# Exponential decay
**Exponential decay** is the process in which a quantity decreases at a rate proportional to its current value, so that it falls as an exponential function of time, N(t) = N₀·e^(−λt). The proportionality constant λ, called the decay constant, has units of inverse time (s⁻¹) and measures the probability of decay per unit time for each individual unit of the quantity[1](https://gnssn.iaea.org/main/bptc/BPTC%20Module%20Documents/Module01%20Nuclear%20physics%20and%20reactor%20theory.pdf). The defining property is memorylessness: the instantaneous probability of decay is independent of how long the unit has already existed[2](https://nvlpubs.nist.gov/nistpubs/sp958-lide/html/116-119.html). The most prominent physical example is radioactive decay, and the same mathematics describes [Newton's law of cooling](https://www.edgepedia.org/newtons-law-of-cooling), the discharge of a capacitor, and the exponential distribution of waiting times[3](https://openstax.org/books/calculus-volume-2/pages/2-8-exponential-growth-and-decay) • [4](https://itl.nist.gov/div898/handbook/apr/section1/apr161.htm).
| Key fact | Detail |
|---|---|
| Decay law | N(t) = N₀·e^(−λt); equivalently N(t) = N₀·2^(−t/t₁/₂)[5](https://openstax.org/books/college-physics/pages/31-5-half-life-and-activity) • [1](https://gnssn.iaea.org/main/bptc/BPTC%20Module%20Documents/Module01%20Nuclear%20physics%20and%20reactor%20theory.pdf) |
| Half-life | t₁/₂ = ln 2 / λ ≈ 0.693/λ; the time for the quantity to fall to half its initial value[5](https://openstax.org/books/college-physics/pages/31-5-half-life-and-activity) |
| Mean lifetime | τ = 1/λ = t₁/₂ / ln 2 ≈ 1.44·t₁/₂, the time to fall to 1/e ≈ 36.8% of the initial value[6](https://www-pub.iaea.org/mtcd/publications/pdf/pub1617web-1294055.pdf) |
| Activity | A = λN, measured in becquerels (1 Bq = 1 decay/s); the older curie is 1 Ci = 3.7×10¹⁰ Bq[7](https://openstax.org/books/university-physics-volume-3/pages/10-3-radioactive-decay) |
| Range of half-lives | From about 10⁻²³ s for the most unstable nuclides to more than 10¹⁶ y for the least unstable, about 46 orders of magnitude[5](https://openstax.org/books/college-physics/pages/31-5-half-life-and-activity) |
| Statistical basis | Each nucleus has a 50% chance of surviving one half-life, regardless of its age; the law is exact only for large numbers[5](https://openstax.org/books/college-physics/pages/31-5-half-life-and-activity) |
## The decay law
The differential equation behind exponential decay states that the rate of loss is proportional to the amount present. For radioactive nuclei, the number lost in a time interval dt is −dN = λN·dt, where the minus sign records that the number of undecayed nuclei decreases[7](https://openstax.org/books/university-physics-volume-3/pages/10-3-radioactive-decay). Integrating from an initial count N₀ at t = 0 gives the decay law[7](https://openstax.org/books/university-physics-volume-3/pages/10-3-radioactive-decay):
\[ N(t) = N_0 \, e^{-\lambda t} \]
The same solution follows from the half-life form n(t) = n₀·2^(−t/t₁/₂), since 2^(−t/t₁/₂) = e^(−λt) with λ = ln 2 / t₁/₂[1](https://gnssn.iaea.org/main/bptc/BPTC%20Module%20Documents/Module01%20Nuclear%20physics%20and%20reactor%20theory.pdf). Taking the natural logarithm of the decay law gives ln A = −λt + ln A₀, so a plot of ln A against t is a straight line with slope −λ; measuring activity over time is therefore a direct way to determine the decay constant[7](https://openstax.org/books/university-physics-volume-3/pages/10-3-radioactive-decay).
