Calculate remaining quantity after radioactive decay, number of half-lives elapsed, decay constant (Ξ»), and activity. Supports carbon-14 dating, Tc-99m, radon, and custom isotopes.
Radioactive decay is one of the most fundamental processes in nuclear physics, chemistry, and medicine. Whether you are a student working through a GCSE or AP Physics problem, a scientist calculating isotope activity, or simply curious about how carbon-14 dating works, our free half-life calculator gives you instant, accurate results with full step-by-step breakdowns.
This tool supports every major time unit β from seconds (for fast-decaying medical isotopes like Tc-99m) to billions of years (for uranium-238) β and includes quick presets for the most commonly studied isotopes in US and UK curricula.
Half-life (symbol TΒ½ or T) is the time required for exactly half of a radioactive substance to undergo nuclear decay. After one half-life, 50% of the original quantity remains. After two half-lives, 25% remains. After ten half-lives, less than 0.1% of the original quantity is left.
The concept was first formally described by Ernest Rutherford in 1907 and has since become central to nuclear physics, geochemistry, pharmacology, and environmental science. Half-life is a property of each specific isotope and never changes regardless of temperature, pressure, or chemical state β making it exceptionally reliable for dating and dosing calculations.
The fundamental equation governing radioactive decay is:
N = Nβ Γ (0.5)^(t/T)
This can be rewritten using the natural exponential: N = Nβ Γ e^(βΞ»t), where the decay constant Ξ» = ln(2)/T β 0.6931/T.
The decay constant Ξ» (lambda) represents the probability of a single atom decaying per unit time. The relationship between Ξ» and T is:
Ξ» = ln(2) / T = 0.693147 / T
Activity (A) β measured in becquerels (Bq) in SI, or curies (Ci) in the older CGS system β represents how many atoms decay per second:
A = Ξ» Γ N
1 becquerel = 1 decay per second. 1 curie = 3.7 Γ 10ΒΉβ° Bq. In the UK, the becquerel is standard; in the US, both Bq and Ci are used depending on the field.
Carbon-14 has a half-life of 5,730 years. An archaeologist finds a sample with 25% of its original C-14 remaining. How old is the sample?
This is the basis of radiocarbon dating, developed by Willard Libby at the University of Chicago (Nobel Prize 1960). In the UK, the Oxford Radiocarbon Accelerator Unit conducts thousands of C-14 dates annually for archaeological research.
Tc-99m is the world's most widely used medical radioisotope, with over 30 million procedures performed annually. Its 6-hour half-life is ideal: long enough to image the patient, short enough to minimise radiation exposure. A hospital might prepare a 400 MBq dose for a bone scan. After 6 hours, 200 MBq remains; after 30 hours (5 half-lives), less than 13 MBq β effectively cleared from the patient.
Radon-222 (half-life 3.82 days) is a naturally occurring gas produced by uranium decay in rocks. Public Health England (now UKHSA) estimates radon causes approximately 1,100 lung cancer deaths per year in the UK. High-risk areas include Cornwall, Devon, and Derbyshire. In the US, the EPA considers radon above 4 pCi/L (148 Bq/mΒ³) actionable. Fitting radon sumps and improving ventilation dramatically reduces exposure.
| Isotope | Half-Life | Application |
|---|---|---|
| Carbon-14 | 5,730 years | Archaeological dating (UK & US) |
| Technetium-99m | 6 hours | Nuclear medicine worldwide |
| Iodine-131 | 8.02 days | Thyroid treatment |
| Radon-222 | 3.82 days | Indoor air hazard UK/US |
| Caesium-137 | 30.17 years | Nuclear fallout, post-Chernobyl |
| Plutonium-239 | 24,100 years | Nuclear waste storage |
| Uranium-238 | 4.47 billion years | Geological age dating |
| Half-Lives Elapsed | Fraction | % Remaining |
|---|---|---|
| 0 | 1/1 | 100% |
| 1 | 1/2 | 50% |
| 2 | 1/4 | 25% |
| 3 | 1/8 | 12.5% |
| 5 | 1/32 | 3.125% |
| 7 | 1/128 | 0.78% |
| 10 | 1/1,024 | 0.098% |
In the UK, half-life is covered at GCSE Physics (AQA, OCR, Edexcel) under Atomic Structure and Radioactivity at age 14β16. A-level Physics extends to decay equations, activity, and medical/industrial applications. In the US, half-life first appears in middle school physical science, with full mathematical treatment in AP Chemistry (first-order kinetics, rate constant k) and AP Physics.
