This article is missing information about the history of the term .half-life(July 2019) |

Number of half-lives elapsed | Fraction remaining | Percentage remaining | |
---|---|---|---|

0 | ^{1}⁄_{1} | 100 | |

1 | ^{1}⁄_{2} | 50 | |

2 | ^{1}⁄_{4} | 25 | |

3 | ^{1}⁄_{8} | 12 | .5 |

4 | ^{1}⁄_{16} | 6 | .25 |

5 | ^{1}⁄_{32} | 3 | .125 |

6 | ^{1}⁄_{64} | 1 | .5625 |

7 | ^{1}⁄_{128} | 0 | .78125 |

... | ... | ... | |

n | ^{1}/_{2n} | ^{100}/_{2n} |

**Half-life** (symbol ** t_{1⁄2}**) is the time required for a quantity to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo, or how long stable atoms survive, radioactive decay. The term is also used more generally to characterize any type of exponential or non-exponential decay. For example, the medical sciences refer to the biological half-life of drugs and other chemicals in the human body. The converse of half-life is doubling time.

- Probabilistic nature
- Formulas for half-life in exponential decay
- Decay by two or more processes
- Examples
- In non-exponential decay
- In biology and pharmacology
- See also
- References
- External links

The original term, *half-life period*, dating to Ernest Rutherford's discovery of the principle in 1907, was shortened to *half-life* in the early 1950s.^{ [1] } Rutherford applied the principle of a radioactive element's half-life to studies of age determination of rocks by measuring the decay period of radium to lead-206.

Half-life is constant over the lifetime of an exponentially decaying quantity, and it is a characteristic unit for the exponential decay equation. The accompanying table shows the reduction of a quantity as a function of the number of half-lives elapsed.

A half-life usually describes the decay of discrete entities, such as radioactive atoms. In that case, it does not work to use the definition that states "half-life is the time required for exactly half of the entities to decay". For example, if there is just one radioactive atom, and its half-life is one second, there will *not* be "half of an atom" left after one second.

Instead, the half-life is defined in terms of probability: "Half-life is the time required for exactly half of the entities to decay * on average *". In other words, the *probability* of a radioactive atom decaying within its half-life is 50%.^{ [2] }

For example, the image on the right is a simulation of many identical atoms undergoing radioactive decay. Note that after one half-life there are not *exactly* one-half of the atoms remaining, only *approximately*, because of the random variation in the process. Nevertheless, when there are many identical atoms decaying (right boxes), the law of large numbers suggests that it is a *very good approximation* to say that half of the atoms remain after one half-life.

Various simple exercises can demonstrate probabilistic decay, for example involving flipping coins or running a statistical computer program.^{ [3] }^{ [4] }^{ [5] }

An exponential decay can be described by any of the following three equivalent formulas:^{ [6] }^{:109–112}

where

*N*_{0}is the initial quantity of the substance that will decay (this quantity may be measured in grams, moles, number of atoms, etc.),*N*(*t*) is the quantity that still remains and has not yet decayed after a time*t*,*t*_{1⁄2}is the half-life of the decaying quantity,- τ is a positive number called the mean lifetime of the decaying quantity,
- λ is a positive number called the decay constant of the decaying quantity.

The three parameters *t*_{1⁄2}, τ, and λ are all directly related in the following way:

where ln(2) is the natural logarithm of 2 (approximately 0.693).^{ [6] }^{:112}

Some quantities decay by two exponential-decay processes simultaneously. In this case, the actual half-life *T*_{1⁄2} can be related to the half-lives *t*_{1} and *t*_{2} that the quantity would have if each of the decay processes acted in isolation:

For three or more processes, the analogous formula is:

For a proof of these formulas, see Exponential decay § Decay by two or more processes.

There is a half-life describing any exponential-decay process. For example:

- As noted above, in radioactive decay the half-life is the length of time after which there is a 50% chance that an atom will have undergone nuclear decay. It varies depending on the atom type and isotope, and is usually determined experimentally. See List of nuclides.
- The current flowing through an RC circuit or RL circuit decays with a half-life of ln(2)
*RC*or ln(2)*L/R*, respectively. For this example the term half time tends to be used, rather than "half-life", but they mean the same thing. - In a chemical reaction, the half-life of a species is the time it takes for the concentration of that substance to fall to half of its initial value. In a first-order reaction the half-life of the reactant is ln(2)/λ, where λ is the reaction rate constant.

