decayed amount is 1000 − 125 = 875 g, or 87.5%. Notice that the remaining and decayed percentages always add to 100.

When the number of half-lives is not a whole number

Nothing changes. If 25 years have passed instead of 30, then n = 2.5, and N = 1000 × (1/2)^2.5 ≈ 176.8 g. Your calculator's yx key handles fractional exponents perfectly well, and so does the tool above.

How to calculate the decay constant

The decay constant λ is the probability per unit time that any one nucleus decays. It comes straight from the half-life:

λ = ln(2) / T½ ≈ 0.693 / T½

For a half-life of 5 years, λ = 0.693 ÷ 5 = 0.1386 per year, usually written 0.1386 year⁻¹. The unit matters: a decay constant is always a reciprocal time, so it inherits whatever unit the half-life was given in. Quote λ in s⁻¹ if the half-life was in seconds, in year⁻¹ if it was in years, and convert deliberately rather than by habit.

Mean lifetime

A related quantity is the mean lifetime τ (tau), the average time a nucleus survives before decaying:

τ = 1 / λ = T½ / ln(2) ≈ 1.44 × T½

The mean lifetime is always longer than the half-life, by about 44%. That feels wrong until you notice that the average is dragged upwards by the long tail of nuclei that survive far beyond the halfway point. The calculator reports τ alongside λ.

Half-life vs decay constant

These describe the same physics from opposite directions, and switching between them is a common exam task.

They are inversely related: a fast decay means a short half-life and a large decay constant. If you ever calculate a large half-life and a large decay constant, you have made an arithmetic error.

Understanding exponential decay

Exponential decay is any process where the rate of decrease is proportional to the amount currently present. Written as a differential equation, that sentence is:

dN/dt = −λN

Read it as: the number of decays per second is proportional to how many undecayed atoms are left. Solving it gives N(t) = N₀e^−λt, which is where the exponential form comes from. The minus sign simply says the population is shrinking.

The same mathematics governs a surprising range of things — the cooling of a hot drink, the discharge of a capacitor, the clearance of a drug from the bloodstream, the absorption of light passing through a filter. In each case something falls by a constant fraction per unit of time, so the half-life concept transfers directly, which is why pharmacologists talk about a drug's half-life in exactly the same way.

What distinguishes exponential decay from linear decrease is that the curve never becomes a straight line down to zero. It approaches the axis asymptotically, always getting closer, never mathematically arriving.

Reading the radioactive decay graph

The graph in this tool plots the remaining quantity against the number of half-lives elapsed. A few features are worth pointing out because they are frequently examined:

• The curve is steepest at the start. The most decay per unit time happens when the most undecayed atoms are present.
• It flattens but never touches the axis. Mathematically the value is always positive, however small.
• Equal horizontal steps give equal ratios, not equal drops. Every one unit to the right halves the height, whether you are going from 100% to 50% or from 1% to 0.5%.
• The shape never changes. Plotted in half-lives, carbon-14 over 50,000 years and a medical isotope over a few hours give exactly the same curve. Only the axis labels differ.

Switch the graph to % decayed and you get the mirror image: a curve rising towards 100% but never reaching it. Both views describe the same sample; use whichever matches the question being asked.

Plotting the natural logarithm of N against time instead gives a straight line of gradient −λ. That trick is how decay constants are actually measured in the laboratory, because a straight-line fit is far easier to do accurately than fitting a curve.

Carbon-14 and radiocarbon dating

Carbon-14 is the best-known application of half-life. It forms continuously in the upper atmosphere when cosmic-ray neutrons strike nitrogen-14, and it mixes into atmospheric carbon dioxide. Living things exchange carbon with their surroundings constantly, so while an organism is alive the proportion of carbon-14 in its tissues stays roughly in step with the atmosphere.

When the organism dies, that exchange stops. No new carbon-14 arrives, and the carbon-14 already present carries on decaying with a half-life of 5,730 years. Measuring how much is left tells you how long ago the exchange stopped.