Nuclear power produces byproducts with vastly different half-lives. Iodine-131 (8 days) was the primary health concern after Chernobyl and Fukushima. Caesium-137 (30 years) persists for decades. Plutonium-239 (24,100 years) requires geological-scale containment, which is why the UK's proposed deep geological disposal facility at Theddlethorpe must be engineered to contain waste for a million years.
Half-life applies beyond nuclear physics. In pharmacokinetics, biological half-life describes how long a drug takes to reduce to half its original concentration in the body. Aspirin: ~3.5 hours. Diazepam (Valium): 20β100 hours. Physicians use half-life data to set safe dosing intervals β a drug with a 6-hour half-life typically needs redosing every 6β8 hours.
N = Nβ Γ (0.5)^(t/T), where Nβ is the initial quantity, t is elapsed time, T is the half-life, and t/T is the number of half-lives elapsed. Both t and T must use the same time unit. Example: 800 g with T = 10 yr after t = 20 yr β N = 800 Γ 0.25 = 200 g.
Ξ» = ln(2) / T = 0.693147 / T. For Carbon-14 (T = 5730 yr): Ξ» = 0.693147 / 5730 = 0.0001210 yrβ»ΒΉ. This means 0.0121% of any C-14 sample decays each year.
Carbon-14 has a half-life of 5,730 years. Living organisms maintain a fixed C-14 ratio by exchanging carbon with the atmosphere. After death, C-14 decays predictably. This allows dating of organic materials from ~300 to ~50,000 years old with precision of Β±20β300 years using modern AMS techniques.
Tc-99m has a 6-hour half-life, emits 140 keV gamma rays ideal for gamma cameras, produces no alpha or beta radiation, and is generated in portable "moly cow" generators. Over 30 million nuclear medicine scans use Tc-99m annually worldwide, covering bone, heart, kidney, and liver imaging.
After 7 half-lives, less than 0.78% of the original quantity remains. After 10 half-lives, less than 0.098%. Nuclear medicine considers a substance effectively cleared after 5 half-lives (3.125% remains). Regulatory guidelines for nuclear waste typically require 10 half-lives of isolation.
Activity (A) is the number of decay events per second, measured in becquerels (Bq) in the SI system. A = Ξ» Γ N. As N decreases over time, so does activity. 1 Bq = 1 decay/second; 1 Ci = 3.7 Γ 10ΒΉβ° Bq (used in US medical and industrial settings).
Yes. Radon-222 gas seeps from uranium-bearing rocks and soil. Its decay products (Po-218, Pb-214, Bi-214) are solid particles that lodge in lung tissue. The UK UKHSA estimates ~1,100 lung cancer deaths per year in the UK from radon exposure. The US EPA estimates ~21,000. Mitigation with sub-slab depressurisation reduces indoor levels by 50β90%.
Modern Accelerator Mass Spectrometry (AMS) C-14 dating achieves Β±20β50 years for samples up to 10,000 years old. Uncertainty increases for older samples, reaching Β±500+ years near the 50,000-year limit. Calibration curves like IntCal23 (based on tree rings, corals, and cave deposits) correct for past fluctuations in atmospheric C-14.
Results are for educational purposes only and are not professional scientific, medical, or radiation safety advice. Radioactive materials are subject to strict regulatory controls in the USA (NRC) and UK (ONR/EA). Always consult qualified professionals for any work involving radioactive substances.