**Half-life and mean life.** The half-life t₁/₂ is the time in which half the nuclei disintegrate; after two half-lives a quarter remains, after three an eighth, and so on[1](https://gnssn.iaea.org/main/bptc/BPTC%20Module%20Documents/Module01%20Nuclear%20physics%20and%20reactor%20theory.pdf). Setting N = N₀/2 in the decay law yields t₁/₂ = ln 2 / λ ≈ 0.693/λ[5](https://openstax.org/books/college-physics/pages/31-5-half-life-and-activity). The mean (average) life τ is the average time from formation to decay, defined as the time for the quantity to fall to 1/e = 0.368 of its initial value; it satisfies λτ = 1, so τ = 1/λ = 1.44·t₁/₂[6](https://www-pub.iaea.org/mtcd/publications/pdf/pub1617web-1294055.pdf). As a worked example, a carbon-14 nucleus with t₁/₂ = 5730 years has a mean life of 8267 years, and a sample's activity falls by 1‰ in about 8 years[8](https://gnssn-qa.iaea.org/main/ncp/Tunisia/lrae/documents/tracers/Volume_I.pdf).
## Activity
The **activity** of a source is the number of disintegrations per unit time, A = λN, the product of the number of radioactive nuclei and the decay constant[1](https://gnssn.iaea.org/main/bptc/BPTC%20Module%20Documents/Module01%20Nuclear%20physics%20and%20reactor%20theory.pdf). Because λ is constant, activity decreases exponentially exactly as the number of nuclei does[1](https://gnssn.iaea.org/main/bptc/BPTC%20Module%20Documents/Module01%20Nuclear%20physics%20and%20reactor%20theory.pdf). The SI unit is the becquerel, 1 Bq = 1 decay per second; the older curie was defined as the activity of 1 g of radium-226 and equals 3.7×10¹⁰ Bq[7](https://openstax.org/books/university-physics-volume-3/pages/10-3-radioactive-decay). Activity per unit mass is the specific activity[1](https://gnssn.iaea.org/main/bptc/BPTC%20Module%20Documents/Module01%20Nuclear%20physics%20and%20reactor%20theory.pdf).
## Statistical character
[Radioactive decay](https://www.edgepedia.org/radioactive-decay) is a random process at the level of single nuclei: the moment when a particular nucleus decays cannot be predicted, but for a large number of identical nuclei the fraction decaying per unit time is predictable[1](https://gnssn.iaea.org/main/bptc/BPTC%20Module%20Documents/Module01%20Nuclear%20physics%20and%20reactor%20theory.pdf). A more precise statement of half-life is probabilistic: each nucleus has a 50% chance of surviving a time equal to one half-life, and if it survives, it again has a 50% chance of surviving the next half-life, like repeated coin flips[5](https://openstax.org/books/college-physics/pages/31-5-half-life-and-activity). The exponential law is therefore a large-number average; individual decay events follow Poisson statistics, and the waiting times between counts in a detector are exponentially distributed[9](https://www.nist.gov/publications/some-case-studies-skewed-and-other-ab-normal-data-distributions-arising-low-level).
## Quantum-mechanical basis
Under simplifying but widely applicable assumptions, Viktor Weisskopf and [Eugene Wigner](https://www.edgepedia.org/eugene-wigner), physicists known for foundational work in quantum theory, showed that quantum mechanics predicts the intensity of radiation emitted by an excited state to follow I(t) ∝ e^(−Γt), where Γ is a rate constant equal to the inverse of the state's lifetime[2](https://nvlpubs.nist.gov/nistpubs/sp958-lide/html/116-119.html). The law expresses the same memorylessness: the instantaneous probability of decay of a given atom is independent of when it is observed[2](https://nvlpubs.nist.gov/nistpubs/sp958-lide/html/116-119.html). The frequency spectrum of such a decaying state has a Lorentzian line shape whose width is inversely proportional to the lifetime, an expression of the energy–time uncertainty relation[2](https://nvlpubs.nist.gov/nistpubs/sp958-lide/html/116-119.html).
