Half-Life Calculator
Convert between half-life, decay constant, and mean lifetime. Find time to reach any target fraction of a radioactive sample.
⏳ What is Half-Life?
The half-life (symbol t½) of a radioactive nuclide is the time required for exactly half of a given number of atoms to undergo spontaneous nuclear decay. It is the most widely used measure of radioactive decay rate and was first introduced by Ernest Rutherford in 1907. Half-life is a fixed, intrinsic property of each nuclide - it cannot be changed by temperature, pressure, chemical state, or any other external condition.
Half-lives span an enormous range in nature. Polonium-214 has a half-life of 164 microseconds - it vanishes almost instantaneously. Carbon-14 has a half-life of 5,730 years - useful for dating organic material up to ~50,000 years old. Uranium-238 has a half-life of 4.468 billion years - comparable to the age of the Solar System, which is why primordial uranium still exists on Earth. Tellurium-128 holds the record for the longest measured half-life: approximately 2.2 × 10²⁴ years.
Half-life is directly related to two other decay rate parameters. The decay constant λ (lambda) is the probability per unit time that any nucleus will decay: λ = ln(2)/t½ ≈ 0.6931/t½. The mean lifetime τ (tau) is the average time a nucleus survives before decaying: τ = 1/λ = t½/ln(2) ≈ 1.4427 × t½. All three are equivalent descriptions of the same physical process - this calculator converts freely among them.
Half-life has critical applications across science and medicine. In radiocarbon dating, archaeologists use C-14's t½ = 5,730 yr to calculate the age of organic samples. In nuclear medicine, physicians choose isotopes with half-lives matched to the procedure - short enough to minimise patient dose, long enough to complete imaging. In nuclear waste management, the half-lives of fission products determine how long waste must be safely stored. In geochronology, the half-lives of U-238, K-40, and Rb-87 are used to date rocks billions of years old.
📐 Formula
Common Isotope Half-Life Reference Table
| Isotope | Symbol | Half-Life | Use / Context |
|---|---|---|---|
| Technetium-99m | ⁹⁹ᵐTc | 6.01 hr | Medical imaging (SPECT) |
| Fluorine-18 | ¹⁸F | 109.8 min | PET scanning |
| Iodine-131 | ¹³¹I | 8.02 days | Thyroid therapy |
| Carbon-14 | ¹⁴C | 5,730 yr | Radiocarbon dating |
| Tritium (H-3) | ³H | 12.32 yr | Nuclear weapons, fusion |
| Caesium-137 | ¹³⁷Cs | 30.17 yr | Fission product, Chernobyl |
| Strontium-90 | ⁹⁰Sr | 28.8 yr | Fission product, RTGs |
| Cobalt-60 | ⁶⁰Co | 5.27 yr | Radiation therapy, sterilisation |
| Plutonium-239 | ²³⁹Pu | 24,110 yr | Nuclear weapons, reactor fuel |
| Radium-226 | ²²⁶Ra | 1,600 yr | Curie's discovery, historical |
| Potassium-40 | ⁴⁰K | 1.248 Gyr | K-Ar geochronology |
| Uranium-235 | ²³⁵U | 703.8 Myr | Reactor fuel, U-Pb dating |
| Uranium-238 | ²³⁸U | 4.468 Gyr | U-Pb dating, primordial |
| Thorium-232 | ²³²Th | 14.05 Gyr | Thorium fuel cycle |
📖 How to Use This Calculator
💡 Example Calculations
Example 1 - Carbon-14 (known half-life, find λ and τ)
C-14 has t½ = 5,730 years. Find λ and mean lifetime τ.
Example 2 - Technetium-99m (medical imaging isotope)
Tc-99m has t½ = 6.0058 hours. Find λ in s⁻¹ and mean lifetime.
Example 3 - From decay constant to half-life (Cs-137)
Caesium-137 has λ = 7.30 × 10−10 s−1. Find t½ and τ.
Frequently Asked Questions
🔗 Related Calculators
What is the half-life of a radioactive element and what does it measure?
The half-life (t½) is the time required for exactly half of the atoms in a radioactive sample to undergo decay. It is the most intuitive measure of a nuclide's decay rate. Short half-lives (milliseconds to days) indicate highly radioactive, quickly decaying nuclides. Long half-lives (thousands to billions of years) indicate slowly decaying, persistently radioactive nuclides. Half-life is independent of sample size - 1 atom of C-14 and 1 mole of C-14 both have t½ = 5,730 years.
