Old page wikitext, before the edit (old_wikitext) | '{{Short description|Time for exponential decay to remove half of a quantity}}
{{About|the scientific and mathematical concept}}
{| class="wikitable" align=right
! Number of<br/>half-lives<br/>elapsed !! Fraction<br/>remaining !! colspan=2| Percentage<br/>remaining
|-
| 0 || {{frac|1|1}} ||align=right style="border-right-width: 0; padding-right:0"| 100||style="border-left-width: 0"|
|-
| 1 || {{1/2}} ||align=right style="border-right-width: 0; padding-right:0"| 50||style="border-left-width: 0"|
|-
| 2 || {{1/4}} ||align=right style="border-right-width: 0; padding-right:0"| 25||style="border-left-width: 0"|
|-
| 3 || {{frac|1|8}} ||align=right style="padding-right:0; border-right-width: 0"| 12||style="border-left-width: 0; padding-left:0"|.5
|-
| 4 || {{frac|1|16}} ||align=right style="border-right-width: 0; padding-right:0"| 6||style="border-left-width: 0; padding-left:0"|.25
|-
| 5 || {{frac|1|32}} || align=right style="border-right-width: 0; padding-right:0"|3||style="border-left-width: 0; padding-left:0"|.125
|-
| 6 || {{frac|1|64}} || align=right style="border-right-width: 0; padding-right:0"|1||style="border-left-width: 0; padding-left:0"|.5625
|-
| 7 || {{frac|1|128}} ||align=right style="border-right-width: 0; padding-right:0"| 0||style="border-left-width: 0; padding-left:0"|.78125
|-
|-
| {{mvar|n}} ||{{frac|1|2<sup>{{mvar|n}}</sup>}} || colspan=2|{{frac|100|2<sup>{{mvar|n}}</sup>}}
|}
{{e (mathematical constant)}}
'''Half-life''' (symbol {{math|'''''t''{{sub|½}}'''}}) is the time required for a quantity (of substance) to reduce to half of its initial value. The term is commonly used in [[nuclear physics]] to describe how quickly unstable [[atom]]s undergo [[radioactive decay]] or how long stable atoms survive. The term is also used more generally to characterize any type of [[exponential decay|exponential]] (or, rarely, [[rate law|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 (in exponential growth) 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.<ref>John Ayto, ''20th Century Words'' (1989), Cambridge University Press.</ref> Rutherford applied the principle of a radioactive [[chemical element|element's]] half-life in 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.
==Probabilistic nature==
[[File:Halflife-sim.gif|thumb|right|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 often 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 ''[[expected value|on average]]''". In other words, the ''probability'' of a radioactive atom decaying within its half-life is 50%.<ref name=PTFP>{{cite book|title=Physics and Technology for Future Presidents|url=https://archive.org/details/physicstechnolog00mull|url-access=limited|author=Muller, Richard A.|author-link=Richard A. Muller|publisher=[[Princeton University Press]]|date=April 12, 2010|pages=[https://archive.org/details/physicstechnolog00mull/page/n138 128]–129|isbn=9780691135045}}</ref>
For example, the accompanying image 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]].<ref>{{cite web |url=http://www.madsci.org/posts/archives/Mar2003/1047912974.Ph.r.html |title=Re: What happens during half-lifes [sic] when there is only one atom left?|publisher=MADSCI.org|author=Chivers, Sidney |date=March 16, 2003}}</ref><ref>{{cite web |url=https://www.exploratorium.edu/snacks/radioactive-decay-model |title=Radioactive-Decay Model|publisher=Exploratorium.edu |access-date=2012-04-25}}</ref><ref>{{cite web |url=http://astro.gmu.edu/classes/c80196/hw2.html |title=Assignment #2: Data, Simulations, and Analytic Science in Decay |publisher=Astro.GLU.edu |date=September 1996 |author=Wallin, John |url-status=unfit |archive-url=https://web.archive.org/web/20110929005007/http://astro.gmu.edu/classes/c80196/hw2.html |archive-date=2011-09-29}}</ref>
==Formulas for half-life in exponential decay==
{{Main|Exponential decay}}
An exponential decay can be described by any of the following four equivalent formulas:<ref name=ln(2)/>{{rp|109–112}}<big><math display="block">\begin{align}
N(t) &= N_0 \left(\frac {1}{2}\right)^{\frac{t}{t_{1/2}}} \\
N(t) &= N_0 2^{-\frac{t}{t_{1/2}}} \\
N(t) &= N_0 e^{-\frac{t}{\tau}} \\
N(t) &= N_0 e^{-\lambda t}
\end{align}</math></big>
where
*{{math|''N''{{sub|0}}}} is the initial quantity of the substance that will decay (this quantity may be measured in grams, [[Mole (unit)|mole]]s, number of atoms, etc.),
*{{math|''N''(''t'')}} is the quantity that still remains and has not yet decayed after a time {{mvar|t}},
*{{math|''t''{{sub|½}}}} is the half-life of the decaying quantity,
*{{mvar|τ}} is a [[positive number]] called the [[mean lifetime]] of the decaying quantity,
*{{mvar|λ}} is a positive number called the [[decay constant]] of the decaying quantity.
The three parameters {{math|''t''{{sub|½}}}}, {{mvar|τ}}, and {{mvar|λ}} are directly related in the following way:<math display="block">t_{1/2} = \frac{\ln (2)}{\lambda} = \tau \ln(2)</math>where {{math|ln(2)}} is the [[natural logarithm of 2]] (approximately 0.693).<ref name="ln(2)">{{cite book|title=Nuclear- and Radiochemistry: Introduction|last=Rösch|first=Frank|publisher=[[Walter de Gruyter]]|date=September 12, 2014|volume=1|isbn=978-3-11-022191-6}}</ref>{{rp|112}}
=== Half-life and reaction orders ===
In [[chemical kinetics]], the value of the half-life depends on the [[Rate equation|reaction order]]:
====Zero order kinetics====
The rate of this kind of reaction does not depend on the substrate [[concentration]], {{math|[A]}}. Thus the concentration decreases linearly.
:<math display="block" chem="">d[\ce A]/dt = - k</math>The integrated [[rate law]] of zero order kinetics is:
<math display="block" chem="">[\ce A] = [\ce A]_0 - kt</math>In order to find the half-life, we have to replace the concentration value for the initial concentration divided by 2: <math display="block" chem="">[\ce A]_{0}/2 = [\ce A]_0 - kt_{1/2}</math>and isolate the time:<math display="block" chem="">t_{1/2} = \frac{[\ce A]_0}{2k}</math>This {{math|''t''{{sub|½}}}} formula indicates that the half-life for a zero order reaction depends on the initial concentration and the rate constant.
====First order kinetics====
In first order reactions, the rate of reaction will be proportional to the concentration of the reactant. Thus the concentration will decrease exponentially.
<math display="block" chem="">[\ce A] = [\ce A]_0 \exp(-kt)</math>as time progresses until it reaches zero, and the half-life will be constant, independent of concentration.
The time {{math|''t''{{sub|½}}}} for {{math|[A]}} to decrease from {{math|[A]{{sub|0}}}} to {{math|{{sfrac|1|2}}[A]{{sub|0}}}} in a first-order reaction is given by the following equation:<math display="block" chem="">[\ce A]_0 /2 = [\ce A]_0 \exp(-kt_{1/2})</math>It can be solved for<math display="block" chem="">kt_{1/2} = -\ln \left(\frac{[\ce A]_0 /2}{[\ce A]_0}\right) = -\ln\frac{1}{2} = \ln 2</math>For a first-order reaction, the half-life of a reactant is independent of its initial concentration. Therefore, if the concentration of {{math|A}} at some arbitrary stage of the reaction is {{math|[A]}}, then it will have fallen to {{math|{{sfrac|1|2}}[A]}} after a further interval of {{tmath|\tfrac{\ln 2}{k}.}} Hence, the half-life of a first order reaction is given as the following:</p><math display="block">t_{1/2} = \frac{\ln 2}{k}</math>The half-life of a first order reaction is independent of its initial concentration and depends solely on the reaction rate constant, {{mvar|k}}.
====Second order kinetics====
In second order reactions, the rate of reaction is proportional to the square of the concentration. By integrating this rate, it can be shown that the concentration {{math|[A]}} of the reactant decreases following this formula:
<math display="block" chem>\frac{1}{[\ce A]} = kt + \frac{1}{[\ce A]_0}</math>We replace {{math|[A]}} for {{math|{{sfrac|1|2}}{{math|[A]}}{{sub|0}}}} in order to calculate the half-life of the reactant {{math|A}} <math display="block" chem="">\frac{1}{[\ce A]_0 /2} = kt_{1/2} + \frac{1}{[\ce A]_0}</math>and isolate the time of the half-life ({{math|''t''{{sub|½}}}}):<math display="block" chem="">t_{1/2} = \frac{1}{[\ce A]_0 k}</math>This shows that the half-life of second order reactions depends on the initial concentration and [[Reaction rate constant|rate constant]].
===Decay by two or more processes===
Some quantities decay by two exponential-decay processes simultaneously. In this case, the actual half-life {{math|''T''{{sub|½}}}} can be related to the half-lives {{math|''t''{{sub|1}}}} and {{math|''t''{{sub|2}}}} that the quantity would have if each of the decay processes acted in isolation:
<math display="block">\frac{1}{T_{1/2}} = \frac{1}{t_1} + \frac{1}{t_2}</math>
For three or more processes, the analogous formula is:
<math display="block">\frac{1}{T_{1/2}} = \frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} + \cdots</math>
For a proof of these formulas, see [[exponential decay#Decay by two or more processes|Exponential decay § Decay by two or more processes]].
