Some processes are defined by how fast they disappear rather than how much they grow, and none is more famous than radioactive decay. A lump of radioactive material does not vanish all at once; it sheds a fixed fraction of itself over a fixed period, called its half-life. The Half-Life Calculator takes an initial amount, a half-life and an elapsed time, and returns how much of the substance remains, what fraction that is, how much has decayed and the underlying decay constant.
What a Half-Life Is
The half-life of a substance is the time required for half of its atoms to decay. It is a statistical property of the isotope, not of the size of the sample: the same isotope always has the same half-life, whether you hold a gram of it or a thousand tons. Carbon-14, the isotope behind radiocarbon dating, has a half-life of about 5,730 years. Uranium-238, the fuel of nuclear reactors and the anchor of geological dating, has a half-life of roughly 4.5 billion years. Iodine-131, used in medical treatments, decays in about eight days. That range, from days to billions of years, is what makes the concept both powerful and easy to misunderstand.
Because decay is statistical, a half-life means that after one half-life you have half the original, after two you have a quarter, after three an eighth, and so on. It never reaches zero in theory, which is one of the most counterintuitive facts about radioactivity: a sample is never technically gone, it just becomes a smaller and smaller fraction of what it was. The calculator models exactly this, showing the amount remaining at whatever elapsed time you choose.
The Decay Formula
The mathematics is a clean exponential: the amount remaining equals the initial amount times one-half raised to the power of elapsed time divided by half-life. In symbols, N(t) equals N-zero times 0.5 to the power of t over T-half. The fraction t divided by T-half counts how many half-lives have passed, so after exactly one half-life the exponent is one and the result is one-half, after two half-lives the exponent is two and the result is one-quarter, and so on. Any time between those milestones produces a value between them, which is where a calculator earns its keep, because nobody computes a fractional exponent like a thousand divided by 5730 in their head.
The calculator applies this formula directly. Start with 100 grams of carbon-14 with a 5,730 year half-life, wait 1,000 years, and the formula gives about 88.6 grams remaining, roughly 88.6 percent of the original. Those are the numbers the tool returns, and they are the numbers worth knowing for everything from archaeology to nuclear waste planning.
Reading the Fraction Remaining
The second output expresses the same result as a percentage of the original amount. It answers the intuitive question, what share of the original is still here, without making you divide the remaining grams by the initial grams yourself. For the carbon-14 example it reads about 88.6 percent. Because this value always falls between zero and one hundred, it is reported directly as a percentage, and it is the number most people quote when they describe how far along a decay process is. It also makes comparisons easy: a sample at 25 percent remaining has decayed through exactly two half-lives, regardless of how large or small the sample started out.
How Much Has Decayed
The third output, the amount decayed, is simply the original amount minus the amount remaining. It is the quantity that has already transformed into its decay products, the atoms that no longer count as the original isotope. This number matters in practical settings such as estimating how much of a radioisotope has been consumed, how much daughter material has accumulated, or how much of a medical dose has cleared by a given time. The calculator computes it in one step, so you never have to remember which of the two numbers to subtract from which.
The Decay Constant
The fourth output is the decay constant, the probability per unit time that an individual atom will decay, expressed here per year. It is connected to the half-life by a simple relationship: the decay constant equals the natural logarithm of two divided by the half-life, approximated here as 0.693147 divided by the half-life in years. The constant is the form of the decay model used in physics and engineering equations, so it is the bridge between the everyday half-life and the professional literature. For carbon-14 it works out to about 0.000121 per year, a tiny number that captures just how slow the process is.
Short Half-Lives and Fractional Years
The calculator works in years for both the half-life and the elapsed time, so the two always share a unit. Isotopes with very short half-lives are simply entered as fractional years. Iodine-131 with an eight-day half-life is about 0.022 years, and radon-222 at roughly 3.8 days is about 0.010 years. Entering those fractions lets the same formula handle everything from medical isotopes to geological ones, with the article values listed for reference. The exponential model itself does not care about the unit; it only requires that the half-life and the elapsed time are expressed in the same one.
The Decay Curve
The line chart plots the amount remaining against elapsed time at your chosen half-life, showing the classic decay curve: steep at first, then flattening toward zero. The curve makes the diminishing returns of decay visible, because each successive half-life removes a smaller absolute amount even though it removes the same fraction. This is why the phrase "half-life" is so important and so often abused in everyday language, the property is always the same fraction per period, never the same absolute amount. Watching the curve drop from half to a quarter to an eighth gives you the picture that numbers alone can blur.
Real-World Uses
Half-life arithmetic sits behind some of the most important measurements in science and medicine. Radiocarbon dating uses the carbon-14 half-life to date organic remains, relating the current fraction of carbon-14 to the time since the organism died. Medical treatments rely on the half-life of a radioisotope to know how long it stays active in the body. Nuclear waste management plans are built around the half-lives of the isotopes in spent fuel, which is why the discussion is framed in decades and millennia. Even in non-radioactive fields, the exponential-decay model describes everything from the concentration of a drug in the blood to the discharge of a capacitor, so the mathematics here generalizes far beyond its famous example.
A Worked Example
Consider 200 grams of a substance with a half-life of 100 years, checked after 300 years. The elapsed time is exactly three half-lives, so the fraction remaining is one-half cubed, which is one-eighth. That leaves 25 grams of the original 200, a fraction of 12.5 percent, with 175 grams decayed. Now enter a non-integer case: the same 200 grams, but only 250 years elapsed. The exponent is 2.5 half-lives, and the formula gives roughly 35.4 grams, about 17.7 percent, with about 164.6 grams decayed. These are precisely the numbers the calculator returns, and the contrast between the tidy three-half-life case and the awkward fractional one shows exactly when the tool becomes necessary.
Common Isotope Half-Lives Worth Knowing
A handful of half-lives anchors most everyday discussions of radioactivity. Carbon-14 sits at 5,730 years and drives radiocarbon dating. Uranium-238 decays over about 4.5 billion years, roughly the age of the Earth, which is why it can date the oldest rocks. Iodine-131, a common medical isotope, clears in about eight days, while iodine-125, used in some treatments and imaging, has a half-life of about 60 days. Cesium-137, a major concern in nuclear accidents, persists for about 30 years, and tritium, used in exit signs and some watches, lasts about 12 years. Keeping these reference points in mind makes the calculator's half-life slider easier to set for any isotope you encounter.
Assumptions to Keep in Mind
- Decay is modeled as pure exponential decay, halving the amount each half-life.
- Half-life and elapsed time are both entered in years.
- The decay constant is ln(2) divided by the half-life, using 0.693147.
- The sample is treated as closed, ignoring ingoing material and decay chains.
- The fraction remaining is reported as a percentage between 0 and 100.
Half-life is one of those concepts that sounds simple and behaves almost like magic, until you have to compute a fraction of a half-life and the simplicity evaporates. The Half-Life Calculator keeps the exponential decay model precise and painless, turning a famous formula into four clear numbers you can use, whether you are dating a fossil or just satisfying a curiosity about how fast things fall apart. Try sliding the half-life and elapsed time in opposite directions to see how sensitive the remaining amount is to both, and the shape of the process will stay with you.
Disclaimer
Results are provided as estimates for informational purposes only and may be inaccurate. Always verify outcomes with a qualified professional before making financial or personal decisions based on these calculations.