Half-Life Calculator

The half-life calculator solves the radioactive decay equation for any of its four variables. Whether you need to know how much of a sample remains after a given time, how long until a sample decays to a target amount, or what half-life an unknown isotope must have to produce the observed decay, this tool handles all four cases in one place.

Half-life is a fundamental concept in physics, chemistry, pharmacology, archaeology, and nuclear engineering. The mathematics is identical in every field, exponential decay following first-order kinetics, but the applications vary widely. This page explains the theory, the formula, and the practical use of the calculator.

How to use the half-life calculator

The calculator has four inputs and a "solve for" selector. The relationship between the inputs is fixed by the decay equation, so you must provide three of them and let the tool compute the fourth.

1. Select what you want to solve for. Choose one of: half-life (t½), elapsed time (t), remaining amount (A), or initial amount (A₀). The selected variable is hidden from the inputs (or rather, its result becomes the output).
2. Enter the other three values in the form fields. All values must be positive numbers; the calculator will reject zero or negative inputs.
3. Pick a consistent time unit. If your half-life is in years, your elapsed time must also be in years. The dropdown covers seconds, minutes, hours, days, and years.
4. Click Calculate. The result box shows the solved value, the formula used, and a summary block with the half-lives elapsed, fraction remaining, percent remaining, and percent decayed.
5. Optional, load a preset. Click any of the isotope buttons (Carbon-14, Uranium-238, Potassium-40, Cobalt-60, Caesium-137, Iodine-131, Radon-222) to populate the half-life field with a real, tabulated value. The calculator will automatically switch to "solve for remaining amount" so you can see what fraction of a sample survives after any elapsed time.

The result panel uses exponential notation when the values are very small or very large, so the calculator remains readable for both a 30-year caesium half-life and a 4.5-billion-year uranium half-life.

What is half-life?

The half-life of a substance is the time required for half of a sample to decay. If you start with 100 g of a radioactive isotope that has a half-life of 5 years, you'll have 50 g remaining after 5 years, 25 g after 10 years, 12.5 g after 15 years, and so on. The amount never quite reaches zero, it gets halved again and again, getting smaller but never vanishing, but for practical purposes, after about ten half-lives the amount is less than 0.1% of the original.

Half-life is a statistical property. It does not mean that any particular atom is guaranteed to decay within one half-life interval. The decay of any individual atom is random and unpredictable. What the half-life describes is the average behaviour of a large population: given a large number of atoms, half of them will have decayed in the time equal to one half-life. With 10²⁰ atoms, the statistical average is extraordinarily precise.

The half-life is also constant for a given isotope. It does not depend on temperature, pressure, chemical form, age of the sample, or any other physical variable (with the very narrow exception of electrons captured by fully ionised atoms in extreme conditions). This is what makes radioactive dating reliable, a sample of uranium buried in rock for a billion years decays at the same rate as uranium in a lab on the surface today.

The radioactive decay formula

The decay follows first-order kinetics. The amount of substance remaining at time t is:

A(t) = A₀ × (½)^(t / t½)

Where A(t) is the amount at time t, A₀ is the initial amount, t is the elapsed time, and t½ is the half-life. This can also be written using the natural exponential:

A(t) = A₀ × e^(-λt)

where λ = ln(2) / t½ ≈ 0.693 / t½ is the decay constant. The two forms are mathematically equivalent because 2 = e^(ln 2).

From these two equivalent forms, we can rearrange to solve for any variable:

The calculator uses the exact rearrangement for the chosen target. Both logs (log₂ and logₑ) are used; the natural log appears in the derivation, and the base-2 log is the direct rearrangement of the half-life form.

Real-world applications

Radiocarbon dating. Carbon-14 is produced continuously in the upper atmosphere by cosmic-ray spallation. Living organisms absorb it (through CO₂ in photosynthesis for plants, through the food chain for animals) at roughly the same rate as the background atmospheric concentration. When an organism dies, intake stops and the C-14 begins to decay with a half-life of 5,730 years (the "Cambridge half-life," the internationally agreed value since 1962). By measuring the ratio of C-14 to stable C-12 in a sample, archaeologists can determine when the organism died. The method is reliable back to about 50,000 years (about nine half-lives); beyond that, the remaining C-14 is too small to measure precisely.

Nuclear medicine. Short-lived radioisotopes are used for diagnosis and treatment because they deliver a controlled radiation dose that decays to safe levels quickly. Iodine-131 (8.02 days) concentrates in the thyroid and is used to treat hyperthyroidism and thyroid cancer. Technetium-99m (6 hours) is the most widely used medical radioisotope, used in roughly 80% of nuclear medicine procedures. Fluorine-18 (110 minutes) is used in PET scans. The short half-life is a feature: the patient gets the diagnostic or therapeutic benefit, and the radiation clears the body in hours to weeks.