## Decay chains and the Bateman equations
When a radioactive parent decays into a radioactive daughter, which decays in turn, the simple exponential no longer describes each member of the chain. The daughter's population changes at a rate equal to the supply from the parent's decay minus its own loss, dN₂/dt = λ₁N₁ − λ₂N₂, and the Bateman equations, a method of setting up such differential equations for each member of a decay chain, give the full time dependence[10](https://www-pub.iaea.org/MTCD/Publications/PDF/Pub1579_web.pdf). The nuclear chart contains more than 110 parent/daughter pairs of practical interest, from ²⁸Mg/²⁸Al to ²³⁴Th/²³⁴ᵐPa[10](https://www-pub.iaea.org/MTCD/Publications/PDF/Pub1579_web.pdf). A related case is production under irradiation: coupling the decay law to a constant activation rate R gives A(t) = Σ·Φ·[1 − e^(−λt)], which approaches a saturation activity Aₛ = Σ·Φ at which production and decay rates are equal; after five half-lives of irradiation the activity already reaches 97% of saturation[11](https://nucleus.iaea.org/sites/connect/RRIHpublic/CompendiumDB/Shared%20Documents/Germany%20TU%20Dresden/Protocols%20in%20PDF/2015_0515%20Lab.Prot.%20Activation%20and%20Decay.pdf).
## Applications
**Radiometric dating.** [Carbon-14](https://www.edgepedia.org/carbon-14) decays to nitrogen-14 with a half-life of 5730 years, so the ratio of remaining carbon-14 to the original amount fixes the age of organic material[3](https://openstax.org/books/calculus-volume-2/pages/2-8-exponential-growth-and-decay). An artifact whose original 100 g of carbon now contains 10 g is about 19,000 years old[3](https://openstax.org/books/calculus-volume-2/pages/2-8-exponential-growth-and-decay). Isotopes with much longer half-lives date older materials; uranium-238, with a half-life of 4.5×10⁹ years, shows that the oldest rocks on Earth solidified about 3.5×10⁹ years ago[5](https://openstax.org/books/college-physics/pages/31-5-half-life-and-activity).
**Effective half-life in the body.** When a radionuclide inside an organism is lost both by radioactive decay and by biological elimination, the two independent exponential processes combine. The effective half-life satisfies 1/T_eff = 1/T_physical + 1/T_biological, so the effective half-life is always shorter than either component[12](https://gnssn.iaea.org/main/ansn/Activity%20Documents%20%20Public/IAEA%E2%80%93KINS%20Basic%20Professional%20Training%20Course%20on%20Nuclear%20Safety/D1_2_Radiation%20Protection%20of%20nuclear%20facilities%20_WKCHO_241021_23.pdf). The IAEA safety glossary generalizes this to any set of simultaneous exponential processes, 1/T_eff = Σᵢ 1/Tᵢ[13](https://gnssn.iaea.org/CSN/relevant_documents/IAEA_Documents/Shared%20Documents/Safety%20Fundamentals%20and%20Safety%20Glossary/SAFETY%20GLOSSARY%202007%20Edition-Terminology%20used%20in%20Nuclear%20Safety%20and%20Radiation%20Protection.pdf).
**Other exponential processes.** Newton's law of cooling, in which an object cools at a rate proportional to the temperature difference from its surroundings, reduces to the same differential equation after a change of variables, giving T(t) = (T₀ − Tₐ)·e^(−kt) + Tₐ[3](https://openstax.org/books/calculus-volume-2/pages/2-8-exponential-growth-and-decay). In reliability engineering, the exponential distribution with constant failure rate h(t) = λ, mean 1/λ, and median ln 2/λ models components whose hazard does not change with age[4](https://itl.nist.gov/div898/handbook/apr/section1/apr161.htm).
## Constancy of the decay constant
The decay constant of a nuclide is characteristic of that nuclide and, in the standard account, independent of external factors such as pressure, temperature, and chemical state[1](https://gnssn.iaea.org/main/bptc/BPTC%20Module%20Documents/Module01%20Nuclear%20physics%20and%20reactor%20theory.pdf) • [7](https://openstax.org/books/university-physics-volume-3/pages/10-3-radioactive-decay). Experiments have found small exceptions tied to the atomic environment, chiefly for decay modes that involve the electrons: the electron-capture half-life of ⁷Be was measured 0.8% longer when the nuclei were embedded in metals cooled to 12 K, while the β⁺ half-life of ²²Na was 1.2% shorter and the β⁻ half-life of ¹⁹⁸Au 4.0% longer under the same conditions[14](https://www.researchgate.net/publication/241259169_Environmental_effects_on_nuclear_decay_rates). A dedicated search for variation of decay rates in an intense antineutrino flux, about 3×10¹² antineutrinos·cm⁻²·s⁻¹ at 6.5 m from a reactor core, found no observable effect and set cross-section upper limits about 10⁴ times more sensitive than earlier experiments[15](https://arxiv.org/abs/2111.05983).
