Half-life - Wikipedia Half-life From Wikipedia, the free encyclopedia Jump to navigation Jump to search Scientific and mathematical term This article is about the scientific and mathematical concept. For the video game, see Half-Life (video game). For other uses, see Half-Life (disambiguation). This article is missing information about the history of the term half-life. Please expand the article to include this information. Further details may exist on the talk page. (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 t1⁄2) 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 radioactive decay or how long stable atoms survive. 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. 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. Contents 1 Probabilistic nature 2 Formulas for half-life in exponential decay 2.1 Half-life and reaction orders 2.2 Decay by two or more processes 2.3 Examples 3 In non-exponential decay 4 In biology and pharmacology 5 See also 6 References 7 External links Probabilistic nature[edit] Simulation of many identical atoms undergoing radioactive decay, starting with either 4 atoms per box (left) or 400 (right). The number at the top is how many half-lives have elapsed. Note the consequence of the law of large numbers: with more atoms, the overall decay is more regular and more predictable. 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] Formulas for half-life in exponential decay[edit] Main article: Exponential decay An exponential decay can be described by any of the following three equivalent formulas:[6]:109–112 N ( t ) = N 0 ( 1 2 ) t t 1 / 2 N ( t ) = N 0 e − t τ N ( t ) = N 0 e − λ t {\displaystyle {\begin{aligned}N(t)&=N_{0}\left({\frac {1}{2}}\right)^{\frac {t}{t_{1/2}}}\\N(t)&=N_{0}e^{-{\frac {t}{\tau }}}\\N(t)&=N_{0}e^{-\lambda t}\end{aligned}}} where N0 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, t1⁄2 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 t1⁄2, τ, and λ are all directly related in the following way: t 1 / 2 = ln ⁡ ( 2 ) λ = τ ln ⁡ ( 2 ) {\displaystyle t_{1/2}={\frac {\ln(2)}{\lambda }}=\tau \ln(2)} where ln(2) is the natural logarithm of 2 (approximately 0.693).[6]:112 Half-life and reaction orders[edit] The value of the half-life depends on the reaction order: Zero order kinetics: The rate of this kind of reaction does not depend on the substrate concentration. The rate law of zero order kinetics is as follows: [ A ] = [ A ] 0 − k t {\displaystyle [A]=[A]_{0}-kt} In order to find the half life we have to replace the concentration value for the initial concentration divided by 2 and isolate the time. If we do it, we find the equation of the half life of the zero order reaction: t 1 / 2 = [ A ] 0 k 2 {\displaystyle t_{1/2}={\frac {[A]_{0}}{k2}}} The t1/2 formula for a zero order reaction suggests the half-life depends on the amount of initial concentration and rate constant. First order kinetics: In first order reactions, the concentration of the reaction will continue to decrease as time progresses until it reaches zero, and the length of half-life will be constant, independent of concentration. The time for [A] to decrease from [A]0 to ½ [A]0 in a first-order reaction is given by the following equation: k t 1 / 2 = − ln ⁡ ( 1 / 2 [ A ] 0 [ A ] 0 ) = − ln ⁡ 1 2 = ln ⁡ 2 {\displaystyle kt_{1/2}=-\ln {\biggl (}{\frac {1/2[A]_{0}}{[A]_{0}}}{\biggr )}=-\ln {\frac {1}{2}}=\ln 2} For a first-order reaction, the half-life of a reactant is independent of its initial concentration. Therefore, if the concentration of A at some arbitrary stage of the reaction is [A], then it will have fallen to ½ [A] after a further interval of (\ln 2)/k. Hence, the half-life of a first order reaction is given as the following: t 1 / 2 = ln ⁡ 2 k {\displaystyle t_{1/2}={\frac {\ln 2}{k}}} The half-life of a first order reaction is independent of its initial concentration and depends solely on the reaction rate constant, k. Second order kinetics: In the second order reactions, the concentration of the reactant decrease following this formula: 1 [ A ] = k t + 1 [ A ] 0 {\displaystyle {\frac {1}{[A]}}=kt+{\frac {1}{[A]_{0}}}} Then, we replace [A] for [A]0 divided by 2 in order to calculate the half-life of the reactant A and isolate the time of the half-life (t1/2): t 1 / 2 = 1 [ A ] 0 k {\displaystyle t_{1/2}={\frac {1}{[A]_{0}k}}} As you can see, the half-life of the second order reactions depends on the initial concentration and rate constant. Decay by two or more processes[edit] Some quantities decay by two exponential-decay processes simultaneously. In this case, the actual half-life T1⁄2 can be related to the half-lives t1 and t2 that the quantity would have if each of the decay processes acted in isolation: 1 T 1 / 2 = 1 t 1 + 1 t 2 {\displaystyle {\frac {1}{T_{1/2}}}={\frac {1}{t_{1}}}+{\frac {1}{t_{2}}}} For three or more processes, the analogous formula is: 1 T 1 / 2 = 1 t 1 + 1 t 2 + 1 t 3 + ⋯ {\displaystyle {\frac {1}{T_{1/2}}}={\frac {1}{t_{1}}}+{\frac {1}{t_{2}}}+{\frac {1}{t_{3}}}+\cdots } For a proof of these formulas, see Exponential decay § Decay by two or more processes. Examples[edit] Half-life demonstrated using dice in a classroom experiment Further information: Exponential decay § Applications and examples 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. In non-exponential decay[edit] 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] In biology and pharmacology[edit] See also: Biological half-life 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 epidemiology, the concept of half-life can refer to the length of time for the number of incident cases in a disease outbreak to drop by half, particularly if the dynamics of the outbreak can be modeled exponentially.