===Examples===
[[File:Dice half-life decay.jpg|thumb|Half-life demonstrated using dice in a [[v:Physics and Astronomy Labs/Radioactive decay with dice|classroom experiment]]]]
{{Further|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 [[atomic nucleus|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 {{math|ln(2)''RC''}} or {{math|ln(2)''L''/''R''}}, respectively. For this example the term [[half time (physics)|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 {{math|ln(2)/''λ''}}, where {{mvar|λ}} (also denoted as {{mvar|k}}) is the [[reaction rate constant]].
==In non-exponential decay==
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.<ref>{{cite book|title=Chemistry for the Biosciences: The Essential Concepts |author1=Jonathan Crowe |author2=Tony Bradshaw |page=568 |url=https://books.google.com/books?id=VxMNBAAAQBAJ&pg=PA568|isbn=9780199662883 |year=2014|publisher=OUP Oxford }}</ref>
==In biology and pharmacology==
{{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 [[tissue (biology)|tissues]], active [[metabolite]]s, and [[receptor (biochemistry)|receptor]] interactions.<ref name="SCM">{{cite book|title=Spinal cord medicine|author1=Lin VW|author2=Cardenas DD|publisher=Demos Medical Publishing, LLC|page=251|url=https://books.google.com/books?id=3anl3G4No_oC&pg=PA251|year=2003|isbn=978-1-888799-61-3}}</ref>
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,<ref>{{cite book|last1=Pang|first1=Xiao-Feng|title=Water: Molecular Structure and Properties|date=2014|publisher=World Scientific|location=New Jersey|isbn=9789814440424|page=451}}</ref> 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 [[pesticide]]s in [[plant]]s,<ref name=tebuau>{{cite web|last1=Australian Pesticides and Veterinary Medicines Authority|title=Tebufenozide in the product Mimic 700 WP Insecticide, Mimic 240 SC Insecticide|url=https://apvma.gov.au/node/14051|publisher=Australian Government|access-date=30 April 2018|language=en|date=31 March 2015}}</ref> and certain authors maintain that [[environmental impact of pesticides|pesticide risk and impact assessment models]] rely on and are sensitive to information describing dissipation from plants.<ref name=acs>{{cite journal|last1=Fantke|first1=Peter|last2=Gillespie|first2=Brenda W.|last3=Juraske|first3=Ronnie|last4=Jolliet|first4=Olivier|title=Estimating Half-Lives for Pesticide Dissipation from Plants|journal=Environmental Science & Technology|date=11 July 2014|volume=48|issue=15|pages=8588–8602|doi=10.1021/es500434p|pmid=24968074|bibcode=2014EnST...48.8588F|doi-access=free|hdl=20.500.11850/91972|hdl-access=free}}</ref>
In [[Basic reproduction number|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 [[Exponential decay#Applications and examples|exponentially]].<ref name = "Balkew">{{cite thesis |last=Balkew |first=Teshome Mogessie |date=December 2010 |title=The SIR Model When S(t) is a Multi-Exponential Function |publisher=East Tennessee State University |url=https://dc.etsu.edu/etd/1747 }}</ref><ref name = "Ireland">{{cite book |editor-first=MW|editor-last=Ireland |date=1928 |title=The Medical Department of the United States Army in the World War, vol. IX: Communicable and Other Diseases |location=Washington: U.S. |publisher=U.S. Government Printing Office |pages=116–7}}</ref>
==See also==
*[[Half time (physics)]]
*[[List of radioactive nuclides by half-life]]
*[[Mean lifetime]]
*[[Median lethal dose]]
==References==
{{Reflist}}
==External links==
{{Wiktionary|half-life}}
{{Commons category|Half times}}
*https://www.calculator.net/half-life-calculator.html Comprehensive half-life calculator
*[https://web.archive.org/web/20160306234718/http://www.nucleonica.net/wiki/index.php?title=Help%3ADecay_Engine%2B%2B wiki: Decay Engine], Nucleonica.net (archived 2016)
*[https://web.archive.org/web/20060617205436/http://www.facstaff.bucknell.edu/mastascu/elessonshtml/SysDyn/SysDyn3TCBasic.htm System Dynamics – Time Constants], Bucknell.edu
*[https://www.nikhef.nl/en/news/researchers-nikhef-and-uva-measure-slowest-radioactive-decay-ever/ Researchers Nikhef and UvA measure slowest radioactive decay ever: Xe-124 with 18 billion trillion years]
*https://academo.org/demos/radioactive-decay-simulator/ Interactive radioactive decay simulator demonstrating how half-life is related to the rate of decay
{{Radiation}}
{{Authority control}}
{{DEFAULTSORT:Half-Life}}
[[Category:Chemical kinetics]]
[[Category:Radioactivity]]
[[Category:Nuclear fission]]
[[Category:Temporal exponentials]]' |
New page wikitext, after the edit (new_wikitext) | '{{Short description|Time for exponential decay to remove half of a quantity}}
{{About|the scientific and mathematical concept}}
{| class="wikitable" align=right
! Number of<br/>half-lives<br/>elapsed !! Fraction<br/>remaining !! colspan=2| Percentage<br/>remaining
|-
| 0 || {{frac|1|1}} ||align=right style="border-right-width: 0; padding-right:0"| 100||style="border-left-width: 0"|
|-
| 1 || {{1/2}} ||align=right style="border-right-width: 0; padding-right:0"| 50||style="border-left-width: 0"|
|-
| 2 || {{1/4}} ||align=right style="border-right-width: 0; padding-right:0"| 25||style="border-left-width: 0"|
|-
| 3 || {{frac|1|8}} ||align=right style="padding-right:0; border-right-width: 0"| 12||style="border-left-width: 0; padding-left:0"|.5
|-
| 4 || {{frac|1|16}} ||align=right style="border-right-width: 0; padding-right:0"| 6||style="border-left-width: 0; padding-left:0"|.25
|-
| 5 || {{frac|1|32}} || align=right style="border-right-width: 0; padding-right:0"|3||style="border-left-width: 0; padding-left:0"|.125
|-
| 6 || {{frac|1|64}} || align=right style="border-right-width: 0; padding-right:0"|1||style="border-left-width: 0; padding-left:0"|.5625
|-
| 7 || {{frac|1|128}} ||align=right style="border-right-width: 0; padding-right:0"| 0||style="border-left-width: 0; padding-left:0"|.78125
|-
|-
| {{mvar|n}} ||{{frac|1|2<sup>{{mvar|n}}</sup>}} || colspan=2|{{frac|100|2<sup>{{mvar|n}}</sup>}}
|}
{{e (mathematical constant)}}
'''Half-life''' (symbol {{math|'''''t''{{sub|½}}'''}}) is the time required for a quantity (of substance) to reduce to half of its initial value. The term is commonly used in [[nuclear physics]] to describe how quickly unstable [[atom]]s undergo [[radioactive decay]] or how long stable atoms survive. The term is also used more generally to characterize any type of [[exponential decay|exponential]] (or, rarely, [[rate law|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 (in exponential growth) 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.<ref>John Ayto, ''20th Century Words'' (1989), Cambridge University Press. HALF LIFE IS WAY BETTER THAN MAYK IT. MAYK IT? MORE LIKE NAKED, HEHE</ref> Rutherford applied the principle of a radioactive [[chemical element|element's]] half-life in 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.
==Probabilistic nature==
[[File:Halflife-sim.gif|thumb|right|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 often 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 ''[[expected value|on average]]''". In other words, the ''probability'' of a radioactive atom decaying within its half-life is 50%.<ref name=PTFP>{{cite book|title=Physics and Technology for Future Presidents|url=https://archive.org/details/physicstechnolog00mull|url-access=limited|author=Muller, Richard A.|author-link=Richard A. Muller|publisher=[[Princeton University Press]]|date=April 12, 2010|pages=[https://archive.org/details/physicstechnolog00mull/page/n138 128]–129|isbn=9780691135045}}</ref>
For example, the accompanying image 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]].<ref>{{cite web |url=http://www.madsci.org/posts/archives/Mar2003/1047912974.Ph.r.html |title=Re: What happens during half-lifes [sic] when there is only one atom left?|publisher=MADSCI.org|author=Chivers, Sidney |date=March 16, 2003}}</ref><ref>{{cite web |url=https://www.exploratorium.edu/snacks/radioactive-decay-model |title=Radioactive-Decay Model|publisher=Exploratorium.edu |access-date=2012-04-25}}</ref><ref>{{cite web |url=http://astro.gmu.edu/classes/c80196/hw2.html |title=Assignment #2: Data, Simulations, and Analytic Science in Decay |publisher=Astro.GLU.edu |date=September 1996 |author=Wallin, John |url-status=unfit |archive-url=https://web.archive.org/web/20110929005007/http://astro.gmu.edu/classes/c80196/hw2.html |archive-date=2011-09-29}}</ref>
==Formulas for half-life in exponential decay==
{{Main|Exponential decay}}
An exponential decay can be described by any of the following four equivalent formulas:<ref name=ln(2)/>{{rp|109–112}}<big><math display="block">\begin{align}
N(t) &= N_0 \left(\frac {1}{2}\right)^{\frac{t}{t_{1/2}}} \\
N(t) &= N_0 2^{-\frac{t}{t_{1/2}}} \\
N(t) &= N_0 e^{-\frac{t}{\tau}} \\
N(t) &= N_0 e^{-\lambda t}
\end{align}</math></big>
where
*{{math|''N''{{sub|0}}}} is the initial quantity of the substance that will decay (this quantity may be measured in grams, [[Mole (unit)|mole]]s, number of atoms, etc.),
*{{math|''N''(''t'')}} is the quantity that still remains and has not yet decayed after a time {{mvar|t}},
*{{math|''t''{{sub|½}}}} is the half-life of the decaying quantity,
*{{mvar|τ}} is a [[positive number]] called the [[mean lifetime]] of the decaying quantity,
*{{mvar|λ}} is a positive number called the [[decay constant]] of the decaying quantity.