How is half-life related to the decay constant λ?
They are related by t½ = ln(2)/λ ≈ 0.6931/λ. The decay constant λ is the probability that any given nucleus will decay per unit time. A large λ means fast decay (short half-life); small λ means slow decay (long half-life). The relationship comes from solving N(t) = N₀/2 = N₀e^(−λt½), which gives λt½ = ln(2).
What is the mean lifetime of a radioactive nucleus and how does it differ from the half-life?
The mean (average) lifetime τ = 1/λ = t½/ln(2) ≈ 1.4427 × t½. It is the average time a nucleus survives before decaying, computed by integrating t·λe^(−λt) from 0 to ∞. After one mean lifetime τ, the fraction remaining is 1/e ≈ 36.79%, not 50% as after one half-life. Mean lifetime is commonly used in particle physics and in deriving the exponential decay law.
What are the half-lives of common isotopes used in medicine?
Technetium-99m: 6.0058 hours (SPECT imaging, most widely used). Fluorine-18: 109.8 minutes (PET scanning). Iodine-131: 8.02 days (thyroid ablation, cancer therapy). Iodine-123: 13.22 hours (thyroid imaging). Thallium-201: 73.01 hours (cardiac SPECT). Gallium-67: 3.26 days (infection/tumor imaging). Lutetium-177: 6.65 days (targeted radionuclide therapy).
What are the half-lives of isotopes used in geology and geochronology?
Uranium-238: 4.468 × 10⁹ years (U-Pb dating, oldest rocks). Uranium-235: 7.04 × 10⁸ years (U-Pb dating). Potassium-40: 1.248 × 10⁹ years (K-Ar dating). Rubidium-87: 4.923 × 10¹⁰ years (Rb-Sr dating). Carbon-14: 5,730 years (radiocarbon dating, organic material up to ~50,000 yr). Samarium-147: 1.07 × 10¹¹ years (Sm-Nd dating of ancient rocks).
How do you calculate the time to reach a given fraction remaining?
From N(t)/N₀ = e^(−λt), solving for t: t = −ln(N/N₀) / λ = −t½ × log₂(N/N₀). For example, to find when 10% remains: t = −t½ × log₂(0.1) = t½ × log₂(10) ≈ t½ × 3.322. So it takes 3.322 half-lives for 90% of the sample to decay.
What is the biological half-life and how does it differ from the physical half-life?
The physical half-life (t½phy) is the nuclear decay rate - fixed and unalterable. The biological half-life (t½bio) is the time for the body to eliminate half of a substance through metabolic processes - depends on physiology, not nuclear physics. The effective half-life combines both: 1/t½eff = 1/t½phy + 1/t½bio, or equivalently t½eff = (t½phy × t½bio) / (t½phy + t½bio). Effective half-life is always shorter than either individual half-life.
Why can't the half-life of a radioactive element be changed?
The half-life is determined by the nuclear force binding protons and neutrons inside the nucleus. Chemical state, temperature, pressure, and physical form have no significant effect on nuclear structure - the binding energy scale (~MeV) is a million times larger than chemical energy scales (~eV). Tiny effects exist for some electron-capture isotopes (where the decay involves orbital electrons), but these are negligible for practical purposes.
What is the shortest known half-life?
The shortest confirmed half-life is that of hydrogen-7 (⁷H), measured at approximately 23 × 10⁻²⁴ seconds (23 yoctoseconds) - it exists for only a few nuclear diameters' worth of time before decaying. Many other nuclides far from the stability line have half-lives in the picosecond to femtosecond range. In contrast, the longest known half-lives are for naturally occurring primordial nuclides like Te-128 (t½ ≈ 2.2 × 10²⁴ years).
How is the half-life measured experimentally?
For short half-lives (seconds to days): measure the activity A(t) = λN over time and fit an exponential decay curve; the time for activity to halve is t½. For long half-lives (millennia to billions of years): measure current activity A = λN, determine N (number of atoms via mass spectrometry), then λ = A/N and t½ = ln(2)/λ. Carbon-14's half-life was first measured precisely by Willard Libby in 1949 using this second method.