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. This half-life calculator works in one direction: enter a decay rate and it returns the 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

1. Find the decay constant. Look up λ for the substance, or derive it from measured data, keeping careful track of its time unit.
2. Divide ln(2) by λ. The result, t½ = ln(2)/λ, is the half-life in the same time unit as λ.
3. 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. The decay-rate box accepts this in e-notation, typed as 1.54e-10.

What This Calculator Shows

The calculator takes two inputs. The starting amount is the quantity present at time zero, in any unit the user chooses, and it must be at least 0.1. The decay rate is λ, given per unit of time.

The result is the half-life, printed to four decimal places, in the same time unit as λ. Below it, a sentence restates the result as the time needed for the starting amount to fall to half. A decay rate of zero or less is refused, because ln(2)/0 has no value.

The chart plots N(t) = N₀ × e^(−λt) from time zero to five half-lives, using 101 evenly spaced points. Five half-lives leave 1/32 of the starting amount, about 3.1%. The curve can be downloaded as a spreadsheet at a step the user picks.

Worked example

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

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