The three parameters {{math|''t''{{sub|½}}}}, {{mvar|τ}}, and {{mvar|λ}} are directly related in the following way:<math display="block">t_{1/2} = \frac{\ln (2)}{\lambda} = \tau \ln(2)</math>where {{math|ln(2)}} is the [[natural logarithm of 2]] (approximately 0.693).<ref name="ln(2)">{{cite book|title=Nuclear- and Radiochemistry: Introduction|last=Rösch|first=Frank|publisher=[[Walter de Gruyter]]|date=September 12, 2014|volume=1|isbn=978-3-11-022191-6}}</ref>{{rp|112}}
=== Half-life and reaction orders ===
In [[chemical kinetics]], the value of the half-life depends on the [[Rate equation|reaction order]]:
====Zero order kinetics====
The rate of this kind of reaction does not depend on the substrate [[concentration]], {{math|[A]}}. Thus the concentration decreases linearly.
:<math display="block" chem="">d[\ce A]/dt = - k</math>The integrated [[rate law]] of zero order kinetics is:
<math display="block" chem="">[\ce A] = [\ce A]_0 - kt</math>In order to find the half-life, we have to replace the concentration value for the initial concentration divided by 2: <math display="block" chem="">[\ce A]_{0}/2 = [\ce A]_0 - kt_{1/2}</math>and isolate the time:<math display="block" chem="">t_{1/2} = \frac{[\ce A]_0}{2k}</math>This {{math|''t''{{sub|½}}}} formula indicates that the half-life for a zero order reaction depends on the initial concentration and the rate constant.
====First order kinetics====
In first order reactions, the rate of reaction will be proportional to the concentration of the reactant. Thus the concentration will decrease exponentially.
<math display="block" chem="">[\ce A] = [\ce A]_0 \exp(-kt)</math>as time progresses until it reaches zero, and the half-life will be constant, independent of concentration.
The time {{math|''t''{{sub|½}}}} for {{math|[A]}} to decrease from {{math|[A]{{sub|0}}}} to {{math|{{sfrac|1|2}}[A]{{sub|0}}}} in a first-order reaction is given by the following equation:<math display="block" chem="">[\ce A]_0 /2 = [\ce A]_0 \exp(-kt_{1/2})</math>It can be solved for<math display="block" chem="">kt_{1/2} = -\ln \left(\frac{[\ce A]_0 /2}{[\ce A]_0}\right) = -\ln\frac{1}{2} = \ln 2</math>For a first-order reaction, the half-life of a reactant is independent of its initial concentration. Therefore, if the concentration of {{math|A}} at some arbitrary stage of the reaction is {{math|[A]}}, then it will have fallen to {{math|{{sfrac|1|2}}[A]}} after a further interval of {{tmath|\tfrac{\ln 2}{k}.}} Hence, the half-life of a first order reaction is given as the following:</p><math display="block">t_{1/2} = \frac{\ln 2}{k}</math>The half-life of a first order reaction is independent of its initial concentration and depends solely on the reaction rate constant, {{mvar|k}}.
====Second order kinetics====
In second order reactions, the rate of reaction is proportional to the square of the concentration. By integrating this rate, it can be shown that the concentration {{math|[A]}} of the reactant decreases following this formula:
<math display="block" chem>\frac{1}{[\ce A]} = kt + \frac{1}{[\ce A]_0}</math>We replace {{math|[A]}} for {{math|{{sfrac|1|2}}{{math|[A]}}{{sub|0}}}} in order to calculate the half-life of the reactant {{math|A}} <math display="block" chem="">\frac{1}{[\ce A]_0 /2} = kt_{1/2} + \frac{1}{[\ce A]_0}</math>and isolate the time of the half-life ({{math|''t''{{sub|½}}}}):<math display="block" chem="">t_{1/2} = \frac{1}{[\ce A]_0 k}</math>This shows that the half-life of second order reactions depends on the initial concentration and [[Reaction rate constant|rate constant]].
===Decay by two or more processes===
Some quantities decay by two exponential-decay processes simultaneously. In this case, the actual half-life {{math|''T''{{sub|½}}}} can be related to the half-lives {{math|''t''{{sub|1}}}} and {{math|''t''{{sub|2}}}} that the quantity would have if each of the decay processes acted in isolation:
<math display="block">\frac{1}{T_{1/2}} = \frac{1}{t_1} + \frac{1}{t_2}</math>
For three or more processes, the analogous formula is:
<math display="block">\frac{1}{T_{1/2}} = \frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} + \cdots</math>
For a proof of these formulas, see [[exponential decay#Decay by two or more processes|Exponential decay § Decay by two or more processes]].
===Examples===
[[File:Dice half-life decay.jpg|thumb|Half-life demonstrated using dice in a [[v:Physics and Astronomy Labs/Radioactive decay with dice|classroom experiment]]]]
{{Further|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 [[atomic nucleus|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 {{math|ln(2)''RC''}} or {{math|ln(2)''L''/''R''}}, respectively. For this example the term [[half time (physics)|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 {{math|ln(2)/''λ''}}, where {{mvar|λ}} (also denoted as {{mvar|k}}) is the [[reaction rate constant]].
==In non-exponential decay==
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.<ref>{{cite book|title=Chemistry for the Biosciences: The Essential Concepts |author1=Jonathan Crowe |author2=Tony Bradshaw |page=568 |url=https://books.google.com/books?id=VxMNBAAAQBAJ&pg=PA568|isbn=9780199662883 |year=2014|publisher=OUP Oxford }}</ref>
==In biology and pharmacology==
{{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 [[tissue (biology)|tissues]], active [[metabolite]]s, and [[receptor (biochemistry)|receptor]] interactions.<ref name="SCM">{{cite book|title=Spinal cord medicine|author1=Lin VW|author2=Cardenas DD|publisher=Demos Medical Publishing, LLC|page=251|url=https://books.google.com/books?id=3anl3G4No_oC&pg=PA251|year=2003|isbn=978-1-888799-61-3}}</ref>
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,<ref>{{cite book|last1=Pang|first1=Xiao-Feng|title=Water: Molecular Structure and Properties|date=2014|publisher=World Scientific|location=New Jersey|isbn=9789814440424|page=451}}</ref> 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 [[pesticide]]s in [[plant]]s,<ref name=tebuau>{{cite web|last1=Australian Pesticides and Veterinary Medicines Authority|title=Tebufenozide in the product Mimic 700 WP Insecticide, Mimic 240 SC Insecticide|url=https://apvma.gov.au/node/14051|publisher=Australian Government|access-date=30 April 2018|language=en|date=31 March 2015}}</ref> and certain authors maintain that [[environmental impact of pesticides|pesticide risk and impact assessment models]] rely on and are sensitive to information describing dissipation from plants.<ref name=acs>{{cite journal|last1=Fantke|first1=Peter|last2=Gillespie|first2=Brenda W.|last3=Juraske|first3=Ronnie|last4=Jolliet|first4=Olivier|title=Estimating Half-Lives for Pesticide Dissipation from Plants|journal=Environmental Science & Technology|date=11 July 2014|volume=48|issue=15|pages=8588–8602|doi=10.1021/es500434p|pmid=24968074|bibcode=2014EnST...48.8588F|doi-access=free|hdl=20.500.11850/91972|hdl-access=free}}</ref>
In [[Basic reproduction number|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 [[Exponential decay#Applications and examples|exponentially]].<ref name = "Balkew">{{cite thesis |last=Balkew |first=Teshome Mogessie |date=December 2010 |title=The SIR Model When S(t) is a Multi-Exponential Function |publisher=East Tennessee State University |url=https://dc.etsu.edu/etd/1747 }}</ref><ref name = "Ireland">{{cite book |editor-first=MW|editor-last=Ireland |date=1928 |title=The Medical Department of the United States Army in the World War, vol. IX: Communicable and Other Diseases |location=Washington: U.S. |publisher=U.S. Government Printing Office |pages=116–7}}</ref>
==See also==
*[[Half time (physics)]]
*[[List of radioactive nuclides by half-life]]
*[[Mean lifetime]]
*[[Median lethal dose]]
==References==
{{Reflist}}
==External links==
{{Wiktionary|half-life}}
{{Commons category|Half times}}
*https://www.calculator.net/half-life-calculator.html Comprehensive half-life calculator
*[https://web.archive.org/web/20160306234718/http://www.nucleonica.net/wiki/index.php?title=Help%3ADecay_Engine%2B%2B wiki: Decay Engine], Nucleonica.net (archived 2016)
*[https://web.archive.org/web/20060617205436/http://www.facstaff.bucknell.edu/mastascu/elessonshtml/SysDyn/SysDyn3TCBasic.htm System Dynamics – Time Constants], Bucknell.edu
*[https://www.nikhef.nl/en/news/researchers-nikhef-and-uva-measure-slowest-radioactive-decay-ever/ Researchers Nikhef and UvA measure slowest radioactive decay ever: Xe-124 with 18 billion trillion years]
*https://academo.org/demos/radioactive-decay-simulator/ Interactive radioactive decay simulator demonstrating how half-life is related to the rate of decay
{{Radiation}}
{{Authority control}}
{{DEFAULTSORT:Half-Life}}
[[Category:Chemical kinetics]]
[[Category:Radioactivity]]
[[Category:Nuclear fission]]
[[Category:Temporal exponentials]]' |
Parsed HTML source of the new revision (new_html) | '<div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Time for exponential decay to remove half of a quantity</div>
<style data-mw-deduplicate="TemplateStyles:r1033289096">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the scientific and mathematical concept. For other uses, see <a href="/wiki/Half-life_(disambiguation)" class="mw-disambig" title="Half-life (disambiguation)">Half-life (disambiguation)</a>.</div>
<table class="wikitable" align="right">
<tbody><tr>
<th>Number of<br />half-lives<br />elapsed</th>
<th>Fraction<br />remaining</th>
<th colspan="2">Percentage<br />remaining
</th></tr>
<tr>
<td>0</td>
<td><style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac"><span class="num">1</span>⁄<span class="den">1</span></span></td>
<td align="right" style="border-right-width: 0; padding-right:0">100</td>
<td style="border-left-width: 0">
</td></tr>
<tr>