## References
1. [IAEA Basic Professional Training Course, Module 1: Nuclear Physics and Reactor Theory](https://gnssn.iaea.org/main/bptc/BPTC%20Module%20Documents/Module01%20Nuclear%20physics%20and%20reactor%20theory.pdf)
2. [Exponential decay of excited states (Weisskopf–Wigner), NIST](https://nvlpubs.nist.gov/nistpubs/sp958-lide/html/116-119.html)
3. [Exponential Growth and Decay, Calculus Volume 2, OpenStax](https://openstax.org/books/calculus-volume-2/pages/2-8-exponential-growth-and-decay)
4. [Exponential distribution, NIST/SEMATECH e-Handbook of Statistical Methods](https://itl.nist.gov/div898/handbook/apr/section1/apr161.htm)
5. [Half-Life and Activity, College Physics, OpenStax](https://openstax.org/books/college-physics/pages/31-5-half-life-and-activity)
6. [Nuclear Medicine Physics, IAEA](https://www-pub.iaea.org/mtcd/publications/pdf/pub1617web-1294055.pdf)
7. [Radioactive Decay, University Physics Volume 3, OpenStax](https://openstax.org/books/university-physics-volume-3/pages/10-3-radioactive-decay)
8. [Equations of Radioactive Decay, IAEA Environmental tracers manual](https://gnssn-qa.iaea.org/main/ncp/Tunisia/lrae/documents/tracers/Volume_I.pdf)
9. [Skewed data distributions in low-level environmental research, NIST (L. A. Currie)](https://www.nist.gov/publications/some-case-studies-skewed-and-other-ab-normal-data-distributions-arising-low-level)
10. [Radiotracer Generators for Industrial Applications, IAEA Radiation Technology Series No. 5](https://www-pub.iaea.org/MTCD/Publications/PDF/Pub1579_web.pdf)
11. [Laboratory protocol: Activation and Decay, TU Dresden / IAEA](https://nucleus.iaea.org/sites/connect/RRIHpublic/CompendiumDB/Shared%20Documents/Germany%20TU%20Dresden/Protocols%20in%20PDF/2015_0515%20Lab.Prot.%20Activation%20and%20Decay.pdf)
12. [Radiation Protection of Nuclear Facilities, IAEA–KINS Basic Professional Training Course](https://gnssn.iaea.org/main/ansn/Activity%20Documents%20%20Public/IAEA%E2%80%93KINS%20Basic%20Professional%20Training%20Course%20on%20Nuclear%20Safety/D1_2_Radiation%20Protection%20of%20nuclear%20facilities%20_WKCHO_241021_23.pdf)
13. [IAEA Safety Glossary (2007 Edition), half-life and effective half-life](https://gnssn.iaea.org/CSN/relevant_documents/IAEA_Documents/Shared%20Documents/Safety%20Fundamentals%20and%20Safety%20Glossary/SAFETY%20GLOSSARY%202007%20Edition-Terminology%20used%20in%20Nuclear%20Safety%20and%20Radiation%20Protection.pdf)
14. [Environmental effects on nuclear decay rates, S.-H. Zhou, Chinese Physics C 35, 449 (2011)](https://www.researchgate.net/publication/241259169_Environmental_effects_on_nuclear_decay_rates)
15. [Test of Nuclear Decay Rate Variation due to an Antineutrino Flux, arXiv:2111.05983](https://arxiv.org/abs/2111.05983)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Particles and nuclei › Nuclear physics › Radioactivity and nuclear decay › Decay kinetics and decay chains › Exponential decay law and decay constants*
*Initially written Sep 17, 2026 · Reviewed: Sep 17, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*
License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license