Nuclear waste storage. Spent fuel rods and other high-level waste contain a mixture of radioactive isotopes with half-lives ranging from seconds to hundreds of thousands of years. Caesium-137 (30.17 years) and strontium-90 (28.8 years) are the main radiation hazards in the first few centuries after the fuel is removed from a reactor. Plutonium-239 (24,100 years) is the dominant long-term hazard. The fact that some isotopes have half-lives comparable to recorded human history is why deep geological disposal (e.g. Finland's Onkalo repository, designed to contain waste for 100,000 years) is the only currently accepted permanent storage method.

Pharmacokinetics. Many drugs are eliminated from the body by first-order kinetics: a constant fraction of the drug in the bloodstream is removed per unit time, regardless of concentration. The drug's "half-life" in the body is the time for plasma concentration to fall by half. For example, caffeine has a half-life of about 5 hours in adults, drink a cup of coffee at 8 am, and by 1 pm half the caffeine has been metabolised; by 6 pm only 25% remains; by bedtime most of it is gone. The same exponential mathematics that governs radioactive decay governs the rise and fall of drug levels in the body.

Why half-life is constant

The half-life of a radioactive isotope is one of the most precisely constant quantities in nature. Unlike chemical reaction rates, which depend on temperature, pressure, concentration, and catalysts, nuclear decay rates are essentially invariant. A sample of uranium in a deep mine, in a cold ocean trench, in the core of a nuclear reactor, or on the surface of the Moon (in the lunar regolith brought back by Apollo astronauts) decays at the same rate to within experimental precision, about one part in a thousand across all measured conditions.

This constancy is what makes radiometric dating possible. The age of the Earth, the age of the solar system, the timing of evolutionary divergences, the dating of archaeological sites, and the dating of human artefacts all rely on this invariance. Without it, none of the chronology of the deep past would be trustworthy.

The physical reason is that nuclear decay is governed by the weak interaction, one of the four fundamental forces of nature. The probability that a given nucleus will decay in a given interval depends only on the structure of the nucleus itself, not on external conditions. Changing the chemical environment changes the electron cloud, but not the nucleus. Even extreme conditions, temperatures of millions of degrees, pressures of millions of atmospheres, magnetic fields millions of times stronger than Earth's, have no measurable effect on decay rates.

Working with extreme half-lives

The calculator handles both very long and very short half-lives without difficulty. Some examples:

• Uranium-238 has a half-life of 4.468 billion years. Over a human lifetime (say 80 years), the fraction decayed is 80 / 4,468,000,000 × ln(2) ≈ 0.0012%, invisible. But over the age of the Earth (4.54 billion years, about one half-life), half the original U-238 has decayed to thorium-234 and then through a long chain to lead-206.
• Carbon-14 with a half-life of 5,730 years. Useful for dating materials from about 100 years to 50,000 years old. After ten half-lives (57,300 years), only 0.0977% of the original C-14 remains, too little to measure reliably.
• Iodine-131 with a half-life of 8.02 days. Used in medicine because after a few weeks the activity is essentially zero, but long enough to treat the patient.
• Polonium-214 with a half-life of 164 microseconds. Decays so fast that any sample produced in a reaction is gone almost instantly. Useful in some industrial applications, useless for any kind of long-term measurement.

When the elapsed time is many half-lives (say, more than 30), the remaining amount rounds to zero in standard double-precision arithmetic. The result panel will show 0 (or a very small scientific-notation number), which is mathematically correct.

Common mistakes

Mixing time units. The most common error is to enter a half-life in years and an elapsed time in days (or vice versa). The calculator uses the unit selected in the dropdown for both fields, so if you say "years," both numbers are interpreted in years. For mixed units, convert first: 30 days = 30/365.25 years ≈ 0.082 years.

Confusing half-life with mean lifetime. Half-life is the time for half the sample to decay. Mean lifetime (τ) is the average lifetime of a single atom: τ = t½ / ln(2) ≈ 1.4427 × t½. The two are not the same. A 1-year half-life corresponds to a 1.44-year mean lifetime. Both are valid quantities; they answer slightly different questions.

Treating decay as linear. After one half-life, 50% remains. After two half-lives, 25% remains, not 0%. The decay is exponential, not linear. A common error is to assume that ten half-lives leaves 5% remaining; the actual figure is 0.0977%. For real-world "how long until safe" questions, 10 half-lives is the standard rule of thumb for "essentially gone."

Worked examples

Example 1, Carbon-14 dating. A piece of charcoal from an archaeological site contains 25% of the C-14 it would have contained when the tree was alive. With t½ = 5,730 years, the time elapsed is t = 5,730 × log₂(100/25) = 5,730 × 2 = 11,460 years. The site is about 11,500 years old.

Example 2, Medical imaging. A patient is given 10 MBq of technetium-99m (half-life 6 hours). After 24 hours, the remaining activity is A = 10 × (½)^(24/6) = 10 × (½)⁴ = 10 / 16 = 0.625 MBq. After 48 hours, A = 10 / 256 ≈ 0.039 MBq, essentially zero.