<td>1</td>
<td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span></td>
<td align="right" style="border-right-width: 0; padding-right:0">50</td>
<td style="border-left-width: 0">
</td></tr>
<tr>
<td>2</td>
<td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">4</span></span></td>
<td align="right" style="border-right-width: 0; padding-right:0">25</td>
<td style="border-left-width: 0">
</td></tr>
<tr>
<td>3</td>
<td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">8</span></span></td>
<td align="right" style="padding-right:0; border-right-width: 0">12</td>
<td style="border-left-width: 0; padding-left:0">.5
</td></tr>
<tr>
<td>4</td>
<td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">16</span></span></td>
<td align="right" style="border-right-width: 0; padding-right:0">6</td>
<td style="border-left-width: 0; padding-left:0">.25
</td></tr>
<tr>
<td>5</td>
<td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">32</span></span></td>
<td align="right" style="border-right-width: 0; padding-right:0">3</td>
<td style="border-left-width: 0; padding-left:0">.125
</td></tr>
<tr>
<td>6</td>
<td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">64</span></span></td>
<td align="right" style="border-right-width: 0; padding-right:0">1</td>
<td style="border-left-width: 0; padding-left:0">.5625
</td></tr>
<tr>
<td>7</td>
<td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">128</span></span></td>
<td align="right" style="border-right-width: 0; padding-right:0">0</td>
<td style="border-left-width: 0; padding-left:0">.78125
</td></tr>
<tr>
<td><span class="texhtml mvar" style="font-style:italic;">n</span></td>
<td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2<sup><span class="texhtml mvar" style="font-style:italic;">n</span></sup></span></span></td>
<td colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">100</span>⁄<span class="den">2<sup><span class="texhtml mvar" style="font-style:italic;">n</span></sup></span></span>
</td></tr></tbody></table>
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style="font-size:130%;">mathematical constant <span class="texhtml mvar" style="font-style:italic;"><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e</a></span></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:Euler%27s_formula.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/180px-Euler%27s_formula.svg.png" decoding="async" width="180" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/270px-Euler%27s_formula.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/360px-Euler%27s_formula.svg.png 2x" data-file-width="700" data-file-height="700" /></a></span></td></tr><tr><th class="sidebar-heading" style="border-top:#aaa 1px solid;">
Properties</th></tr><tr><td class="sidebar-content">
<ul><li><a href="/wiki/Natural_logarithm" title="Natural logarithm">Natural logarithm</a></li>
<li><a href="/wiki/Exponential_function" title="Exponential function">Exponential function</a></li></ul></td>
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Applications</th></tr><tr><td class="sidebar-content">
<ul><li><a href="/wiki/Compound_interest" title="Compound interest">compound interest</a></li>
<li><a href="/wiki/Euler%27s_identity" title="Euler's identity">Euler's identity</a></li>
<li><a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a></li>
<li><a class="mw-selflink selflink">half-lives</a>
<ul><li>exponential <a href="/wiki/Exponential_growth" title="Exponential growth">growth</a> and <a href="/wiki/Exponential_decay" title="Exponential decay">decay</a></li></ul></li></ul></td>
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Defining <span class="texhtml mvar" style="font-style:italic;">e</span></th></tr><tr><td class="sidebar-content">
<ul><li><a href="/wiki/Proof_that_e_is_irrational" title="Proof that e is irrational">proof that <span class="texhtml mvar" style="font-style:italic;">e</span> is irrational</a></li>
<li><a href="/wiki/List_of_representations_of_e" title="List of representations of e">representations of <span class="texhtml mvar" style="font-style:italic;">e</span></a></li>
<li><a href="/wiki/Lindemann%E2%80%93Weierstrass_theorem" title="Lindemann–Weierstrass theorem">Lindemann–Weierstrass theorem</a></li></ul></td>
</tr><tr><th class="sidebar-heading" style="border-top:#aaa 1px solid;">
People</th></tr><tr><td class="sidebar-content">
<ul><li><a href="/wiki/John_Napier" title="John Napier">John Napier</a></li>
<li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a><br /></li></ul></td>
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Related topics</th></tr><tr><td class="sidebar-content">
<ul><li><a href="/wiki/Schanuel%27s_conjecture" title="Schanuel's conjecture">Schanuel's conjecture</a></li></ul></td>
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<p><b>Half-life</b> (symbol <span class="texhtml"><b><i>t</i><sub>½</sub></b></span>) is the time required for a quantity (of substance) to reduce to half of its initial value. The term is commonly used in <a href="/wiki/Nuclear_physics" title="Nuclear physics">nuclear physics</a> to describe how quickly unstable <a href="/wiki/Atom" title="Atom">atoms</a> undergo <a href="/wiki/Radioactive_decay" title="Radioactive decay">radioactive decay</a> or how long stable atoms survive. The term is also used more generally to characterize any type of <a href="/wiki/Exponential_decay" title="Exponential decay">exponential</a> (or, rarely, <a href="/wiki/Rate_law" class="mw-redirect" title="Rate law">non-exponential</a>) decay. For example, the medical sciences refer to the <a href="/wiki/Biological_half-life" title="Biological half-life">biological half-life</a> of drugs and other chemicals in the human body. The converse of half-life (in exponential growth) is <a href="/wiki/Doubling_time" title="Doubling time">doubling time</a>.
</p><p>The original term, <i>half-life period</i>, dating to <a href="/wiki/Ernest_Rutherford" title="Ernest Rutherford">Ernest Rutherford</a>'s discovery of the principle in 1907, was shortened to <i>half-life</i> in the early 1950s.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup> Rutherford applied the principle of a radioactive <a href="/wiki/Chemical_element" title="Chemical element">element's</a> half-life in studies of age determination of rocks by measuring the decay period of <a href="/wiki/Radium" title="Radium">radium</a> to <a href="/wiki/Lead-206" class="mw-redirect" title="Lead-206">lead-206</a>.
</p><p>Half-life is constant over the lifetime of an exponentially decaying quantity, and it is a <a href="/wiki/Characteristic_unit" class="mw-redirect" title="Characteristic unit">characteristic unit</a> for the exponential decay equation. The accompanying table shows the reduction of a quantity as a function of the number of half-lives elapsed.
</p>
<div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Probabilistic_nature"><span class="tocnumber">1</span> <span class="toctext">Probabilistic nature</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#Formulas_for_half-life_in_exponential_decay"><span class="tocnumber">2</span> <span class="toctext">Formulas for half-life in exponential decay</span></a>
<ul>
<li class="toclevel-2 tocsection-3"><a href="#Half-life_and_reaction_orders"><span class="tocnumber">2.1</span> <span class="toctext">Half-life and reaction orders</span></a>
<ul>
<li class="toclevel-3 tocsection-4"><a href="#Zero_order_kinetics"><span class="tocnumber">2.1.1</span> <span class="toctext">Zero order kinetics</span></a></li>
<li class="toclevel-3 tocsection-5"><a href="#First_order_kinetics"><span class="tocnumber">2.1.2</span> <span class="toctext">First order kinetics</span></a></li>
<li class="toclevel-3 tocsection-6"><a href="#Second_order_kinetics"><span class="tocnumber">2.1.3</span> <span class="toctext">Second order kinetics</span></a></li>
</ul>
</li>
<li class="toclevel-2 tocsection-7"><a href="#Decay_by_two_or_more_processes"><span class="tocnumber">2.2</span> <span class="toctext">Decay by two or more processes</span></a></li>
<li class="toclevel-2 tocsection-8"><a href="#Examples"><span class="tocnumber">2.3</span> <span class="toctext">Examples</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-9"><a href="#In_non-exponential_decay"><span class="tocnumber">3</span> <span class="toctext">In non-exponential decay</span></a></li>
<li class="toclevel-1 tocsection-10"><a href="#In_biology_and_pharmacology"><span class="tocnumber">4</span> <span class="toctext">In biology and pharmacology</span></a></li>
<li class="toclevel-1 tocsection-11"><a href="#See_also"><span class="tocnumber">5</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-12"><a href="#References"><span class="tocnumber">6</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-13"><a href="#External_links"><span class="tocnumber">7</span> <span class="toctext">External links</span></a></li>
</ul>
</div>
<h2><span class="mw-headline" id="Probabilistic_nature">Probabilistic nature</span><span class="mw-editsection">
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<figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Halflife-sim.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/3/3f/Halflife-sim.gif" decoding="async" width="100" height="188" class="mw-file-element" data-file-width="100" data-file-height="188" /></a><figcaption>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 <a href="/wiki/Law_of_large_numbers" title="Law of large numbers">law of large numbers</a>: with more atoms, the overall decay is more regular and more predictable.</figcaption></figure>
<p>A half-life often 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 <i>not</i> be "half of an atom" left after one second.
