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Calculate radioactive decay, remaining quantity, or elapsed time.
N(t) = N₀ × (1/2)^(t/t½)
1000 units with 10-unit half-life after 30 units of time: 125 remain.
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Calculate remaining quantity, initial quantity, elapsed time, or half-life for radioactive decay and exponential decay processes.
N(t) = N₀ × (½)^(t / t½)
where λ = ln(2) / t½ | N(t) = N₀ × e^(−λt)
Any unit (atoms, grams, Bq…)
| Isotope | Half-Life | Application |
|---|---|---|
| Carbon-14 | 5,730 years | Archaeological dating |
| Uranium-238 | 4.47 billion yr | Geological dating |
| Iodine-131 | 8.02 days | Medical imaging/therapy |
| Cesium-137 | 30.17 years | Nuclear fallout tracer |
| Radon-222 | 3.82 days | Radon gas monitoring |
| Tritium (H-3) | 12.32 years | Self-luminous devices |
| Cobalt-60 | 5.27 years | Radiation therapy |
| Strontium-90 | 28.8 years | Nuclear waste concern |
| Polonium-210 | 138.4 days | Alpha radiation source |
| Technetium-99m | 6.01 hours | Medical diagnostics |
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Reviewed by CalculatorApp.me Engineering Editorial Team · Updated March 2026 · 12 min read
Half-life (t½) is the time required for exactly half of a given quantity of a substance to undergo decay, transformation, or disappearance. It is the defining parameter of any exponential decay process. Whether describing radioactive nuclides, drug plasma concentrations, or attenuating signals, the same mathematical framework applies.
For radioactive isotopes, half-life is an intrinsic nuclear property that depends solely on the identity of the nucleus — it cannot be altered by temperature, pressure, chemical state, or any other external physical condition. Each nucleus decays independently; the half-life reflects the statistical behaviour of a large ensemble.
After one half-life, 50% remains. After two half-lives, 25%. After ten half-lives, less than 0.1%. The substance never mathematically reaches zero, but practically vanishes after sufficient time.
N(t) = N₀ × (½)^(t / t½)
Example: 1000 g Carbon-14 after 11,460 yr (2 half-lives)
N = 1000 × (½)² = 1000 × 0.25 = 250 g
N(t) = N₀ × e^(−λt)
λ = ln(2) / t½ ≈ 0.6931 / t½
λ is the decay constant — probability of decay per unit time
Mean lifetime τ = 1/λ = t½ / ln(2) ≈ 1.443 × t½
t = t½ × log(N(t)/N₀) / log(½)
= t½ × ln(N₀/N(t)) / ln(2)
Use when you know initial & remaining quantities and need to find time elapsed — key for radiocarbon dating.
t½ = t × ln(2) / ln(N₀/N(t))
= t × log(½) / log(N(t)/N₀)
Experimentally determine half-life by measuring initial and remaining quantities at a known elapsed time.
| Decay Type | Particle Emitted | Penetrating Power | Example Isotope | Application |
|---|---|---|---|---|
| Alpha (α) | Helium-4 nucleus (2p + 2n) | Low — stopped by paper | Polonium-210 (t½ = 138 d) | Smoke detectors (Am-241) |
| Beta⁻ (β⁻) | Electron + antineutrino | Medium — stopped by 1 cm Al | Carbon-14 (t½ = 5,730 yr) | Radiocarbon dating |
| Beta⁺ (β⁺) | Positron + neutrino | Medium — stopped by 1 cm Al | Fluorine-18 (t½ = 110 min) | PET scans (medical imaging) |
| Gamma (γ) | High-energy photon | High — needs cm of lead | Cobalt-60 (t½ = 5.27 yr) | Radiation therapy |
| Electron Capture | Orbital electron captured | Low external radiation | Iodine-125 (t½ = 59.4 d) | Brachytherapy |
| Spontaneous Fission | Two daughter nuclei + neutrons | Very high — needs thick shielding | Californium-252 (t½ = 2.65 yr) | Neutron source |
Source: National Nuclear Data Center (NNDC), Brookhaven National Laboratory
| Isotope | Decay Type | Half-Life | Field | Application |
|---|---|---|---|---|
| Tc-99m | Gamma (isomeric transition) | 6.01 hours | Medicine | SPECT diagnostic imaging |
| F-18 | Beta⁺ / Positron | 109.8 minutes | Medicine | PET scans (oncology, neurology) |
| I-131 | Beta⁻ + Gamma | 8.02 days | Medicine | Thyroid cancer treatment |
| Co-60 | Beta⁻ + Gamma | 5.27 years | Medicine/Ind. | Radiation therapy; food sterilisation |
| Cs-137 | Beta⁻ + Gamma | 30.17 years | Environment | Nuclear fallout tracer; calibration |
