Half-life and exponential radioactive decay
Half-life is the time required for an exponentially decaying population to fall to one-half of its initial amount. After n half-lives, the expected remaining fraction is (1/2)ⁿ.
Radioactive decay is stochastic for individual nuclei, while the exponential equation describes the expected behavior of a sufficiently large population. The same mathematics can represent other ideal first-order processes when a constant half-life is appropriate.
How to use the half-life calculator
- Choose the unknown: Select initial quantity, remaining quantity, elapsed time, or half-life.
- Use consistent units: Match both quantity units and both time units.
- Enter positive values: For inverse solutions, remaining quantity cannot exceed the initial quantity.
- Interpret the model: Confirm constant-rate first-order decay is appropriate.
Formula and variables
Elapsed time t and half-life T½ must use the same unit. Quantities N₀ and N(t) must use the same count, mass, activity, or amount unit.
N(t) = N₀(1/2)^(t/T½)- N₀ — Initial quantity
- Quantity at time zero (quantity)
- N(t) — Remaining quantity
- Expected quantity after elapsed time (same as N₀)
- t — Elapsed time
- Nonnegative time after the initial observation (time)
- T½ — Half-life
- Positive time for quantity to halve (same as t)
One carbon-14 half-life
An initial quantity of 100 remains for 5,730 years with a half-life of 5,730 years.
- Initial quantity
- 100
- Time and half-life
- 5,730 years each
- N = 100 × (1/2)^(5730/5730)
Result: The expected remaining quantity is 50.
One half-life reduces the expected population to one-half of its starting amount.
Understanding your results
Half-life describes an exponential expectation
The model approaches zero but does not reach exactly zero at a finite time.
- After two half-lives, 25% remains.
- Equal half-life intervals remove equal fractions, not equal absolute amounts.
- A solved time of zero is valid when remaining quantity equals initial quantity.
- Small numbers of nuclei can show substantial random variation around the expectation.
Assumptions
- The decay constant and half-life remain constant.
- The population is closed except for the modeled decay process.
- Initial and remaining quantities are directly comparable.
Limitations
- Does not model decay chains, daughter products, mixtures of isotopes, background, detection efficiency, or uncertainty.
- Does not perform radiometric age calibration or contamination correction.
- Not a radiation safety, dose, or medical decision tool.
Common mistakes
- Mixing time units between elapsed time and half-life.
- Subtracting a fixed amount each half-life instead of multiplying by one-half.
- Entering remaining quantity greater than initial quantity for a decay-only model.
- Treating expected fraction as an exact count for a small sample.
Practical use cases
Radioactive decay education
Explore expected remaining fractions across multiple half-lives.
Ideal first-order processes
Solve the same mathematical model when a constant half-life has been independently established.
Frequently asked questions
Does a substance disappear after several half-lives?
The mathematical expectation approaches zero continuously; it does not become exactly zero at a finite time.
Can I use grams instead of atom count?
Yes, if the measured mass is proportional to the number of undecayed nuclei and both quantity fields use the same unit.
How is decay constant related to half-life?
For first-order decay, λ = ln(2)/T½.
Sources and review
- Radioactive Decay — OpenStax University Physics Volume 3. Accessed 2026-07-13.
Reviewed 2026-07-13.