</p><p>Instead, the half-life is defined in terms of <a href="/wiki/Probability" title="Probability">probability</a>: "Half-life is the time required for exactly half of the entities to decay <i><a href="/wiki/Expected_value" title="Expected value">on average</a></i>". In other words, the <i>probability</i> of a radioactive atom decaying within its half-life is 50%.<sup id="cite_ref-PTFP_2-0" class="reference"><a href="#cite_note-PTFP-2">[2]</a></sup>
</p><p>For example, the accompanying image is a simulation of many identical atoms undergoing radioactive decay. Note that after one half-life there are not <i>exactly</i> one-half of the atoms remaining, only <i>approximately</i>, because of the random variation in the process. Nevertheless, when there are many identical atoms decaying (right boxes), the <a href="/wiki/Law_of_large_numbers" title="Law of large numbers">law of large numbers</a> suggests that it is a <i>very good approximation</i> to say that half of the atoms remain after one half-life.
</p><p>Various simple exercises can demonstrate probabilistic decay, for example involving flipping coins or running a statistical <a href="/wiki/Computer_program" title="Computer program">computer program</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3">[3]</a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4">[4]</a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5">[5]</a></sup>
</p>
<h2><span class="mw-headline" id="Formulas_for_half-life_in_exponential_decay">Formulas for half-life in exponential decay</span><span class="mw-editsection">
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<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1033289096"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Exponential_decay" title="Exponential decay">Exponential decay</a></div>
<p>An exponential decay can be described by any of the following four equivalent formulas:<sup id="cite_ref-ln(2)_6-0" class="reference"><a href="#cite_note-ln(2)-6">[6]</a></sup><sup class="reference nowrap"><span title="Page / location: 109–112">: 109–112 </span></sup><big><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}N(t)&=N_{0}\left({\frac {1}{2}}\right)^{\frac {t}{t_{1/2}}}\\N(t)&=N_{0}2^{-{\frac {t}{t_{1/2}}}}\\N(t)&=N_{0}e^{-{\frac {t}{\tau }}}\\N(t)&=N_{0}e^{-\lambda t}\end{aligned}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}N(t)&=N_{0}\left({\frac {1}{2}}\right)^{\frac {t}{t_{1/2}}}\\N(t)&=N_{0}2^{-{\frac {t}{t_{1/2}}}}\\N(t)&=N_{0}e^{-{\frac {t}{\tau }}}\\N(t)&=N_{0}e^{-\lambda t}\end{aligned}}}</annotation>
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</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61e3eb6b36ff3ce4f7d19ca1bc791d65fa40bd4c" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -9.171ex; width:20.588ex; height:19.509ex;" alt="{\displaystyle {\begin{aligned}N(t)&=N_{0}\left({\frac {1}{2}}\right)^{\frac {t}{t_{1/2}}}\\N(t)&=N_{0}2^{-{\frac {t}{t_{1/2}}}}\\N(t)&=N_{0}e^{-{\frac {t}{\tau }}}\\N(t)&=N_{0}e^{-\lambda t}\end{aligned}}}"></div></big>
where
</p>
<ul><li><span class="texhtml"><i>N</i><sub>0</sub></span> is the initial quantity of the substance that will decay (this quantity may be measured in grams, <a href="/wiki/Mole_(unit)" title="Mole (unit)">moles</a>, number of atoms, etc.),</li>
<li><span class="texhtml"><i>N</i>(<i>t</i>)</span> is the quantity that still remains and has not yet decayed after a time <span class="texhtml mvar" style="font-style:italic;">t</span>,</li>
<li><span class="texhtml"><i>t</i><sub>½</sub></span> is the half-life of the decaying quantity,</li>
<li><span class="texhtml mvar" style="font-style:italic;">τ</span> is a <a href="/wiki/Positive_number" class="mw-redirect" title="Positive number">positive number</a> called the <a href="/wiki/Mean_lifetime" class="mw-redirect" title="Mean lifetime">mean lifetime</a> of the decaying quantity,</li>
<li><span class="texhtml mvar" style="font-style:italic;">λ</span> is a positive number called the <a href="/wiki/Decay_constant" class="mw-redirect" title="Decay constant">decay constant</a> of the decaying quantity.</li></ul>
<p>The three parameters <span class="texhtml"><i>t</i><sub>½</sub></span>, <span class="texhtml mvar" style="font-style:italic;">τ</span>, and <span class="texhtml mvar" style="font-style:italic;">λ</span> are directly related in the following way:<div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1/2}={\frac {\ln(2)}{\lambda }}=\tau \ln(2)}">
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<annotation encoding="application/x-tex">{\displaystyle t_{1/2}={\frac {\ln(2)}{\lambda }}=\tau \ln(2)}</annotation>
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</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff22414b8bce156bc7c4569acc248dd19cf98886" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.982ex; height:5.843ex;" alt="{\displaystyle t_{1/2}={\frac {\ln(2)}{\lambda }}=\tau \ln(2)}"></div>where <span class="texhtml">ln(2)</span> is the <a href="/wiki/Natural_logarithm_of_2" title="Natural logarithm of 2">natural logarithm of 2</a> (approximately 0.693).<sup id="cite_ref-ln(2)_6-1" class="reference"><a href="#cite_note-ln(2)-6">[6]</a></sup><sup class="reference nowrap"><span title="Page / location: 112">: 112 </span></sup>
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<h3><span class="mw-headline" id="Half-life_and_reaction_orders">Half-life and reaction orders</span><span class="mw-editsection">
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<p>In <a href="/wiki/Chemical_kinetics" title="Chemical kinetics">chemical kinetics</a>, the value of the half-life depends on the <a href="/wiki/Rate_equation" title="Rate equation">reaction order</a>:
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<h4><span class="mw-headline" id="Zero_order_kinetics">Zero order kinetics</span><span class="mw-editsection">
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<p>The rate of this kind of reaction does not depend on the substrate <a href="/wiki/Concentration" title="Concentration">concentration</a>, <span class="texhtml">[A]</span>. Thus the concentration decreases linearly.
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<dl><dd><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d[{\ce {A}}]/dt=-k}">
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</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0d11a976526a07f717c25cd724e2544a4ee5649" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.588ex; height:2.843ex;" alt="{\displaystyle d[{\ce {A}}]/dt=-k}"></div>The integrated <a href="/wiki/Rate_law" class="mw-redirect" title="Rate law">rate law</a> of zero order kinetics is:</dd></dl>
<p><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\ce {A}}]=[{\ce {A}}]_{0}-kt}">
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</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a8d0e16353886b89240f30b9f54cdc7a4149786" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.118ex; height:2.843ex;" alt="{\displaystyle [{\ce {A}}]=[{\ce {A}}]_{0}-kt}"></div>In order to find the half-life, we have to replace the concentration value for the initial concentration divided by 2: <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\ce {A}}]_{0}/2=[{\ce {A}}]_{0}-kt_{1/2}}">
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<annotation encoding="application/x-tex">{\displaystyle [{\ce {A}}]_{0}/2=[{\ce {A}}]_{0}-kt_{1/2}}</annotation>
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</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/826087aba9042219c8eef066b73bc9c2fefcb904" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.195ex; height:3.176ex;" alt="{\displaystyle [{\ce {A}}]_{0}/2=[{\ce {A}}]_{0}-kt_{1/2}}"></div>and isolate the time:<div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1/2}={\frac {[{\ce {A}}]_{0}}{2k}}}">
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<annotation encoding="application/x-tex">{\displaystyle t_{1/2}={\frac {[{\ce {A}}]_{0}}{2k}}}</annotation>
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</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4eca8d1ce264cb32692257294eca4fcc00c0866" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.563ex; height:5.843ex;" alt="{\displaystyle t_{1/2}={\frac {[{\ce {A}}]_{0}}{2k}}}"></div>This <span class="texhtml"><i>t</i><sub>½</sub></span> formula indicates that the half-life for a zero order reaction depends on the initial concentration and the rate constant.
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<h4><span class="mw-headline" id="First_order_kinetics">First order kinetics</span><span class="mw-editsection">
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<p>In first order reactions, the rate of reaction will be proportional to the concentration of the reactant. Thus the concentration will decrease exponentially.
<div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\ce {A}}]=[{\ce {A}}]_{0}\exp(-kt)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>A</mtext>
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<mo stretchy="false">]</mo>
<mo>=</mo>
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>A</mtext>
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<mo stretchy="false">]</mo>
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<mn>0</mn>
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<mi>exp</mi>
<mo>⁡<!-- --></mo>
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<annotation encoding="application/x-tex">{\displaystyle [{\ce {A}}]=[{\ce {A}}]_{0}\exp(-kt)}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e66d0ac1acaa8904f02364bd2a42163fd0c45757" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.834ex; height:2.843ex;" alt="{\displaystyle [{\ce {A}}]=[{\ce {A}}]_{0}\exp(-kt)}"></div>as time progresses until it reaches zero, and the half-life will be constant, independent of concentration.
