Half-Life Calculator: Radioactive Decay & Elimination

What Is Half-Life?

Half-life is the time it takes for a quantity to fall to half of its starting value. A half-life calculator finds this time from a decay rate, or finds the decay rate from a known half-life. The idea applies to radioactive isotopes, drugs leaving the body, and any other process that decays exponentially.

Half-life does not depend on how much material is present. Uranium-238 has a half-life of about 4.5 billion years whether the sample is one kilogram or one atom's worth of a large population. Only the decay rate of the substance sets the half-life.

Half-Life Formula

The standard formula is:

t½ = ln(2) / λ

t½ is the half-life, in whatever time unit λ uses.
ln(2) is the natural logarithm of 2, about 0.6931.
λ (lambda) is the decay constant, the fraction of the substance that decays per unit of time.

This formula comes from the exponential decay equation:

N(t) = N₀ × e^(−λt)

Here N₀ is the starting quantity, N(t) is the quantity left after time t, and e is Euler's number, about 2.718. Setting N(t) equal to N₀/2 and solving for t gives t½ = ln(2)/λ. A higher decay rate means a shorter half-life; a lower decay rate means a longer one.

Decay rate units

The decay rate λ is measured in inverse time, such as per second, per day, or per year. It must match the time unit wanted for the answer. A rate given as "per day" and read as "per hour" will make the result wrong by a factor of 24, so checking units first avoids errors before they spread through a calculation.

How to Calculate Half-Life

Find the decay constant. Look up λ for the substance, or derive it from measured data, keeping careful track of its time unit.
Divide ln(2) by λ. The result, t½ = ln(2)/λ, is the half-life in the same time unit as λ.
Check the size of the answer. A medical isotope should not produce a million-year half-life; a geological one should not produce a half-life measured in days. A result far outside the expected range usually points to a unit mismatch.

For very long-lived isotopes, λ is often written in scientific notation, such as 1.54 × 10⁻¹⁰ per year for uranium-238. Most calculators, including this one, accept this either in exponent form or as "1.54e-10".

Worked Examples

Example 1: Carbon-14 dating

Carbon-14 has a decay rate of about 1.21 × 10⁻⁴ per year.

t½ = ln(2) / (1.21 × 10⁻⁴) ≈ 5,730 years

Living organisms exchange carbon with the atmosphere continuously, keeping their carbon-14 level constant. After death, that exchange stops, and the carbon-14 already present begins to decay on this schedule. Comparing the remaining carbon-14 to stable carbon-12 lets scientists estimate how long ago an organism died. The method becomes unreliable past about 60,000 years, since too little carbon-14 remains to measure accurately by then.

Example 2: Iodine-131 in medicine

Iodine-131, used in thyroid treatment and imaging, has a decay rate of about 0.0862 per day.

t½ = ln(2) / 0.0862 ≈ 8.04 days

After about 8 days, half of an administered dose of iodine-131 has decayed. Ten half-lives, roughly 80 days, brings the radioactivity down to about a thousandth of the original level, which hospitals use when planning patient safety measures and room turnover.

Example 3: A drug elimination rate

A medication is eliminated from the body at a rate of 0.2 per hour.

t½ = ln(2) / 0.2 ≈ 3.47 hours

Half of the drug leaves the bloodstream in about 3.5 hours. Doctors use this figure to space doses so the drug stays within a therapeutic range: too short a gap risks a buildup toward toxic levels, and too long a gap lets the level fall below effectiveness.

Related Quantities

A few other terms describe the same decay process from different angles.

Decay constant (λ) is the input to the half-life formula. It equals ln(2)/t½ and gives the probability per unit time that a given atom decays.
Mean lifetime (τ) is the average time a particle exists before decaying, equal to 1/λ, or t½/ln(2). It runs about 1.44 times longer than the half-life.
Activity, measured in becquerels or curies, is the number of decays per second in an actual sample. Unlike λ, activity depends on how much material is present: a gram of a short-lived isotope has far higher activity than a gram of a long-lived one.
Effective half-life applies inside living organisms, where a substance is removed by both physical decay and biological processes such as excretion. It is found from 1/t_eff = 1/t_physical + 1/t_biological, and it is always shorter than the physical half-life alone.

Applications of Half-Life

Radiometric dating. Carbon-14 dates organic material up to about 60,000 years old. Longer-lived isotopes date older material: uranium-238 decaying into lead-206, with a half-life near 4.5 billion years, is used to date rock formed since the origin of the Earth.

Nuclear waste management. Spent fuel contains a mix of isotopes with very different half-lives. Iodine-131 becomes safe within months, while plutonium-239, with a half-life near 24,000 years, needs storage on a scale of hundreds of thousands of years.

Pharmacology. A drug's half-life sets its dosing schedule. Regular dosing reaches a stable blood concentration after about five half-lives, and clearing a drug from the body to a negligible level also takes about five half-lives.

Nuclear medicine. Technetium-99m, with a half-life of about 6 hours, is common in diagnostic imaging because it stays active long enough to produce an image but decays quickly enough to limit a patient's radiation exposure.

Environmental tracing. Cesium-137 (about 30 years) and strontium-90 (about 29 years) are used to track contamination from nuclear accidents, since their half-lives are long enough to persist for decades but short enough to eventually decay away.

Frequently Asked Questions

What is half-life in simple terms?

Half-life is the time needed for half of a quantity to disappear or convert into something else. Starting with 100 grams of a radioactive isotope, one half-life later leaves 50 grams, and one more half-life leaves 25 grams. This time is the same no matter how much material is present at the start.

How is half-life calculated from a decay rate?

Divide ln(2), about 0.693, by the decay rate λ: t½ = ln(2)/λ. The result comes out in whatever time unit λ was measured in, so a rate per year gives a half-life in years.

Does temperature or pressure change a half-life?

No. Radioactive half-lives are set by processes inside the atomic nucleus and are not affected by temperature, pressure, or chemical state. This is different from biological or effective half-lives, which can shift with factors like metabolism.

How many half-lives until a substance is essentially gone?

Exponential decay never reaches exactly zero, but after 10 half-lives only about 0.1% of the original amount remains, a level usually treated as negligible in practice.

Does half-life apply to things that are not radioactive?

Yes. Any quantity that decays exponentially has a half-life, including caffeine leaving the bloodstream, drugs clearing the body, and some pollutants breaking down in soil or water. The same formula, t½ = ln(2)/λ, applies in each case.

How accurate is carbon-14 dating?

For samples younger than about 30,000 years, results are typically accurate to within a few hundred years, after calibration against tree-ring records to account for past changes in atmospheric carbon-14 levels. Beyond about 60,000 years, too little carbon-14 remains for a reliable measurement.

References

L'Annunziata, Michael F. (2016). Radioactivity: Introduction and History, From the Quantum to Quarks. Elsevier Science. ISBN 978-0444634979.
Krane, Kenneth S. (1988). Introductory Nuclear Physics. Wiley. ISBN 978-0471805533.
National Institute of Standards and Technology. "Radionuclide Half-Life Measurements." https://www.nist.gov/pml/radionuclide-half-life-measurements
International Atomic Energy Agency. "Live Chart of Nuclides." https://www-nds.iaea.org/relnsd/vcharthtml/VChartHTML.html

Half-Life Calculator: Radioactive Decay & Elimination