| Sr-90 | Beta⁻ | 28.8 years | Environment | Nuclear waste concern; bone seeker |
| H-3 (Tritium) | Beta⁻ | 12.32 years | Science/Defense | Self-luminous devices; NMR labelling |
| C-14 | Beta⁻ | 5,730 years | Archaeology | Radiocarbon dating (up to ~50,000 yr) |
| Ra-226 | Alpha + Gamma | 1,600 years | History | Early cancer treatment (now obsolete) |
| Pu-239 | Alpha | 24,110 years | Nuclear energy | Reactor fuel; weapons-grade material |
| U-235 | Alpha | 703.8 million yr | Geology/Energy | Nuclear fuel; uranium-lead dating |
| U-238 | Alpha | 4.47 billion years | Geology | Uranium-lead radiometric dating |
Data: IAEA Nuclear Data Section & NNDC Chart of Nuclides
French physicist Henri Becquerel accidentally discovers that uranium salts emit penetrating radiation capable of fogging photographic plates — the discovery of natural radioactivity. He shares the 1903 Nobel Prize in Physics with Pierre and Marie Curie.
Marie and Pierre Curie isolate two new radioactive elements — polonium (named after Marie's homeland) and radium — demonstrating that radioactivity is an atomic property. Their meticulous experiments establish quantitative methods for measuring decay rates.
Ernest Rutherford and Frederick Soddy publish their transformation theory of radioactivity, proposing that radioactive decay involves the spontaneous transmutation of one element into another. They identify alpha and beta particles as distinct radiation types.
Rutherford formally defines "half-life" as the time for half of any radioactive sample to decay. This simple, temperature-independent constant revolutionises nuclear physics and forms the basis for all subsequent decay calculations.
Willard Frank Libby at the University of Chicago develops radiocarbon dating using Carbon-14's 5,730-year half-life to determine the age of organic materials. This technique transforms archaeology and geology. Libby receives the 1960 Nobel Prize in Chemistry.
Short-lived isotopes like Technetium-99m (t½ = 6 h) and Fluorine-18 (t½ = 110 min) are engineered into radiopharmaceuticals. Their precisely known half-lives allow doctors to image organ function with minimal radiation exposure, revolutionising diagnostic medicine.
Comprehensive IAEA guidance on radiation protection principles, dose limits, and the use of half-life in safety calculations for occupational workers.
Peer-reviewed calibration of the ¹⁴C half-life and atmospheric correction curves, essential for accurate archaeological and geological dating.
Authoritative reference database of all known isotopes, including half-lives, decay modes, and energies maintained by Brookhaven National Laboratory.
Review of how short-lived isotopes are harnessed in PET and SPECT imaging, leveraging precise half-life control for diagnostic accuracy.
U.S. Environmental Protection Agency consumer guide explaining half-life concepts and their role in radiation risk assessment and cleanup standards.
WHO / IARC analysis of ionising radiation and cancer risk, including dose-response relationships based on isotope half-life and exposure duration.
After one half-life, the substance is gone
After one half-life, exactly 50% remains. The substance undergoes continuous exponential decay — it theoretically never reaches absolute zero. Practical "safety" thresholds are typically 10 half-lives (≈0.1% remaining).
Heating or cooling a radioactive substance changes its half-life
Nuclear decay rates are governed by quantum mechanical tunneling through the nuclear potential barrier — a process completely unaffected by thermal energy changes at any achievable laboratory temperature. t½ is constant.
All radioactive materials are equally dangerous
Danger depends on decay type (alpha, beta, gamma), energy, half-life, and exposure route. A short half-life means rapid decay and high activity but short duration. A long half-life means lower activity but persists longer.
Half-life only applies to nuclear physics
Half-life governs any first-order exponential decay: drug plasma concentrations in pharmacokinetics, capacitor discharge in electronics, and light signal attenuation in fiber optics all follow the same t½ equation.
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