</p><p>
The time <span class="texhtml"><i>t</i><sub>½</sub></span> for <span class="texhtml">[A]</span> to decrease from <span class="texhtml">[A]<sub>0</sub></span> to <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac"><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span></span>[A]<sub>0</sub></span> in a first-order reaction is given by the following equation:<div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\ce {A}}]_{0}/2=[{\ce {A}}]_{0}\exp(-kt_{1/2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>A</mtext>
</mrow>
<msub>
<mo stretchy="false">]</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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<mo>=</mo>
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<mo stretchy="false">]</mo>
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<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle [{\ce {A}}]_{0}/2=[{\ce {A}}]_{0}\exp(-kt_{1/2})}</annotation>
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</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ca0db5cf3bfaac58a98000354a6a9d3bd72f5eb" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.911ex; height:3.176ex;" alt="{\displaystyle [{\ce {A}}]_{0}/2=[{\ce {A}}]_{0}\exp(-kt_{1/2})}"></div>It can be solved for<div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle kt_{1/2}=-\ln \left({\frac {[{\ce {A}}]_{0}/2}{[{\ce {A}}]_{0}}}\right)=-\ln {\frac {1}{2}}=\ln 2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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<mn>2</mn>
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<mo>=</mo>
<mo>−<!-- − --></mo>
<mi>ln</mi>
<mo>⁡<!-- --></mo>
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<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
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<mo>)</mo>
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<mo>=</mo>
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<annotation encoding="application/x-tex">{\displaystyle kt_{1/2}=-\ln \left({\frac {[{\ce {A}}]_{0}/2}{[{\ce {A}}]_{0}}}\right)=-\ln {\frac {1}{2}}=\ln 2}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/082b3f6dc924c53343aacf4e0a58112958cd6824" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.861ex; height:6.509ex;" alt="{\displaystyle kt_{1/2}=-\ln \left({\frac {[{\ce {A}}]_{0}/2}{[{\ce {A}}]_{0}}}\right)=-\ln {\frac {1}{2}}=\ln 2}"></div>For a first-order reaction, the half-life of a reactant is independent of its initial concentration. Therefore, if the concentration of <span class="texhtml">A</span> at some arbitrary stage of the reaction is <span class="texhtml">[A]</span>, then it will have fallen to <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac"><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span></span>[A]</span> after a further interval of <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\ln 2}{k}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mrow>
<mi>ln</mi>
<mo>⁡<!-- --></mo>
<mn>2</mn>
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<mi>k</mi>
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<mo>.</mo>
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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\ln 2}{k}}.}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e497ed96efd6944a2786ad0eb504c8532a2f999a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.063ex; height:3.843ex;" alt="{\displaystyle {\tfrac {\ln 2}{k}}.}"></span></span> Hence, the half-life of a first order reaction is given as the following:</p><p class="mw-empty-elt"></p><p><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1/2}={\frac {\ln 2}{k}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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<mo>/</mo>
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<mo>=</mo>
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<mfrac>
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<mi>ln</mi>
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<mi>k</mi>
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<annotation encoding="application/x-tex">{\displaystyle t_{1/2}={\frac {\ln 2}{k}}}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d0454df2c3b7a529a7192d99baea14bcb59ebf0" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.961ex; height:5.509ex;" alt="{\displaystyle t_{1/2}={\frac {\ln 2}{k}}}"></div>The half-life of a first order reaction is independent of its initial concentration and depends solely on the reaction rate constant, <span class="texhtml mvar" style="font-style:italic;">k</span>.
</p><h4><span class="mw-headline" id="Second_order_kinetics">Second order kinetics</span><span class="mw-editsection">
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<p>In second order reactions, the rate of reaction is proportional to the square of the concentration. By integrating this rate, it can be shown that the concentration <span class="texhtml">[A]</span> of the reactant decreases following this formula:
</p><p><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{[{\ce {A}}]}}=kt+{\frac {1}{[{\ce {A}}]_{0}}}}">
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<mrow class="MJX-TeXAtom-ORD">
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<mo>=</mo>
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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{[{\ce {A}}]}}=kt+{\frac {1}{[{\ce {A}}]_{0}}}}</annotation>
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</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee6d34dbb3bff34a51b42cc06ab317aace3dfa08" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.79ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{[{\ce {A}}]}}=kt+{\frac {1}{[{\ce {A}}]_{0}}}}"></div>We replace <span class="texhtml">[A]</span> for <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac"><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span></span><span class="texhtml">[A]</span><sub>0</sub></span> in order to calculate the half-life of the reactant <span class="texhtml">A</span> <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{[{\ce {A}}]_{0}/2}}=kt_{1/2}+{\frac {1}{[{\ce {A}}]_{0}}}}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{[{\ce {A}}]_{0}/2}}=kt_{1/2}+{\frac {1}{[{\ce {A}}]_{0}}}}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff43700aeca7595315c4fd1c4cf3798288d887b0" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.867ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{[{\ce {A}}]_{0}/2}}=kt_{1/2}+{\frac {1}{[{\ce {A}}]_{0}}}}"></div>and isolate the time of the half-life (<span class="texhtml"><i>t</i><sub>½</sub></span>):<div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1/2}={\frac {1}{[{\ce {A}}]_{0}k}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
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</msub>
<mo>=</mo>
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<mfrac>
<mn>1</mn>
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<mo stretchy="false">[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>A</mtext>
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<msub>
<mo stretchy="false">]</mo>
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<mn>0</mn>
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<mi>k</mi>
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<annotation encoding="application/x-tex">{\displaystyle t_{1/2}={\frac {1}{[{\ce {A}}]_{0}k}}}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/842dfc3b12854a4d7a822b58337efed28b8cc82a" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.775ex; height:6.009ex;" alt="{\displaystyle t_{1/2}={\frac {1}{[{\ce {A}}]_{0}k}}}"></div>This shows that the half-life of second order reactions depends on the initial concentration and <a href="/wiki/Reaction_rate_constant" title="Reaction rate constant">rate constant</a>.
</p>
<h3><span class="mw-headline" id="Decay_by_two_or_more_processes">Decay by two or more processes</span><span class="mw-editsection">
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<p>Some quantities decay by two exponential-decay processes simultaneously. In this case, the actual half-life <span class="texhtml"><i>T</i><sub>½</sub></span> can be related to the half-lives <span class="texhtml"><i>t</i><sub>1</sub></span> and <span class="texhtml"><i>t</i><sub>2</sub></span> that the quantity would have if each of the decay processes acted in isolation:
<div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{T_{1/2}}}={\frac {1}{t_{1}}}+{\frac {1}{t_{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msub>
<mi>T</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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<mn>2</mn>
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</msub>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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</msub>
</mfrac>
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<mo>+</mo>
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<mfrac>
<mn>1</mn>
<msub>
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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{T_{1/2}}}={\frac {1}{t_{1}}}+{\frac {1}{t_{2}}}}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/904bbab9844b592234b5c958b47b82455a7f0aaa" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.291ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{T_{1/2}}}={\frac {1}{t_{1}}}+{\frac {1}{t_{2}}}}"></div>
</p><p>For three or more processes, the analogous formula is:
<div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{T_{1/2}}}={\frac {1}{t_{1}}}+{\frac {1}{t_{2}}}+{\frac {1}{t_{3}}}+\cdots }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msub>
<mi>T</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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<mo>/</mo>
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<mn>2</mn>
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<mn>1</mn>
<msub>
<mi>t</mi>
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<mn>1</mn>
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</mfrac>
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<mo>+</mo>
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<mn>1</mn>
<msub>
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<mn>1</mn>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{T_{1/2}}}={\frac {1}{t_{1}}}+{\frac {1}{t_{2}}}+{\frac {1}{t_{3}}}+\cdots }</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62323ab5558cdb11454def8cc83e0eef880b0baf" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.425ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{T_{1/2}}}={\frac {1}{t_{1}}}+{\frac {1}{t_{2}}}+{\frac {1}{t_{3}}}+\cdots }"></div>
For a proof of these formulas, see <a href="/wiki/Exponential_decay#Decay_by_two_or_more_processes" title="Exponential decay">Exponential decay § Decay by two or more processes</a>.
</p>
<h3><span class="mw-headline" id="Examples">Examples</span><span class="mw-editsection">
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<figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Dice_half-life_decay.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Dice_half-life_decay.jpg/220px-Dice_half-life_decay.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Dice_half-life_decay.jpg/330px-Dice_half-life_decay.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Dice_half-life_decay.jpg/440px-Dice_half-life_decay.jpg 2x" data-file-width="3264" data-file-height="2448" /></a><figcaption>Half-life demonstrated using dice in a <a href="https://en.wikiversity.org/wiki/Physics_and_Astronomy_Labs/Radioactive_decay_with_dice" class="extiw" title="v:Physics and Astronomy Labs/Radioactive decay with dice">classroom experiment</a></figcaption></figure>
<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1033289096"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Exponential_decay#Applications_and_examples" title="Exponential decay">Exponential decay § Applications and examples</a></div>
<p>There is a half-life describing any exponential-decay process. For example:
</p>
<ul><li>As noted above, in <a href="/wiki/Radioactive_decay" title="Radioactive decay">radioactive decay</a> the half-life is the length of time after which there is a 50% chance that an atom will have undergone <a href="/wiki/Atomic_nucleus" title="Atomic nucleus">nuclear</a> decay. It varies depending on the atom type and <a href="/wiki/Isotope" title="Isotope">isotope</a>, and is usually determined experimentally. See <a href="/wiki/List_of_nuclides" title="List of nuclides">List of nuclides</a>.</li>
<li>The current flowing through an <a href="/wiki/RC_circuit" title="RC circuit">RC circuit</a> or <a href="/wiki/RL_circuit" title="RL circuit">RL circuit</a> decays with a half-life of <span class="texhtml">ln(2)<i>RC</i></span> or <span class="texhtml">ln(2)<i>L</i>/<i>R</i></span>, respectively. For this example the term <a href="/wiki/Half_time_(physics)" title="Half time (physics)">half time</a> tends to be used rather than "half-life", but they mean the same thing.</li>
<li>In a <a href="/wiki/Chemical_reaction" title="Chemical reaction">chemical reaction</a>, 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 <span class="texhtml">ln(2)/<i>λ</i></span>, where <span class="texhtml mvar" style="font-style:italic;">λ</span> (also denoted as <span class="texhtml mvar" style="font-style:italic;">k</span>) is the <a href="/wiki/Reaction_rate_constant" title="Reaction rate constant">reaction rate constant</a>.</li></ul>
<h2><span class="mw-headline" id="In_non-exponential_decay">In non-exponential decay</span><span class="mw-editsection">
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<p>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 <a href="/wiki/Biological_half-life" title="Biological half-life">biological half-life</a> 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.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7">[7]</a></sup>
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<h2><span class="mw-headline" id="In_biology_and_pharmacology">In biology and pharmacology</span><span class="mw-editsection">
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<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1033289096"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Biological_half-life" title="Biological half-life">Biological half-life</a></div>
<p>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 <a href="/wiki/Blood_plasma" title="Blood plasma">blood plasma</a> to reach one-half of its steady-state value (the "plasma half-life").
