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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 | .563 |

7 | ^{1}⁄_{128} | 0 | .781 |

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

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

**Half-life** (symbol ** t_{1⁄2}**) is the time required for a quantity to reduce to half 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.

**Nuclear physics** is the field of physics that studies atomic nuclei and their constituents and interactions. Other forms of nuclear matter are also studied. Nuclear physics should not be confused with atomic physics, which studies the atom as a whole, including its electrons.

**Radioactive decay** is the process by which an unstable atomic nucleus loses energy by emitting radiation, such as an alpha particle, beta particle with neutrino or only a neutrino in the case of electron capture, or a gamma ray or electron in the case of internal conversion. A material containing such unstable nuclei is considered **radioactive**. Certain highly excited short-lived nuclear states can decay through neutron emission, or more rarely, proton emission.

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**:

- 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.

**Ernest Rutherford, 1st Baron Rutherford of Nelson**, HFRSE LLD, was a New Zealand-born British physicist who came to be known as the father of nuclear physics. *Encyclopædia Britannica* considers him to be the greatest experimentalist since Michael Faraday (1791–1867).

A **chemical element** is a species of atom having the same number of protons in their atomic nuclei. For example, the atomic number of oxygen is 8, so the element oxygen consists of all atoms which have exactly 8 protons.

**Radium** is a chemical element with symbol **Ra** and atomic number 88. It is the sixth element in group 2 of the periodic table, also known as the alkaline earth metals. Pure radium is silvery-white, but it readily reacts with nitrogen (rather than oxygen) on exposure to air, forming a black surface layer of radium nitride (Ra_{3}N_{2}). All isotopes of radium are highly radioactive, with the most stable isotope being radium-226, which has a half-life of 1600 years and decays into radon gas (specifically the isotope radon-222). When radium decays, ionizing radiation is a product, which can excite fluorescent chemicals and cause radioluminescence.

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%.

**Probability** is the measure of the likelihood that an event will occur. See glossary of probability and statistics. Probability quantifies as a number between 0 and 1, where, loosely speaking, 0 indicates impossibility and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2.

In probability theory, the **expected value** of a random variable, intuitively, is the long-run average value of repetitions of the **same experiment** it represents. For example, the expected value in rolling a six-sided die is 3.5, because the average of all the numbers that come up is 3.5 as the number of rolls approaches infinity. In other words, the law of large numbers states that the arithmetic mean of the values almost surely converges to the expected value as the number of repetitions approaches infinity. The expected value is also known as the **expectation**, **mathematical expectation**, **EV**, **average**, **mean value**, **mean**, or **first moment**.

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.

In probability theory, the **law of large numbers** (**LLN**) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.

There are various simple exercises that demonstrate probabilistic decay, for example involving flipping coins or running a statistical computer program.^{ [2] }^{ [3] }^{ [4] }

A **computer program** is a collection of instructions that performs a specific task when executed by a computer. A computer requires programs to function.

An exponential decay can be described by any of the following three equivalent formulas:

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).

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.^{ [5] }

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.^{ [6] }

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,^{ [7] } though this can be altered by behavior and various 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,^{ [8] } and certain authors maintain that pesticide risk and impact assessment models rely on and are sensitive to information describing dissipation from plants.^{ [9] }

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 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 that describes 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 exhibited when the rate of change—the change per instant or unit of time—of the value of a mathematical function is proportional to the function's current value, resulting in its value at any time being an exponential function of time, i.e., a function in which the time value is the exponent. Exponential decay occurs in the same way when the growth rate is negative. In the case of a discrete domain of definition with equal intervals, it is also called **geometric growth** or **geometric decay**, the function values forming a geometric progression. In either exponential growth or exponential decay, the ratio of the rate of change of the quantity to its current size remains constant over time.

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

An ideal **Bose gas** is a quantum-mechanical phase of matter, analogous to a classical ideal gas. It is composed of bosons, which have an integer value of spin, and obey Bose–Einstein statistics. The statistical mechanics of bosons were developed by Satyendra Nath Bose for a photon gas, and extended to massive particles by Albert Einstein who realized that an ideal gas of bosons would form a condensate at a low enough temperature, unlike a classical ideal gas. This condensate is known as a Bose–Einstein condensate.

**Variational Bayesian methods** are a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning. They are typically used in complex statistical models consisting of observed variables as well as unknown parameters and latent variables, with various sorts of relationships among the three types of random variables, as might be described by a graphical model. As is typical in Bayesian inference, the parameters and latent variables are grouped together as "unobserved variables". Variational Bayesian methods are primarily used for two purposes:

- To provide an analytical approximation to the posterior probability of the unobserved variables, in order to do statistical inference over these variables.
- To derive a lower bound for the marginal likelihood of the observed data. This is typically used for performing model selection, the general idea being that a higher marginal likelihood for a given model indicates a better fit of the data by that model and hence a greater probability that the model in question was the one that generated the data.

In pharmacokinetics, **effective half-life** is the rate of accumulation or elimination of a biochemical or pharmacological substance in an organism; 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 pharmacology, the **clearance** is a pharmacokinetic measurement of the volume of plasma from which a substance is completely removed per unit time; the usual units are mL/min. The quantity reflects the rate of drug elimination divided by plasma concentration.

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

In nuclear physics, the **Geiger–Nuttall law** or **Geiger–Nuttall rule** relates the decay constant of a radioactive isotope with the energy of the alpha particles emitted. Roughly speaking, it states that short-lived isotopes emit more energetic alpha particles than long-lived ones.

The **biological half-life** of a biological substance is the time it takes for half to be removed by biological processes when the rate of removal is roughly exponential. It is often denoted by the abbreviation . Examples include metabolites, drugs, and signalling molecules. Typically, this refers to the body's cleansing through the function of kidneys and liver in addition to excretion functions to eliminate a substance from the body. In a medical context, half-life may also describe the time it takes for the blood plasma concentration of a substance to halve its steady-state. The relationship between the biological and plasma half-lives of a substance can be complex depending on the substance in question, due to factors including accumulation in tissues, active metabolites, and receptor interactions.

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.

**Penetration depth** is a measure of how deep light or any electromagnetic radiation can penetrate into a material. It is defined as the depth at which the intensity of the radiation inside the material falls to 1/e of its original value at the surface.

The **doubling time** is the period of time required for a quantity to double in size or 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.

**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.

- ↑ John Ayto,
*20th Century Words*(1989), Cambridge University Press. - ↑ Chivers, Sidney (March 16, 2003). "Re: What happens durring 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: BOT: original-url status unknown (link)
- ↑ Jonathan Crowe, Tony Bradshaw (2014).
*Chemistry for the Biosciences: The Essential Concepts*. p. 568. ISBN 9780199662883.CS1 maint: Uses authors parameter (link) - ↑ 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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