Half-Life and Radioactive Decay Calculator

Solve for remaining quantity, elapsed time, or half-life. Choose from 20 real isotopes, watch the decay curve, and get activity in becquerels and curies.

Solve for:
Half-Life
T½
Initial Amount
N0
Elapsed Time
t
Remaining Amount
N(t)
units
Enter values above to calculate.

Equations

N(t) = N0 × (½)t / T½ = N0 e−λt
λ = ln(2) / T½     (decay constant)
τ = 1 / λ = T½ / ln(2)     (mean lifetime)
t = T½ × log2(N0 / N)     (age from remaining fraction)
A = λN     (activity, in decays per second = becquerels)
1 Ci = 3.7 × 1010 Bq     1 Bq = 1 decay/s
Note on statistics: Radioactive decay is a random process. These equations describe the expected behaviour of a large number of nuclei, and are extremely accurate for macroscopic samples — even a microgram contains upwards of 1015 atoms. For very small numbers of atoms the actual decay count fluctuates around the predicted value with a standard deviation of approximately √N (Poisson statistics).

About Radioactive Decay

An unstable atomic nucleus will sooner or later shed energy by emitting radiation. What makes this process remarkable is that it is completely memoryless: a nucleus that has already existed for a billion years is exactly as likely to decay in the next second as one created a moment ago. There is no ageing, no wearing out, and no way to predict which particular nucleus will go next. Yet from this pure randomness emerges one of the most precise and reliable laws in physics.

The reason is statistics. Each nucleus has a fixed probability per unit time of decaying — the decay constant λ. With enormous numbers of nuclei present, the aggregate behaviour becomes essentially deterministic, giving the exponential law N(t) = N0e−λt. The half-life is simply the time for half the sample to decay, and it is a constant of the isotope: unaffected by temperature, pressure, chemical bonding, or magnetic fields. Half-lives span an extraordinary range, from under a microsecond to more than 1024 years — vastly longer than the age of the universe.

Radioactivity was discovered by Henri Becquerel in 1896, when uranium salts left in a drawer fogged a photographic plate with no sunlight involved. Marie and Pierre Curie isolated polonium and radium and coined the term radioactivity; Marie Curie remains the only person to win Nobel Prizes in two different sciences. Ernest Rutherford and Frederick Soddy established in 1902 that decay transmutes one element into another and follows an exponential law — the discovery that gave us the concept of half-life.

The applications are everywhere. Radiocarbon dating exploits carbon-14's 5,730-year half-life to date organic remains up to roughly 50,000 years old, while uranium-lead and potassium-argon dating reach back billions of years and give us the age of the Earth. In medicine, technetium-99m is used in tens of millions of diagnostic scans each year, and iodine-125 seeds are implanted directly into tumours for brachytherapy. Most medical isotopes are manufactured in research reactors and cyclotrons — university research reactors such as the one at McMaster University are among the world's principal suppliers of iodine-125. Household smoke detectors rely on a tiny quantity of americium-241, and radon-222 seeping from soil is the second leading cause of lung cancer.

Explore Nuclear Physics Hands-On: Radiation detection and measurement bring these ideas into the lab. Browse Test & Measurement Instruments and Physics Kits at xUmp.com — curated by a physicist.

How to Use This Calculator

Pick an isotope to load its half-life automatically, or choose Custom and type your own. Use the Solve for buttons to choose which quantity you want calculated — the remaining amount after a given time, the elapsed time (useful for dating problems), or the half-life itself (useful when you have measured a decay in the lab). The blue highlighted field is always the calculated one.

Enter the initial amount in grams, moles, atoms, or arbitrary units. If you use a mass or a count, the calculator also reports the activity in becquerels and curies. Drag the time slider or press Animate to watch the sample decay, and switch the vertical axis to Log to see the exponential become a straight line.

Try these: Carbon-14 with 25% remaining (solve for elapsed time) gives about 11,460 years — exactly two half-lives. Technetium-99m over 24 hours shows why hospitals must generate it on site. Iodine-131 after 8 days leaves half the dose.

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