How a Half-Life Calculator Works
A half-life calculator applies the exponential decay equation N(t) = N₀ × (0.5)^(t / t½) to answer three different questions: how much of a substance remains after a given time, how long it took to reach a given quantity, or what the half-life period must be given observed quantities. Enter the values you know and the calculator solves for the unknown.
The calculator supports five time units (seconds, minutes, hours, days, and years) and includes presets for well-known isotopes including Carbon-14, Uranium-235, Iodine-131, Radium-226, and Tritium (H-3). A decay table shows the quantity remaining at each half-life interval from 0 to 10.
The Three Core Formulas
- Find Remaining: N(t) = N₀ × (0.5)^(t / t½)
- Find Elapsed Time: t = t½ × log(N / N₀) / log(0.5)
- Find Half-Life: t½ = t × log(0.5) / log(N / N₀)
What Is Radioactive Decay?
Radioactive decay is the spontaneous process by which an unstable atomic nucleus loses energy by emitting radiation. Each radioactive isotope decays at a characteristic rate described by its half-life — the time required for exactly half of a given sample to decay into a different element or a more stable isotope.
The decay process is random at the level of individual atoms but highly predictable for large numbers of atoms, making the exponential model extremely accurate in practice. Half-lives span an enormous range: from fractions of a second for highly unstable isotopes to billions of years for stable ones like Uranium-238 (4.47 billion years).
Real-World Applications
- Carbon Dating: Carbon-14 (t½ = 5,730 years) lets archaeologists date organic materials up to ~50,000 years old by measuring how much C-14 remains compared to stable C-12.
- Nuclear Medicine: Short-lived isotopes like Iodine-131 (t½ = 8 days) are used for targeted cancer treatment, delivering localised radiation that fades within weeks.
- Nuclear Power: Understanding the half-lives of fission products is essential for managing spent nuclear fuel and assessing long-term radioactive waste storage.
- Radiometric Dating: Uranium-Lead and Potassium-Argon decay systems date geological samples from thousands to billions of years, underpinning the geologic timescale.
- Food Safety: Irradiation using Cobalt-60 extends shelf life; its 5.27-year half-life informs source replacement schedules.
Frequently Asked Questions
Does all of a radioactive substance eventually decay?
Mathematically, the exponential decay function never reaches zero — there is always a tiny fraction remaining. In practice, once the number of atoms falls below a few thousand the statistical model breaks down, and the last atoms decay at unpredictable times. For engineering and safety purposes a substance is typically considered safe after 10 half-lives (~0.1% remaining).
Why does the half-life stay constant over time?
Radioactive decay is governed by quantum mechanics. Each nucleus has an intrinsic probability of decaying per unit time that depends only on its internal structure, not on temperature, pressure, chemical state, or how old the sample is. This constant probability is what produces the characteristic fixed half-life.
How accurate is carbon dating?
Radiocarbon dating is accurate to within a few decades for samples up to about 30,000 years old and can reach ~50,000 years with modern accelerator mass spectrometry. Accuracy depends on knowing the original atmospheric C-14 concentration, which scientists calibrate using tree rings, coral, and other records extending back ~55,000 years.