What this calculator does
Radioactive decay is exponential, so a fixed fraction disappears in each fixed period rather than a fixed amount. After one half-life, half remains; after two, a quarter; after three, an eighth.
The consequence is that decay never quite finishes. Each half-life removes half of what is left, so the amount approaches zero without reaching it. This is why contamination timescales are quoted in multiples of the half-life: ten half-lives leaves about one part in a thousand, and twenty leaves about one in a million.
The formula
The amount remaining is the initial amount multiplied by one half raised to the power of elapsed time over half-life. The decay constant lambda is the natural log of 2 divided by the half-life, and it is the fraction decaying per unit time for very short intervals. The same mathematics applies to any exponential decay, not only radioactivity.
| Term | Meaning |
|---|---|
| Half-life | The time for half the sample to decay. Constant for a given isotope. |
| Decay constant λ | ln2 divided by the half-life, the instantaneous decay rate per nucleus. |
| Exponential decay | A fixed proportion lost per unit time, not a fixed quantity. |
| Carbon-14 | Half-life 5,730 years, the basis of radiocarbon dating. |
The inputs explained
| Field | What to enter |
|---|---|
| Initial amount (units) | Starting amount, in any unit. The answer comes back in the same unit. |
| Elapsed time (years) | Time elapsed, in the same unit as the half-life. |
| Half-life (years) | Half-life of the isotope. Carbon-14 is 5,730 years; iodine-131 is about 8 days. |
When to use it
Radiocarbon dating
The fraction of carbon-14 remaining gives an age, which is the classic use of the relationship run backwards.
Medical isotope planning
Short-lived diagnostic isotopes decay meaningfully during transport, so the dose at use differs from the dose dispatched.
Waste storage timescales
How long material must be isolated is set by multiples of the half-life of its longest-lived component.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How much remains after each half-life?
Elapsed times at whole multiples of the half-life.
| Time elapsed | Amount remaining | Fraction remaining | Half-lives elapsed |
|---|---|---|---|
| 0 years | 100.000 units | 100.0% | 0.000 |
| 5,730 years | 50.000 units | 50.0% | 1.000 |
| 11,460 years | 25.000 units | 25.0% | 2.000 |
| 17,190 years | 12.500 units | 12.5% | 3.000 |
Questions
What is a decay constant?
The natural log of 2 divided by the half-life, about 0.693 over t½. It represents the probability per unit time that any given nucleus decays. It appears in the exponential form of the decay law, N = N₀e^(−λt), which is equivalent to the halving form.
Why does decay never reach zero?
Because each half-life removes a proportion, not an amount. Half of any remaining quantity is still a quantity. In practice a sample eventually contains so few atoms that it is undetectable, but the mathematical curve approaches zero asymptotically.
Does half-life change with temperature or pressure?
No, not for practical purposes. Radioactive decay is a nuclear process and is essentially unaffected by chemical state, temperature or pressure. This is what makes it a reliable clock, unlike almost every other rate process in chemistry.
How long until a radioactive sample is safe?
A common working figure is ten half-lives, leaving about 0.1% of the original activity. What counts as safe depends on the starting activity, the isotope and the type of radiation, so this is a rule of thumb rather than a regulatory standard.
For the decay rate in becquerels, see the radioactive activity calculator. For dating applications, see the radiocarbon dating calculator.