Half-Life Calculator
Calculate radioactive and exponential decay. Solve for remaining amount, initial amount, elapsed time or half-life with 12 built-in isotopes.
Choose what to solve for, fill in the known values, then calculate.
Half-Life Calculator: Model Radioactive and Exponential Decay
Half-life is one of the most important concepts in nuclear physics, chemistry, pharmacology and many other fields. It describes the time required for exactly half of a decaying quantity to disappear, whether radioactive atoms, drug molecules in the bloodstream, or the charge on a capacitor. This calculator solves the complete exponential decay equation for any unknown variable, includes twelve common radioactive isotopes with their accepted half-life values, and reports the decay constant, mean lifetime and percentage remaining for any scenario you enter.
Decay constant: λ = ln(2) / T½ ≈ 0.693 / T½
Mean lifetime: τ = T½ / ln(2) ≈ 1.443 × T½
Alternative form: N(t) = N₀ × e^(−λt)
After 1 half-life: 50% remains
After 2: 25% | After 3: 12.5% | After 10: ~0.1%
The Four Things You Can Solve For
The decay equation links four variables: the initial amount, the remaining amount, the elapsed time and the half-life. Knowing any three lets you compute the fourth. This calculator handles all four cases. Solve for the remaining amount to predict how much of a sample will be left after a given time. Solve for initial amount to work backward from a measurement. Solve for elapsed time to perform radiometric dating. Solve for half-life when you have measured decay in a laboratory.
Decay Constant and Mean Lifetime
The decay constant, written as lambda, is the probability per unit time that a single atom will decay. It connects directly to the half-life: lambda equals 0.693 divided by the half-life. The mean lifetime, written tau, is the average time an individual atom survives before decaying. Because decay is exponential rather than linear, the mean lifetime is longer than the half-life by a factor of about 1.443.
Radiocarbon Dating and Carbon-14
The most famous application of half-life is radiocarbon dating. Living organisms continuously absorb carbon-14 from the atmosphere. When an organism dies, absorption stops and the carbon-14 begins to decay with a half-life of 5,730 years. By measuring how much carbon-14 remains in a sample of bone, wood or cloth and comparing it to the living concentration, archaeologists calculate the age of the specimen, reliable up to roughly 50,000 years.
Medical and Pharmaceutical Applications
Half-life is equally central in medicine. Radioactive isotopes used in nuclear medicine, such as technetium-99m with its six-hour half-life, must decay quickly enough to limit patient exposure while lasting long enough to complete imaging. Iodine-131 treats thyroid conditions. In pharmacology, drug half-life determines dosing schedules.
Beyond Radioactivity
Although half-life originated in nuclear physics, the underlying exponential decay model describes countless natural processes. A discharging capacitor loses voltage exponentially. A hot object cooling toward room temperature follows a similar curve. Populations declining due to a constant mortality rate and the elimination of pollutants all obey the same equation.
Tips & Recommendations
After 10 half-lives, only about 0.1% remains. This is often treated as effectively gone for safety and dosing.
Mean lifetime is always about 1.443 times the half-life, never equal. Do not confuse the two.
Keep time and half-life in the same unit (both years, both days). The calculator does not auto-convert units.
Carbon-14 dating works up to ~50,000 years. Beyond that, use isotopes with longer half-lives like potassium-40.
Frequently Asked Questions
What is half-life?
Half-life is the time required for half of a quantity to decay or reduce. After one half-life, 50% remains; after two, 25%; after three, 12.5%. It is constant for a given radioactive isotope regardless of the starting amount.
How do you calculate half-life?
Use the formula N(t) = N0 times (1/2)^(t/T), where N0 is the initial amount, t is elapsed time, and T is the half-life. Rearrange to solve for any unknown variable.
What is the decay constant?
The decay constant (lambda) equals ln(2) divided by the half-life. It represents the probability of decay per unit time. A larger decay constant means faster decay and a shorter half-life.
What is mean lifetime?
Mean lifetime (tau) is the average time a particle exists before decaying, equal to half-life divided by ln(2), or about 1.443 times the half-life.
How is half-life used in carbon dating?
Carbon-14 has a half-life of 5,730 years. By measuring the remaining C-14 in organic material and comparing it to the original level, scientists calculate how long ago the organism died, up to about 50,000 years.
Does half-life apply to non-radioactive decay?
Yes. The same exponential formula models drug elimination, capacitor discharge, cooling and any process where the rate of decrease is proportional to the current amount.
Recent Calculations
No calculations yet