The term "half-life" is almost exclusively used for decay processes that are exponential (such as radioactive decay or the other examples above), or approximately exponential (such as biological half-life discussed below). In a decay process that is not even close to exponential, the half-life will change dramatically while the decay is happening. In this situation it is generally uncommon to talk about half-life in the first place, but sometimes people will describe the decay in terms of its "first half-life", "second half-life", etc., where the first half-life is defined as the time required for decay from the initial value to 50%, the second half-life is from 50% to 25%, and so on.^{ [7] }

A biological half-life or elimination half-life is the time it takes for a substance (drug, radioactive nuclide, or other) to lose one-half of its pharmacologic, physiologic, or radiological activity. In a medical context, the half-life may also describe the time that it takes for the concentration of a substance in blood plasma to reach one-half of its steady-state value (the "plasma half-life").

The relationship between the biological and plasma half-lives of a substance can be complex, due to factors including accumulation in tissues, active metabolites, and receptor interactions.^{ [8] }

While a radioactive isotope decays almost perfectly according to so-called "first order kinetics" where the rate constant is a fixed number, the elimination of a substance from a living organism usually follows more complex chemical kinetics.

For example, the biological half-life of water in a human being is about 9 to 10 days,^{ [9] } though this can be altered by behavior and other conditions. The biological half-life of caesium in human beings is between one and four months.

The concept of a half-life has also been utilized for pesticides in plants,^{ [10] } and certain authors maintain that pesticide risk and impact assessment models rely on and are sensitive to information describing dissipation from plants.^{ [11] }

In nuclear physics, **beta decay** (*β*-decay) is a type of radioactive decay in which a beta particle is emitted from an atomic nucleus, transforming the original nuclide to an isobar. For example, beta decay of a neutron transforms it into a proton by the emission of an electron accompanied by an antineutrino; or, conversely a proton is converted into a neutron by the emission of a positron with a neutrino in so-called *positron emission*. Neither the beta particle nor its associated (anti-)neutrino exist within the nucleus prior to beta decay, but are created in the decay process. By this process, unstable atoms obtain a more stable ratio of protons to neutrons. The probability of a nuclide decaying due to beta and other forms of decay is determined by its nuclear binding energy. The binding energies of all existing nuclides form what is called the nuclear band or valley of stability. For either electron or positron emission to be energetically possible, the energy release or *Q* value must be positive.

In physics, **optical depth** or **optical thickness**, is the natural logarithm of the ratio of *incident* to *transmitted* radiant power through a material, and **spectral optical depth** or **spectral optical thickness** is the natural logarithm of the ratio of incident to transmitted spectral radiant power through a material. Optical depth is dimensionless, and in particular is not a length, though it is a monotonically increasing function of optical path length, and approaches zero as the path length approaches zero. The use of the term "optical density" for optical depth is discouraged.

In probability theory and statistics, the **exponential distribution** is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

The **decay energy** is the energy released by a radioactive decay. Radioactive decay is the process in which an unstable atomic nucleus loses energy by emitting ionizing particles and radiation. This decay, or loss of energy, results in an atom of one type, called the parent nuclide transforming to an atom of a different type, called the daughter nuclide.

The **curie** is a non-SI unit of radioactivity originally defined in 1910. According to a notice in *Nature* at the time, it was named in honour of Pierre Curie, but was considered at least by some to be in honour of Marie Curie as well.

**Exponential growth** is a specific way that a quantity may increase over time. It occurs when the instantaneous rate of change of a quantity with respect to time is proportional to the quantity itself. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent.

**Radioactive decay** is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is considered **radioactive**. Three of the most common types of decay are alpha decay, beta decay, and gamma decay, all of which involve emitting one or more particles or photons. The weak force is the mechanism that is responsible for beta decay.

In nuclear science, the **decay chain** refers to a series of radioactive decays of different radioactive decay products as a sequential series of transformations. It is also known as a "radioactive cascade". Most radioisotopes do not decay directly to a stable state, but rather undergo a series of decays until eventually a stable isotope is reached.

A quantity is subject to **exponential decay** if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where *N* is the quantity and λ (lambda) is a positive rate called the **exponential decay constant**:

**Specific activity** is the activity per quantity of a radionuclide and is a physical property of that radionuclide.

In particle physics and nuclear physics, the **branching fraction** for a decay is the fraction of particles which decay by an individual decay mode with respect to the total number of particles which decay. It is equal to the ratio of the **partial decay constant** to the overall decay constant. Sometimes a **partial half-life** is given, but this term is misleading; due to competing modes it is not true that half of the particles will decay through a particular decay mode after its partial half-life. The partial half-life is merely an alternate way to specify the partial decay constant λ, the two being related through:

In pharmacokinetics, the **effective half-life** is the rate of accumulation or elimination of a biochemical or pharmacological substance in an organism; it is the analogue of biological half-life when the kinetics are governed by multiple independent mechanisms. This is seen when there are multiple mechanisms of elimination, or when a drug occupies multiple pharmacological compartments. It reflects the cumulative effect of the individual half-lives, as observed by the changes in the actual serum concentration of a drug under a given dosing regimen. The complexity of biological systems means that most pharmacological substances do not have a single mechanism of elimination, and hence the observed or effective half-life does not reflect that of a single process, but rather the summation of multiple independent processes.

