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Half-Life Calculator
Choose what you need: the amount left after a time, the time needed to reach a given amount, or the half-life itself. The tool also gives the decay constant and mean lifetime.
The exponential decay law
Whenever the rate of loss is proportional to the amount present, the amount falls exponentially. The half-life T is the time for the amount to fall to one half. After t units the amount is N = N0 × (1/2)t ÷ T. After one half-life half remains, after two a quarter, after three an eighth.
The same law written with a decay constant is N = N0 × e−λt, where λ = ln 2 ÷ T. The two forms are identical. The tool reports λ and the mean lifetime τ = 1 ÷ λ = T ÷ ln 2, which is the average time before an individual atom decays.
Solving for time or half-life
Rearranging the law with logarithms gives the other two modes. For elapsed time, t = T × ln(N0 ÷ N) ÷ ln 2. For the half-life from two measurements, T = t × ln 2 ÷ ln(N0 ÷ N).
Example: 100 units fall to 12.5 with a half-life of 5 units of time. The ratio N0 ÷ N is 8, which is 2 cubed, so three half-lives have passed and t = 15. Because the amount only approaches zero and never reaches it, the time mode requires a remaining amount above zero.
Where half-life applies
Half-life is best known from radioactive decay, where each nuclide has its own constant value, from fractions of a second to billions of years. Radiocarbon dating uses carbon-14, whose half-life is commonly given as about 5,730 years. The idea also describes first-order chemical reactions, the elimination of many drugs from the body, and the fall of a capacitor's charge in a resistor circuit, where the time constant plays the role of the mean lifetime.
Enter the half-life from a reliable source for your substance; this tool does not store isotope data, so the value is always yours to supply.
Assumptions and cautions
- First-order decay only. The model assumes a constant probability of decay. Zero-order and second-order processes follow different curves.
- Same unit for time and half-life. The unit box is only a label. Convert days to years yourself before you enter them.
- Medicine. Drug half-lives vary between people and doses. This page is for learning, not for deciding doses or timing.
- Safety. Decay arithmetic does not measure radiation risk. Contact a qualified radiation safety professional for real exposure questions.
A quick sanity check helps: if the elapsed time is less than one half-life, more than half should remain; if it is more than one half-life, less than half should. When your result breaks that rule, a unit mismatch is the most likely cause.
Worked example
A sample of 80 grams of a substance with a 5-day half-life decays for 15 days, which is exactly three half-lives.
Inputs
| Solve for | Amount remaining |
|---|---|
| Initial amount | 80 |
| Elapsed time | 15 |
| Half-life | 5 |
| Time unit (label only) | days |
Result
N = N0 × (1/2)^(t ÷ T), with t ÷ T = 3 half-lives.
| Initial amount | 80 |
|---|---|
| Amount remaining | 10 |
| Elapsed time | 15 days |
| Half-life | 5 days |
| Number of half-lives elapsed | 3 |
| Fraction remaining | 12.5% |
| Decay constant (λ = ln 2 ÷ T) | 0.138629 per day |
| Mean lifetime (τ = T ÷ ln 2) | 7.213475 days |
Accuracy note: Results use standard double-precision arithmetic, so very large or very small values may be rounded. Check important results by another method.
Frequently asked questions
What is the half-life formula?
The amount remaining is the initial amount times one half raised to the power of elapsed time divided by half-life. After three half-lives, one eighth remains.
How many half-lives until a substance is mostly gone?
After 3 half-lives 12.5% is left, after 5 about 3.1%, and after 10 about 0.1%. The amount never reaches exactly zero in this model.
What is the decay constant?
The decay constant λ is the fraction of the amount that decays per unit time for small intervals. It equals the natural log of 2 divided by the half-life.
What is mean lifetime and how does it differ from half-life?
Mean lifetime is the average survival time of a single particle and equals the half-life divided by ln 2, about 1.4427 times the half-life.
Can I use percentages as the amounts?
Yes. Set the initial amount to 100 and read the remaining value as a percentage. Only the ratio between initial and remaining amounts matters.
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References and sources
- IUPAC Compendium of Chemical Terminology (the Gold Book), entries on half-life and first-order reaction. Link