Some worked figures, which you can reproduce with the carbon-14 preset above:

• After 5,730 years — one half-life — 50% of the original carbon-14 remains.
• After 11,460 years — two half-lives — 25% remains.
• After 17,190 years — three half-lives — 12.5% remains.
• After 28,650 years — five half-lives — about 3.125% remains.

This is also why radiocarbon dating has a practical ceiling of roughly 50,000 years. Beyond about nine half-lives there is so little carbon-14 left that the measurement is swamped by contamination and background. Real radiocarbon work also applies calibration curves, because the atmospheric carbon-14 level has not been perfectly constant through history — the raw exponential calculation is the starting point, not the final published date.

Half-life in physics, chemistry and nuclear science

In physics

Half-life is the standard way of describing nuclear stability. Known half-lives span an almost absurd range — from tiny fractions of a second for some synthetic superheavy nuclei to figures far exceeding the age of the universe for nearly-stable isotopes. The same equation covers the entire span; only the numbers change.

In chemistry

The term appears beyond the nucleus too. A first-order chemical reaction has a half-life that is independent of the starting concentration, mathematically identical to radioactive decay. Second-order reactions also have half-lives, but there the half-life does depend on concentration, so the simple constant-halving picture no longer applies. If you are using this calculator for reaction kinetics, check that your reaction really is first order.

In nuclear science and medicine

Half-life drives practical decisions. Diagnostic imaging favours isotopes with short half-lives so that the activity falls quickly once the scan is finished. Waste management deals with the opposite problem: long-lived isotopes remain radioactive for geological periods, which is why storage timescales are discussed in thousands of years. In both cases the underlying arithmetic is the equation on this page — though real decisions also involve biological clearance, shielding and regulation, none of which this tool models.

Common half-life calculation mistakes

Almost every wrong answer in a half-life question comes from one of these:

1. Mismatched time units. Dividing 30 days by a half-life of 2 years gives nonsense. Convert first, always. The calculator does this automatically and shows the conversion in the working.
2. Assuming two half-lives means nothing is left. Two half-lives leaves 25%, not 0%. Halving twice is not the same as subtracting twice.
3. Subtracting instead of halving. If 500 g decays in the first period, learners often subtract another 500 g for the second. The correct step is to halve what remains, giving 250 g.
4. Multiplying by the number of half-lives. The count n is an exponent, not a multiplier. It goes in the power, never in front.
5. Rounding n to a whole number. A time of 25 years with a 10-year half-life is 2.5 half-lives, not 2 or 3. Fractional exponents are perfectly valid.
6. Confusing remaining with decayed. Read the question carefully — "how much is left" and "how much has decayed" are complements that sum to 100%.
7. Forgetting the unit on λ. A decay constant without a reciprocal time unit is meaningless. 0.1386 is not an answer; 0.1386 year⁻¹ is.
8. Using this model for a second-order reaction. Constant halving is a property of first-order processes only.

Frequently asked questions

Related ToolAdda tools

If you are working through a physics or chemistry problem set, these pair naturally with the decay calculator. The Scientific Calculator handles the logarithms and fractional powers this page uses, and its log and ln keys are what you need when rearranging for time. The Chemical Equation Balancer solves reaction coefficients with atom-by-atom verification, and the Interactive Periodic Table covers element properties and flags radioactive elements, while the Electron Configuration Builder deals with the electron structure outside the nucleus.

For the arithmetic around your answer, the Percentage Calculator converts between remaining fractions and percentages, and the Unit Converter handles mass and time conversions when a question mixes units.

What this tool does, and what may come later

Available now: remaining-quantity, elapsed-time, half-life and initial-quantity solving; decay constant and mean lifetime; an interactive decay curve with quantity, percent-remaining and percent-decayed modes; a timeline slider measured in half-lives; an educational decay animation with adjustable speed; an automatically generated decay table; step-by-step working; decimal and scientific notation; and export by copy, CSV, PNG or print.

Under consideration for later versions: comparing two isotopes on one chart, entering a custom decay constant directly, activity and becquerel calculations, mass-to-atom-count conversion, decay chain visualisation, a built-in isotope library, and a logarithmic graph mode. These are not implemented yet and are listed here only so the current scope is clear.