[12][13] See also[edit] Half time (physics) List of radioactive nuclides by half-life Mean lifetime Median lethal dose References[edit] ^ 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) ^ a b 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. ^ Balkew, Teshome Mogessie (December 2010). The SIR Model When S(t) is a Multi-Exponential Function (Thesis). East Tennessee State University. ^ Ireland, MW, ed. (1928). The Medical Department of the United States Army in the World War, vol. IX: Communicable and Other Diseases. Washington: U.S.: U.S. Government Printing Office. pp. 116–7. External links[edit] Look up half-life in Wiktionary, the free dictionary. Wikimedia Commons has media related to Half times. Welcome to Nucleonica, Nucleonica.net (archived 2017) wiki: Decay Engine, Nucleonica.net (archived 2016) System Dynamics – Time Constants, Bucknell.edu Researchers Nikhef and UvA measure slowest radioactive decay ever: Xe-124 with 18 billion trillion years v t e Radiation (physics and health) Main articles Non-ionizing radiation Acoustic radiation force Infrared Light Starlight Sunlight Microwave Radio waves Ultraviolet Ionizing radiation Radioactive decay Cluster decay Background radiation Alpha particle Beta particle Gamma ray Cosmic ray Neutron radiation Nuclear fission Nuclear fusion Nuclear reactors Nuclear weapons Particle accelerators Radioactive materials X-ray Earth's energy budget Electromagnetic radiation Synchrotron radiation Thermal radiation Black-body radiation Particle radiation Gravitational radiation Cosmic background radiation Cherenkov radiation Askaryan radiation Bremsstrahlung Unruh radiation Dark radiation Radiation and health Radiation syndrome acute chronic Health physics Dosimetry Electromagnetic radiation and health Laser safety Lasers and aviation safety Medical radiography Mobile phone radiation and health Radiation protection Radiation therapy Radioactivity in the life sciences Radioactive contamination Radiobiology Biological dose units and quantities Wireless electronic devices and health Radiation heat-transfer Related articles Half-life Nuclear physics Radioactive source Radiation hardening List of civilian radiation accidents 1996 Costa Rica accident 1987 Goiânia accident 1984 Moroccan accident 1990 Zaragoza accident See also: the categories Radiation effects, Radioactivity, Radiobiology, and Radiation protection Authority control GND: 4258821-2 MA: 3941253, 113514576 Retrieved from "https://en.wikipedia.org/w/index.php?title=Half-life&oldid=1014132428" Categories: Chemical kinetics Exponentials Radioactivity Hidden categories: CS1 maint: unfit URL Articles with short description Short description is different from Wikidata Articles to be expanded from July 2019 Commons category link is on Wikidata Wikipedia articles with GND identifiers Wikipedia articles with MA identifiers Wikipedia articles with multiple identifiers Navigation menu Personal tools Not logged in Talk Contributions Create account Log in Namespaces Article Talk Variants Views Read Edit View history More Search Navigation Main page Contents Current events Random article About Wikipedia Contact us Donate Contribute Help Learn to edit Community portal Recent changes Upload file Tools What links here Related changes Upload file Special pages Permanent link Page information Cite this page Wikidata item Print/export Download as PDF Printable version In other projects Wikimedia Commons Languages Afrikaans العربية Aragonés Asturianu বাংলা Bân-lâm-gú Беларуская भोजपुरी Български Bosanski Català Чӑвашла Čeština Cymraeg Dansk الدارجة Deutsch Eesti Ελληνικά Español Esperanto Euskara فارسی Français Gaeilge Galego 贛語 한국어 हिन्दी Hrvatski Bahasa Indonesia Íslenska Italiano עברית ಕನ್ನಡ ქართული Қазақша Kiswahili Kreyòl ayisyen Latviešu Lietuvių Limburgs Magyar Македонски മലയാളം Bahasa Melayu Nederlands 日本語 Nordfriisk Norsk bokmål Norsk nynorsk Occitan Oʻzbekcha/ўзбекча پنجابی Plattdüütsch Polski Português Română Runa Simi Русский Simple English Slovenčina Slovenščina کوردی Српски / srpski Srpskohrvatski / српскохрватски Suomi Svenska தமிழ் Татарча/tatarça తెలుగు ไทย Türkçe Українська اردو Tiếng Việt 吴语 粵語 中文 Edit links This page was last edited on 25 March 2021, at 10:46 (UTC). 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