How to use the half-life calculator
- Enter the initial amount of the substance in any unit you prefer (grams, moles, becquerels, etc.).
- Type the half-life using your chosen time unit (seconds, years, days).
- Enter the elapsed time in the exact same time unit as the half-life.
- The remaining amount, number of half-lives passed and percentage left update automatically.
How the decay formula works
The half life calculator applies the exponential decay equation:
N(t) = N₀ × (1/2)^(t / t½)
Here N₀ is the initial quantity, t is the elapsed time and t½ is the half-life. The exponent t / t½ counts how many half-lives have passed. For example, if you start with 100 grams of a substance whose half-life is 10 years, after 30 years the exponent is 30 / 10 = 3. The remaining amount is 100 × (0.5)³ = 100 × 0.125 = 12.5 grams.
This model assumes a large number of atoms where statistical averaging holds. At very small quantities (a handful of atoms), individual decay events become unpredictable, but the expected value still follows this formula.
Common half-lives reference table
Below are approximate half-lives for several well-known isotopes used in medicine, archaeology and energy production. Use these values directly in the calculator above.
| Isotope | Half-life | Common use |
|---|---|---|
| Carbon-14 | 5,730 years | Radiocarbon dating |
| Iodine-131 | 8.02 days | Thyroid treatment |
| Tritium (H-3) | 12.32 years | Luminous dials, fusion research |
| Cobalt-60 | 5.27 years | Cancer radiotherapy |
| Uranium-238 | 4.468 billion years | Geological dating |
| Technetium-99m | 6.01 hours | Medical imaging |
When entering these into the calculator, make sure the elapsed time uses the same unit. If the half-life is in days, enter the elapsed time in days as well.
Practical applications of radioactive decay
Understanding exponential decay matters far beyond physics classrooms. In nuclear medicine, doctors use the half-life of an isotope to determine safe dosage intervals. Iodine-131, with its roughly eight-day half-life, delivers radiation to thyroid tissue and then decays quickly enough to limit long-term exposure.
In archaeology, radiocarbon dating relies on Carbon-14's 5,730-year half-life to estimate the age of organic artifacts. By measuring the ratio of remaining C-14 to stable C-12, researchers calculate how many half-lives have passed since the organism died.
Nuclear waste management also depends on these calculations. Isotopes with short half-lives become safe relatively quickly, while those like Plutonium-239 (24,100 years) require long-term containment strategies. Engineers use decay formulas to plan storage timelines and assess environmental risk.
Common mistakes when calculating decay
The most frequent error is mixing time units. If the half-life is given in years but the elapsed time is in months, the result will be wrong by a factor of twelve. Always convert both values to the same unit before entering them into the half life calculator.
Another mistake is confusing the decay constant (lambda) with the half-life. They are related by t½ = ln(2) / λ, but they are not interchangeable in the formula. This tool uses the half-life form directly, so no conversion is needed.
Finally, remember that the formula gives the expected remaining amount. It does not mean exactly half the atoms vanish at the stroke of the half-life mark. Decay is a continuous probabilistic process, and the exponential curve describes the average behavior of a large population of atoms.
Relationship between half-life and decay constant
The decay constant lambda (λ) represents the probability per unit time that a single atom will decay. It connects to the half-life through the natural logarithm of two:
λ = ln(2) / t½ ≈ 0.693147 / t½
You can rewrite the decay equation using lambda instead: N(t) = N₀ × e^(-λt). Both forms produce identical results. The half-life version is more intuitive because it directly shows how many halving cycles have occurred, while the exponential form is preferred in differential equations and calculus-based derivations.
For quick mental estimates, remember that after 10 half-lives only about 0.1% of the original material remains (1/1024). After 20 half-lives, less than one millionth is left. This rule of thumb helps assess when a radioactive sample becomes negligible without running the full calculation.
Worked examples
Each result below was produced by the same code the tool runs, so you can check your own numbers against them.
| Example | Input | Result |
|---|---|---|
| Carbon-14 dating (2 half-lives) | Initial amount (N₀): 100; Half-life (t½): 5730; Elapsed time (t): 11460 | Remaining amount: 25; Amount decayed: 75; Percentage remaining: 25%; Half-lives elapsed: 2 |
| Iodine-131 medical dose after 24 days | Initial amount (N₀): 50; Half-life (t½): 8.02; Elapsed time (t): 24 | Remaining amount: 6.28249442785; Amount decayed: 43.7175055722; Percentage remaining: 12.5649888557%; Half-lives elapsed: 2.99251870324 |
| Uranium-238 over one million years | Initial amount (N₀): 1000; Half-life (t½): 4468000000; Elapsed time (t): 1000000 | Remaining amount: 999.844876138; Amount decayed: 0.15512386243; Percentage remaining: 99.9844876138%; Half-lives elapsed: 0.000223813786929 |
| Tritium after 5 years | Initial amount (N₀): 10; Half-life (t½): 12.32; Elapsed time (t): 5 | Remaining amount: 7.54794514977; Amount decayed: 2.45205485023; Percentage remaining: 75.4794514977%; Half-lives elapsed: 0.405844155844 |
FAQ
What is a half-life?
A half-life is the time required for exactly half of the atoms in a radioactive sample to undergo decay. It is a constant property of each isotope and does not depend on the initial amount.
Does temperature or pressure affect radioactive half-life?
No. Unlike chemical reactions, nuclear decay rates are essentially unaffected by temperature, pressure or chemical environment. The half-life remains constant under normal conditions.
Can I use this half life calculator for non-radioactive decay?
Yes. Any process that follows first-order exponential decay, such as drug elimination from the bloodstream or capacitor discharge, can use the same mathematical formula.
Why must the time units match?
The formula divides elapsed time by the half-life. If one is in years and the other in days, the ratio is incorrect and the output will be meaningless. Convert both to the same unit first.
What happens after many half-lives?
The remaining amount approaches zero asymptotically but never reaches it mathematically. In practice, after about 10 to 20 half-lives the quantity is too small to measure or poses no hazard.
Is my data sent to a server when I use this tool?
No. All calculations happen locally in your browser. Nothing you type is transmitted or stored.