</p><p>The relationship between the biological and plasma half-lives of a substance can be complex, due to factors including accumulation in <a href="/wiki/Tissue_(biology)" title="Tissue (biology)">tissues</a>, active <a href="/wiki/Metabolite" title="Metabolite">metabolites</a>, and <a href="/wiki/Receptor_(biochemistry)" title="Receptor (biochemistry)">receptor</a> interactions.<sup id="cite_ref-SCM_8-0" class="reference"><a href="#cite_note-SCM-8">[8]</a></sup>
</p><p>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.
</p><p>For example, the biological half-life of water in a human being is about 9 to 10 days,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9">[9]</a></sup> though this can be altered by behavior and other conditions. The biological half-life of <a href="/wiki/Caesium" title="Caesium">caesium</a> in human beings is between one and four months.
</p><p>The concept of a half-life has also been utilized for <a href="/wiki/Pesticide" title="Pesticide">pesticides</a> in <a href="/wiki/Plant" title="Plant">plants</a>,<sup id="cite_ref-tebuau_10-0" class="reference"><a href="#cite_note-tebuau-10">[10]</a></sup> and certain authors maintain that <a href="/wiki/Environmental_impact_of_pesticides" title="Environmental impact of pesticides">pesticide risk and impact assessment models</a> rely on and are sensitive to information describing dissipation from plants.<sup id="cite_ref-acs_11-0" class="reference"><a href="#cite_note-acs-11">[11]</a></sup>
</p><p>In <a href="/wiki/Basic_reproduction_number" title="Basic reproduction number">epidemiology</a>, 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 <a href="/wiki/Exponential_decay#Applications_and_examples" title="Exponential decay">exponentially</a>.<sup id="cite_ref-Balkew_12-0" class="reference"><a href="#cite_note-Balkew-12">[12]</a></sup><sup id="cite_ref-Ireland_13-0" class="reference"><a href="#cite_note-Ireland-13">[13]</a></sup>
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<h2><span class="mw-headline" id="See_also">See also</span><span class="mw-editsection">
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<ul><li><a href="/wiki/Half_time_(physics)" title="Half time (physics)">Half time (physics)</a></li>
<li><a href="/wiki/List_of_radioactive_nuclides_by_half-life" title="List of radioactive nuclides by half-life">List of radioactive nuclides by half-life</a></li>
<li><a href="/wiki/Mean_lifetime" class="mw-redirect" title="Mean lifetime">Mean lifetime</a></li>
<li><a href="/wiki/Median_lethal_dose" title="Median lethal dose">Median lethal dose</a></li></ul>
<h2><span class="mw-headline" id="References">References</span><span class="mw-editsection">
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<style data-mw-deduplicate="TemplateStyles:r1217336898">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist">
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">John Ayto, <i>20th Century Words</i> (1989), Cambridge University Press. HALF LIFE IS WAY BETTER THAN MAYK IT. MAYK IT? MORE LIKE NAKED, HEHE</span>
</li>
<li id="cite_note-PTFP-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-PTFP_2-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1215172403">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a{background-size:contain}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a{background-size:contain}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a{background-size:contain}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#2C882D;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911F}html.skin-theme-clientpref-night .mw-parser-output .cs1-visible-error,html.skin-theme-clientpref-night .mw-parser-output .cs1-hidden-error{color:#f8a397}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-visible-error,html.skin-theme-clientpref-os .mw-parser-output .cs1-hidden-error{color:#f8a397}html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911F}}</style><cite id="CITEREFMuller,_Richard_A.2010" class="citation book cs1"><a href="/wiki/Richard_A._Muller" title="Richard A. Muller">Muller, Richard A.</a> (April 12, 2010). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/physicstechnolog00mull"><i>Physics and Technology for Future Presidents</i></a></span>. <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/physicstechnolog00mull/page/n138">128</a>–129. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780691135045" title="Special:BookSources/9780691135045"><bdi>9780691135045</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+and+Technology+for+Future+Presidents&rft.pages=128-129&rft.pub=Princeton+University+Press&rft.date=2010-04-12&rft.isbn=9780691135045&rft.au=Muller%2C+Richard+A.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fphysicstechnolog00mull&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHalf-life" class="Z3988"></span></span>
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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFChivers,_Sidney2003" class="citation web cs1">Chivers, Sidney (March 16, 2003). <a rel="nofollow" class="external text" href="http://www.madsci.org/posts/archives/Mar2003/1047912974.Ph.r.html">"Re: What happens during half-lifes [sic] when there is only one atom left?"</a>. MADSCI.org.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Re%3A+What+happens+during+half-lifes+%26%2391%3Bsic%26%2393%3B+when+there+is+only+one+atom+left%3F&rft.pub=MADSCI.org&rft.date=2003-03-16&rft.au=Chivers%2C+Sidney&rft_id=http%3A%2F%2Fwww.madsci.org%2Fposts%2Farchives%2FMar2003%2F1047912974.Ph.r.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHalf-life" class="Z3988"></span></span>
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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.exploratorium.edu/snacks/radioactive-decay-model">"Radioactive-Decay Model"</a>. Exploratorium.edu<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-04-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Radioactive-Decay+Model&rft.pub=Exploratorium.edu&rft_id=https%3A%2F%2Fwww.exploratorium.edu%2Fsnacks%2Fradioactive-decay-model&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHalf-life" class="Z3988"></span></span>
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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFWallin,_John1996" class="citation web cs1">Wallin, John (September 1996). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110929005007/http://astro.gmu.edu/classes/c80196/hw2.html">"Assignment #2: Data, Simulations, and Analytic Science in Decay"</a>. Astro.GLU.edu. Archived from the original on 2011-09-29.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Assignment+%232%3A+Data%2C+Simulations%2C+and+Analytic+Science+in+Decay&rft.pub=Astro.GLU.edu&rft.date=1996-09&rft.au=Wallin%2C+John&rft_id=http%3A%2F%2Fastro.gmu.edu%2Fclasses%2Fc80196%2Fhw2.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHalf-life" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_web" title="Template:Cite web">cite web</a>}}</code>: CS1 maint: unfit URL (<a href="/wiki/Category:CS1_maint:_unfit_URL" title="Category:CS1 maint: unfit URL">link</a>)</span></span>
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<li id="cite_note-ln(2)-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-ln(2)_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ln(2)_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFRösch2014" class="citation book cs1">Rösch, Frank (September 12, 2014). <i>Nuclear- and Radiochemistry: Introduction</i>. Vol. 1. <a href="/wiki/Walter_de_Gruyter" class="mw-redirect" title="Walter de Gruyter">Walter de Gruyter</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-11-022191-6" title="Special:BookSources/978-3-11-022191-6"><bdi>978-3-11-022191-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Nuclear-+and+Radiochemistry%3A+Introduction&rft.pub=Walter+de+Gruyter&rft.date=2014-09-12&rft.isbn=978-3-11-022191-6&rft.aulast=R%C3%B6sch&rft.aufirst=Frank&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHalf-life" class="Z3988"></span></span>
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<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFJonathan_CroweTony_Bradshaw2014" class="citation book cs1">Jonathan Crowe; Tony Bradshaw (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VxMNBAAAQBAJ&pg=PA568"><i>Chemistry for the Biosciences: The Essential Concepts</i></a>. OUP Oxford. p. 568. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780199662883" title="Special:BookSources/9780199662883"><bdi>9780199662883</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chemistry+for+the+Biosciences%3A+The+Essential+Concepts&rft.pages=568&rft.pub=OUP+Oxford&rft.date=2014&rft.isbn=9780199662883&rft.au=Jonathan+Crowe&rft.au=Tony+Bradshaw&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVxMNBAAAQBAJ%26pg%3DPA568&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHalf-life" class="Z3988"></span></span>
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<li id="cite_note-SCM-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-SCM_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFLin_VWCardenas_DD2003" class="citation book cs1">Lin VW; Cardenas DD (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3anl3G4No_oC&pg=PA251"><i>Spinal cord medicine</i></a>. Demos Medical Publishing, LLC. p. 251. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-888799-61-3" title="Special:BookSources/978-1-888799-61-3"><bdi>978-1-888799-61-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spinal+cord+medicine&rft.pages=251&rft.pub=Demos+Medical+Publishing%2C+LLC&rft.date=2003&rft.isbn=978-1-888799-61-3&rft.au=Lin+VW&rft.au=Cardenas+DD&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D3anl3G4No_oC%26pg%3DPA251&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHalf-life" class="Z3988"></span></span>
</li>
<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFPang2014" class="citation book cs1">Pang, Xiao-Feng (2014). <i>Water: Molecular Structure and Properties</i>. New Jersey: World Scientific. p. 451. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789814440424" title="Special:BookSources/9789814440424"><bdi>9789814440424</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Water%3A+Molecular+Structure+and+Properties&rft.place=New+Jersey&rft.pages=451&rft.pub=World+Scientific&rft.date=2014&rft.isbn=9789814440424&rft.aulast=Pang&rft.aufirst=Xiao-Feng&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHalf-life" class="Z3988"></span></span>
