Half Life Calculator

A half life calculator solves the exponential decay law for one unknown. Every nucleus in a radioactive sample has the same fixed chance of decaying per unit time, so the quantity falls by half over each half-life no matter how much is left: N(t) = N₀ x (1/2)^{(t / t_1/2)}. Any three of initial quantity, remaining quantity, half-life, and elapsed time fix the fourth.

Initial and remaining quantity can be in any unit (grams, atoms, becquerels of activity, milligrams per liter of a drug) as long as both use the same one, because only their ratio enters the equation. The half-life and the elapsed time each carry their own unit, from seconds to years, and are converted to a common base before the exponent is taken. A year here is the Julian year of 365.25 days, or 31,557,600 seconds. Numbers accept decimals, fractions such as 3/4, mixed numbers such as 5 1/2, and scientific notation such as 1.5e3.

Remaining quantity is N₀ x 0.5^{(t / t_1/2)}, and initial quantity is N(t) / 0.5^{(t / t_1/2)}. Solving for the half-life gives t_1/2 = t x ln(2) / ln(N₀ / N(t)), reported in the elapsed-time unit; solving for elapsed time gives t = t_1/2 x ln(N₀ / N(t)) / ln(2), reported in the half-life unit. Both of those require the initial quantity to exceed the remaining one, since a sample cannot grow by decaying. The decay constant is lambda = ln(2) / t_1/2 = 0.693147 / t_1/2, given per second and switched to e-notation when it is very small. Settings holds the decimal places (0 to 15) and a step-by-step derivation.

The number of half-lives elapsed, t / t_1/2, is the quickest sense check: after one half-life 50 percent remains, after two 25 percent, and after ten about 0.1 percent. The law describes the average behavior of very many nuclei; for a handful of atoms the actual count scatters around it. In pharmacology the same arithmetic holds only while elimination is first-order, which covers most drugs at therapeutic doses but not substances whose elimination saturates, such as alcohol.

A 100 MBq dose of iodine-131, taking its half-life as 8 days, is followed for 24 days. That is 24 / 8 = 3 half-lives, so the remaining activity is 100 x 0.5³ = 12.5 MBq, with 12.5 percent remaining and 87.5 percent decayed. The decay constant is 0.693147 / (8 x 86,400 s) = 1.0028e-6 per second. Working backwards, 12.5 MBq after 24 days with the same half-life gives an initial activity of 12.5 / 0.125 = 100 MBq.

Half-life calculations apply across nuclear physics, chemistry, medicine, archaeology, and environmental science. Any scenario involving exponential decay of a substance or signal can benefit from determining one unknown variable given the other three.

• Nuclear Physics: Determine how much of a radioactive isotope remains after a given period. Cesium-137 has a half-life of about 30.1 years (30.08 in current nuclear data evaluations, 30.17 in older tables), so 1,000 grams falls to roughly 100 grams after 100 years, a figure that matters for assessing long-term contamination.
• Archaeology and Geology: Estimate the age of organic samples using known decay rates. Given an initial Carbon-14 concentration and a measured remaining fraction, solving for elapsed time dates artifacts or geological formations; 25 percent remaining with a 5,730-year half-life is 11,460 years.
• Medical Imaging: Calculate how long a radioactive tracer remains active in a patient. Technetium-99m has a half-life of about 6 hours, so after 24 hours four half-lives have passed and 6.25 percent of the dose remains, which helps schedule follow-up procedures and assess radiation exposure.
• Pharmacology: Model the biological half-life of drugs in the bloodstream. Given an administered dose and a measured plasma concentration at a later time, solve for the drug half-life or predict when the concentration drops below a therapeutic threshold.
• Environmental Science: Assess how long pollutants or radioactive waste remain hazardous. For decommissioning nuclear facilities, strontium-90 (half-life about 28.8 years) takes 28.8 x log2(100), roughly 191 years, to decay to one percent of its original activity.
• Education: Teach exponential decay concepts by changing one value and watching the others respond. The step-by-step derivation shows the substituted values and intermediate results, reinforcing how the same equation rearranges for each unknown.
• Industrial Radiography: Plan source replacement schedules for non-destructive testing equipment. Iridium-192 has a half-life of about 73.8 days, so a source that must not fall below one fifth of its initial activity needs replacing after roughly 171 days.