Half-Life Tool
Explore exponential radioactive decay by solving for remaining quantity, initial quantity, half-life or elapsed time. Examine remaining and decayed percentages, decay constants, interactive curves and tables, with optional isotope presets and downloadable results.
| Half-lives | Time (years) | Remaining | Decayed |
|---|
Model assumes a constant half-life and exponential decay without additional production or removal.
Table of Contents
Highlights
- Solve for remaining amount, starting amount, half-life or elapsed time.
- Interactive decay graph with optional decayed-quantity curve.
- Live half-life explorer with remaining and decayed percentages.
- Isotope presets and adjustable decay-table intervals.
- Export decay data as CSV, chart as PNG or results as JSON.
How Half-Life Tool Works?
Radioactive decay follows an exponential law when each nucleus has a constant probability of decaying. The half-life, T½, is the time required for half of a large starting population to remain. The same mathematical relationship applies to an initial mass, activity, atom count or other quantity undergoing ideal exponential decay.
Here N0 is the starting quantity, N(t) is the quantity remaining after time t, and λ is the decay constant. The tool rearranges this relationship to solve for any one of four quantities when the other necessary values are supplied.
The number of half-lives elapsed is t/T½. The remaining percentage is 100 × N/N0; the decayed percentage is 100 minus that value. For a starting quantity of 100 g with a half-life of 5 years, 10 years represents two half-lives, leaving 25 g and 75% decayed.
| Half-lives elapsed | Remaining | Decayed |
|---|---|---|
| 0 | 100% | 0% |
| 1 | 50% | 50% |
| 2 | 25% | 75% |
| 3 | 12.5% | 87.5% |
| 4 | 6.25% | 93.75% |
Results include the solved quantity, elapsed half-lives, both percentages and the decay constant. An interactive curve can also show cumulative decay, while an adjustable explorer and interval-based table display values at different stages.
Practical Use Cases
Radiochemistry students: Check homework involving a starting atom count, a known half-life and the fraction remaining after several half-lives, then compare the figures with the decay curve.
Medical imaging trainees: Explore how the activity of technetium-99m changes over hours using the illustrative 6.01-hour preset, while remembering that administered activity and patient dose require additional clinical modelling.
Environmental scientists: Examine idealised decay of radon-222 or iodine-131 and estimate how long it takes for a specified fraction of an isolated sample to remain.
Archaeology students: Use the carbon-14 half-life preset to understand the mathematical relationship between a remaining fraction and elapsed time, without treating that simplified result as a calibrated radiocarbon date.
Radiation safety trainees: Compare the fraction of cobalt-60 activity remaining after different intervals and download a decay table for a teaching exercise.
Physics educators: Demonstrate why successive half-lives remove equal fractions rather than equal absolute amounts, using the interactive explorer and remaining-versus-decayed graph.
Accuracy and Limitations
Calculations use double-precision JavaScript arithmetic and the ideal single-component exponential law, so displayed values are approximations subject to rounding, especially at extreme magnitudes or long times. The model assumes a fixed half-life and no additional production or removal, and it does not model decay chains, daughter-product build-up, mixed isotopes, biological clearance, environmental transport or uncertainty in measurements. Isotope half-lives are illustrative presets, not certified reference data. Solving for elapsed time or half-life requires a positive remaining quantity; solving for half-life also requires less remaining than initially present and positive elapsed time. The graph covers a limited range of half-lives, and its table contains sampled intervals rather than a continuous dataset.
