Half-Life Calculator: Radioactive Decay & Elimination
This half-life calculator finds t½ = ln(2)/λ from a decay rate, for radioactive isotopes, drug elimination, and other exponential decay, with worked examples.
Half-Life Calculator
Calculate half-life from decay rate for radioactive isotopes, medications, or any exponentially decaying substance. Half-life is the time it takes for a quantity to reduce to half its initial value.
The half-life is calculated using the following formula:
Where λ (lambda) is the decay constant, which represents the rate at which the substance decays.
Decay Visualization
The graph shows how the quantity decreases over time, following the curve N(t) = N₀ × e^(−λt). The quantity reaches half its initial value at the calculated half-life.
Documentation
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 1.55 × 10⁻¹⁰ per year for uranium-238. Most calculators, including this one, accept this either in exponent form or as "1.55e-10". The calculator shows the half-life to four decimal places and plots the decay curve over five half-lives.
Worked Examples
Example 1: Carbon-14 dating
Carbon-14 has a decay rate of about 1.21 × 10⁻⁴ per year, which corresponds to its measured half-life of 5,730 years.
t½ = ln(2) / (1.21 × 10⁻⁴) ≈ 5,728 years
The small gap from 5,730 comes from rounding λ to three significant figures.
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 is accurate up to about 30,000 years and gives reasonable dates up to about 50,000 years. Past that, too little carbon-14 remains to measure.
Example 2: Iodine-131 in medicine
Iodine-131, used in thyroid treatment and imaging, has a half-life of about 8 days. Its decay rate is 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, bring the radioactivity down to about a thousandth of the original level (1/2¹⁰ = 1/1,024).
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. Too short a gap can let the drug build up toward toxic levels. Too long a gap lets the level fall below the useful range.
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 50,000 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 decays away within months. Plutonium-239 has a half-life of about 24,110 years, so ten half-lives last about 241,000 years.
Pharmacology. A drug's half-life sets its dosing schedule. With regular doses, the blood level builds up over several half-lives and then levels off. After a single dose, five half-lives leave about 3% of it (1/2⁵ = 1/32).
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?
Carbon-14 dating is accurate for samples up to about 30,000 years old and gives reasonably accurate dates up to about 50,000 years. Older samples hold too little carbon-14 to measure reliably.
References
- OpenStax. Chemistry 2e, 21.3 Radioactive Decay. Radioactive Decay — t½ = ln 2/λ, N = N₀e^(−λt), carbon-14 half-life 5,730 years and λ = 1.21 × 10⁻⁴ per year, carbon-14 dating range.
- U.S. Environmental Protection Agency. Radionuclide Basics: Uranium. Radionuclide Basics: Uranium — uranium-238 half-life of 4.47 billion years.
- U.S. Environmental Protection Agency. Radionuclide Basics: Iodine. Radionuclide Basics: Iodine — iodine-131 half-life of 8 days.
- U.S. Environmental Protection Agency. Radionuclide Basics: Plutonium. Radionuclide Basics: Plutonium — plutonium-239 half-life of 24,110 years.
- U.S. Environmental Protection Agency. Radionuclide Basics: Technetium-99. Radionuclide Basics: Technetium-99 — technetium-99m half-life of 6 hours.
- U.S. Environmental Protection Agency. Radionuclide Basics: Cesium-137. Radionuclide Basics: Cesium-137 — cesium-137 half-life of 30.17 years.
- U.S. Environmental Protection Agency. Radionuclide Basics: Strontium-90. Radionuclide Basics: Strontium-90 — strontium-90 half-life of 29 years.