In science, ** e-folding** is the time interval in which an exponentially growing quantity increases by a factor of

In nuclear physics, **secular equilibrium** is a situation in which the quantity of a radioactive isotope remains constant because its production rate is equal to its decay rate.

The **doubling time** is time it takes for a population to double in size/value. It is applied to population growth, inflation, resource extraction, consumption of goods, compound interest, the volume of malignant tumours, and many other things that tend to grow over time. When the relative growth rate is constant, the quantity undergoes exponential growth and has a constant doubling time or period, which can be calculated directly from the growth rate.

In physics and engineering, the **time constant**, usually denoted by the Greek letter τ (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system. The time constant is the main characteristic unit of a first-order LTI system.

**Laser linewidth** is the spectral linewidth of a laser beam.

In nuclear physics, the **Bateman equation** is a mathematical model describing abundances and activities in a decay chain as a function of time, based on the decay rates and initial abundances. The model was formulated by Ernest Rutherford in 1905 and the analytical solution was provided by Harry Bateman in 1910.

**Potassium–calcium dating**, abbreviated **K–Ca dating**, is a radiometric dating method used in geochronology. It is based upon measuring the ratio of a parent isotope of potassium (^{40}K) to a daughter isotope of calcium (^{40}Ca). This form of radioactive decay is accomplished through beta decay.

The **perturbed γ-γ angular correlation**, **PAC** for short or **PAC-Spectroscopy**, is a method of nuclear solid-state physics with which magnetic and electric fields in crystal structures can be measured. In doing so, electrical field gradients and the Larmor frequency in magnetic fields as well as dynamic effects are determined. With this very sensitive method, which requires only about 10-1000 billion atoms of a radioactive isotope per measurement, material properties in the local structure, phase transitions, magnetism and diffusion can be investigated. The PAC method is related to nuclear magnetic resonance and the Mössbauer effect, but shows no signal attenuation at very high temperatures. Today only the time-differential perturbed angular correlation (**TDPAC**) is used.

- ↑ John Ayto,
*20th Century Words*(1989), Cambridge University Press. - ↑ Muller, Richard A. (April 12, 2010).
*Physics and Technology for Future Presidents*. Princeton University Press. pp. 128–129. ISBN 9780691135045. - ↑ Chivers, Sidney (March 16, 2003). "Re: What happens during half-lifes [sic] when there is only one atom left?". MADSCI.org.
- ↑ "Radioactive-Decay Model". Exploratorium.edu. Retrieved 2012-04-25.
- ↑ Wallin, John (September 1996). "Assignment #2: Data, Simulations, and Analytic Science in Decay". Astro.GLU.edu. Archived from the original on 2011-09-29.CS1 maint: unfit url (link)
- 1 2 Rösch, Frank (September 12, 2014).
*Nuclear- and Radiochemistry: Introduction*.**1**. Walter de Gruyter. ISBN 978-3-11-022191-6. - ↑ Jonathan Crowe; Tony Bradshaw (2014).
*Chemistry for the Biosciences: The Essential Concepts*. p. 568. ISBN 9780199662883. - ↑ Lin VW; Cardenas DD (2003).
*Spinal cord medicine*. Demos Medical Publishing, LLC. p. 251. ISBN 978-1-888799-61-3. - ↑ Pang, Xiao-Feng (2014).
*Water: Molecular Structure and Properties*. New Jersey: World Scientific. p. 451. ISBN 9789814440424. - ↑ Australian Pesticides and Veterinary Medicines Authority (31 March 2015). "Tebufenozide in the product Mimic 700 WP Insecticide, Mimic 240 SC Insecticide". Australian Government. Retrieved 30 April 2018.
- ↑ Fantke, Peter; Gillespie, Brenda W.; Juraske, Ronnie; Jolliet, Olivier (11 July 2014). "Estimating Half-Lives for Pesticide Dissipation from Plants".
*Environmental Science & Technology*.**48**(15): 8588–8602. Bibcode:2014EnST...48.8588F. doi:10.1021/es500434p. PMID 24968074.

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