Example 3, Nuclear waste. A spent fuel rod contains 1 kg of caesium-137. After 100 years (about 3.32 half-lives), the remaining caesium is 1 × (½)^3.32 ≈ 0.10 kg. The site will remain hazardous for roughly 10 half-lives (300 years) before Cs-137 activity drops below regulatory thresholds.

Example 4, Caffeine. A 200 mg dose of caffeine, with a half-life of 5 hours. After 5 hours: 100 mg remaining. After 10 hours: 50 mg. After 15 hours: 25 mg. After 24 hours: 200 × (½)^(24/5) = 200 × 0.0718 ≈ 14.4 mg. By bedtime (say 16 hours after the morning coffee), 200 × (½)^3.2 ≈ 43 mg remain.

Limits of the model

The exponential decay model applies to first-order processes, any process where the rate of decay is proportional to the amount present. This includes radioactive decay, many chemical reactions, drug elimination (in most cases), and charge/discharge of capacitors through a resistor.

It does not apply to:

• Zero-order processes (rate independent of amount), such as alcohol elimination at high concentrations. The body metabolises alcohol at a roughly constant rate per hour, not a constant fraction.
• Saturation kinetics (where the rate depends on the available "machinery" to do the work). Some drugs and many enzymatic reactions fall into this category.
• Branching decay (where a parent isotope decays into a daughter that is itself radioactive). The total activity of the sample then has a more complex time dependence, requiring the Bateman equations.
• Mixed samples (where the measured "amount" is a sum of contributions from multiple isotopes with different half-lives). A nuclear waste sample typically contains dozens of isotopes.

For these more complex cases, the simple exponential model gives an approximation at best. The calculator is exact for single-species first-order decay; for anything more complicated, a more elaborate model is needed.

Frequently Asked Questions

How do I calculate the remaining amount after a given time? Use A = A₀ × (½)^(t / t½). For example, with A₀ = 100 g, t = 10 years, t½ = 5 years, A = 100 × (½)² = 25 g. The calculator does this for you when you select "remaining amount" as the target.

What is the half-life of Carbon-14? 5,730 years. This is the "Cambridge half-life", the value adopted by international convention in 1962 and used in all radiocarbon dating since. The actual physical half-life is closer to 5,730 years; the difference between the old "Libby half-life" of 5,568 years and the modern value reflects refinements in measurement, not a real change in C-14 itself.

Can I use this for drug half-life? Yes, the math is identical. If a drug has a 6-hour half-life and you take 100 mg, after 6 hours you have 50 mg, after 12 hours 25 mg, after 24 hours 6.25 mg. The calculator does not, however, model saturation kinetics, active metabolites, or any other complications, it is a pure exponential decay calculator.

How accurate is the result? The mathematics is exact to floating-point precision (about 15 significant digits). The accuracy of any real-world answer depends on the inputs. Isotope half-lives are tabulated to 4 or more significant figures; for a 5,730-year half-life, the actual uncertainty is about ±40 years. Volume measurements of decay products have their own experimental uncertainties, often larger.

What if my initial and remaining amounts are equal? Then either no time has passed (t = 0) or the half-life is infinite. The calculator will display an error because log₂(1) = 0, and division by zero is undefined. Mathematically, the only way for A₀ = A after a positive time is if t½ is infinite, i.e. the substance is stable.

What is the difference between half-life and mean lifetime? Half-life (t½) is the time for half the sample to decay. Mean lifetime (τ) is the average lifetime of a single atom: τ = t½ / ln(2) ≈ 1.4427 × t½. Both are valid measures. Half-life is more common in chemistry, pharmacology, and everyday use; mean lifetime is more common in physics.

Why does the percentage decayed not equal the percentage remaining plus 100? It does, they always sum to 100%. The result panel shows both as a sanity check. If you see 75% decayed and 25% remaining, that's two half-lives elapsed.

Is half-life the same as the rate constant? No. The rate constant λ = ln(2) / t½ is the proportionality factor in the natural-exponential form A(t) = A₀ × e^(-λt). The two are related: a small λ (long half-life) means slow decay; a large λ (short half-life) means fast decay. They are interchangeable, but you must use the one your equation expects.

Sources and references

• Krane, K. S. (1988). Introductory Nuclear Physics. John Wiley & Sons., Standard undergraduate text covering radioactive decay law, half-life, and decay chains.
• Curie, P. & Curie, M. (1902). Sur la constante de temps caractéristique de la conversion des substances radioactives. Comptes Rendus., Original derivation of the decay constant.
• Libby, W. F. (1962). Radiocarbon Dating. University of Chicago Press., The classic reference on C-14 dating.
• BIPM (2019). SI Brochure: The International System of Units., Defines the second, on which all half-life measurements are based.
• NUBASE / IAEA (2021). Evaluated Nuclear Structure Data File., The standard source for tabulated half-lives of all known isotopes.

References

• HyperPhysics, Half-Life, Georgia State University, the exponential decay treatment. http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/halfli2.html
• IUPAC Gold Book, half-life (radioactive decay). https://goldbook.iupac.org/terms/view/H01316