</li>
<li id="cite_note-tebuau-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-tebuau_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFAustralian_Pesticides_and_Veterinary_Medicines_Authority2015" class="citation web cs1">Australian Pesticides and Veterinary Medicines Authority (31 March 2015). <a rel="nofollow" class="external text" href="https://apvma.gov.au/node/14051">"Tebufenozide in the product Mimic 700 WP Insecticide, Mimic 240 SC Insecticide"</a>. Australian Government<span class="reference-accessdate">. Retrieved <span class="nowrap">30 April</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Tebufenozide+in+the+product+Mimic+700+WP+Insecticide%2C+Mimic+240+SC+Insecticide&rft.pub=Australian+Government&rft.date=2015-03-31&rft.au=Australian+Pesticides+and+Veterinary+Medicines+Authority&rft_id=https%3A%2F%2Fapvma.gov.au%2Fnode%2F14051&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHalf-life" class="Z3988"></span></span>
</li>
<li id="cite_note-acs-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-acs_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFFantkeGillespieJuraskeJolliet2014" class="citation journal cs1">Fantke, Peter; Gillespie, Brenda W.; Juraske, Ronnie; Jolliet, Olivier (11 July 2014). <a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fes500434p">"Estimating Half-Lives for Pesticide Dissipation from Plants"</a>. <i>Environmental Science & Technology</i>. <b>48</b> (15): 8588–8602. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014EnST...48.8588F">2014EnST...48.8588F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fes500434p">10.1021/es500434p</a></span>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/20.500.11850%2F91972">20.500.11850/91972</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/24968074">24968074</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Environmental+Science+%26+Technology&rft.atitle=Estimating+Half-Lives+for+Pesticide+Dissipation+from+Plants&rft.volume=48&rft.issue=15&rft.pages=8588-8602&rft.date=2014-07-11&rft_id=info%3Ahdl%2F20.500.11850%2F91972&rft_id=info%3Apmid%2F24968074&rft_id=info%3Adoi%2F10.1021%2Fes500434p&rft_id=info%3Abibcode%2F2014EnST...48.8588F&rft.aulast=Fantke&rft.aufirst=Peter&rft.au=Gillespie%2C+Brenda+W.&rft.au=Juraske%2C+Ronnie&rft.au=Jolliet%2C+Olivier&rft_id=https%3A%2F%2Fdoi.org%2F10.1021%252Fes500434p&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHalf-life" class="Z3988"></span></span>
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<li id="cite_note-Balkew-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-Balkew_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFBalkew2010" class="citation thesis cs1">Balkew, Teshome Mogessie (December 2010). <a rel="nofollow" class="external text" href="https://dc.etsu.edu/etd/1747"><i>The SIR Model When S(t) is a Multi-Exponential Function</i></a> (Thesis). East Tennessee State University.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=The+SIR+Model+When+S%28t%29+is+a+Multi-Exponential+Function&rft.inst=East+Tennessee+State+University&rft.date=2010-12&rft.aulast=Balkew&rft.aufirst=Teshome+Mogessie&rft_id=https%3A%2F%2Fdc.etsu.edu%2Fetd%2F1747&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHalf-life" class="Z3988"></span></span>
</li>
<li id="cite_note-Ireland-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ireland_13-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1215172403"><cite id="CITEREFIreland1928" class="citation book cs1">Ireland, MW, ed. (1928). <i>The Medical Department of the United States Army in the World War, vol. IX: Communicable and Other Diseases</i>. Washington: U.S.: U.S. Government Printing Office. pp. 116–7.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Medical+Department+of+the+United+States+Army+in+the+World+War%2C+vol.+IX%3A+Communicable+and+Other+Diseases&rft.place=Washington%3A+U.S.&rft.pages=116-7&rft.pub=U.S.+Government+Printing+Office&rft.date=1928&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHalf-life" class="Z3988"></span></span>
</li>
</ol></div></div>
<h2><span class="mw-headline" id="External_links">External links</span><span class="mw-editsection">
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<div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/half-life" class="extiw" title="wiktionary:half-life">half-life</a></b></i> in Wiktionary, the free dictionary.</div></div>
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<div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Half_times" class="extiw" title="commons:Category:Half times">Half times</a></span>.</div></div>
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<ul><li><a rel="nofollow" class="external free" href="https://www.calculator.net/half-life-calculator.html">https://www.calculator.net/half-life-calculator.html</a> Comprehensive half-life calculator</li>
<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160306234718/http://www.nucleonica.net/wiki/index.php?title=Help%3ADecay_Engine%2B%2B">wiki: Decay Engine</a>, Nucleonica.net (archived 2016)</li>
<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060617205436/http://www.facstaff.bucknell.edu/mastascu/elessonshtml/SysDyn/SysDyn3TCBasic.htm">System Dynamics – Time Constants</a>, Bucknell.edu</li>
<li><a rel="nofollow" class="external text" href="https://www.nikhef.nl/en/news/researchers-nikhef-and-uva-measure-slowest-radioactive-decay-ever/">Researchers Nikhef and UvA measure slowest radioactive decay ever: Xe-124 with 18 billion trillion years</a></li>
<li><a rel="nofollow" class="external free" href="https://academo.org/demos/radioactive-decay-simulator/">https://academo.org/demos/radioactive-decay-simulator/</a> Interactive radioactive decay simulator demonstrating how half-life is related to the rate of decay</li></ul>
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<ul><li><a href="/wiki/Acoustic_radiation_force" title="Acoustic radiation force">Acoustic radiation force</a></li>
<li><a href="/wiki/Infrared" title="Infrared">Infrared</a></li>
<li><a href="/wiki/Light" title="Light">Light</a></li>
<li><a href="/wiki/Starlight" title="Starlight">Starlight</a></li>
<li><a href="/wiki/Sunlight" title="Sunlight">Sunlight</a></li>
<li><a href="/wiki/Microwave" title="Microwave">Microwave</a></li>
<li><a href="/wiki/Radio_wave" title="Radio wave">Radio waves</a></li>
<li><a href="/wiki/Ultraviolet" title="Ultraviolet">Ultraviolet</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Ionizing_radiation" title="Ionizing radiation">Ionizing radiation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/wiki/Radioactive_decay" title="Radioactive decay">Radioactive decay</a></li>
<li><a href="/wiki/Cluster_decay" title="Cluster decay">Cluster decay</a></li>
<li><a href="/wiki/Background_radiation" title="Background radiation">Background radiation</a></li>
<li><a href="/wiki/Alpha_particle" title="Alpha particle">Alpha particle</a></li>
<li><a href="/wiki/Beta_particle" title="Beta particle">Beta particle</a></li>
<li><a href="/wiki/Gamma_ray" title="Gamma ray">Gamma ray</a></li>
<li><a href="/wiki/Cosmic_ray" title="Cosmic ray">Cosmic ray</a></li>
<li><a href="/wiki/Neutron_radiation" title="Neutron radiation">Neutron radiation</a></li>
<li><a href="/wiki/Nuclear_fission" title="Nuclear fission">Nuclear fission</a></li>
<li><a href="/wiki/Nuclear_fusion" title="Nuclear fusion">Nuclear fusion</a></li>
<li><a href="/wiki/Nuclear_reactor" title="Nuclear reactor">Nuclear reactors</a></li>
<li><a href="/wiki/Nuclear_weapon" title="Nuclear weapon">Nuclear weapons</a></li>
<li><a href="/wiki/Particle_accelerator" title="Particle accelerator">Particle accelerators</a></li>
<li><a href="/wiki/Radionuclide" title="Radionuclide">Radioactive materials</a></li>
<li><a href="/wiki/X-ray" title="X-ray">X-ray</a></li></ul>
</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/wiki/Earth%27s_energy_budget" title="Earth's energy budget">Earth's energy budget</a></li>
<li><a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">Electromagnetic radiation</a></li>
<li><a href="/wiki/Synchrotron_radiation" title="Synchrotron radiation">Synchrotron radiation</a></li>
<li><a href="/wiki/Thermal_radiation" title="Thermal radiation">Thermal radiation</a></li>
<li><a href="/wiki/Black-body_radiation" title="Black-body radiation">Black-body radiation</a></li>
<li><a href="/wiki/Particle_radiation" title="Particle radiation">Particle radiation</a></li>
<li><a href="/wiki/Gravitational_radiation" class="mw-redirect" title="Gravitational radiation">Gravitational radiation</a></li>
<li><a href="/wiki/Cosmic_background_radiation" title="Cosmic background radiation">Cosmic background radiation</a></li>
<li><a href="/wiki/Cherenkov_radiation" title="Cherenkov radiation">Cherenkov radiation</a></li>
<li><a href="/wiki/Askaryan_radiation" title="Askaryan radiation">Askaryan radiation</a></li>
<li><a href="/wiki/Bremsstrahlung" title="Bremsstrahlung">Bremsstrahlung</a></li>
<li><a href="/wiki/Unruh_radiation" class="mw-redirect" title="Unruh radiation">Unruh radiation</a></li>
<li><a href="/wiki/Dark_radiation" title="Dark radiation">Dark radiation</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Radiation <br />and health</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li>Radiation syndrome
<ul><li><a href="/wiki/Acute_radiation_syndrome" title="Acute radiation syndrome">acute</a></li>
<li><a href="/wiki/Chronic_radiation_syndrome" title="Chronic radiation syndrome">chronic</a></li></ul></li>
<li><a href="/wiki/Health_physics" title="Health physics">Health physics</a></li>
<li><a href="/wiki/Dosimetry" title="Dosimetry">Dosimetry</a></li>
<li><a href="/wiki/Electromagnetic_radiation_and_health" title="Electromagnetic radiation and health">Electromagnetic radiation and health</a></li>
<li><a href="/wiki/Laser_safety" title="Laser safety">Laser safety</a></li>
<li><a href="/wiki/Lasers_and_aviation_safety" title="Lasers and aviation safety">Lasers and aviation safety</a></li>
<li><a href="/wiki/Medical_radiography" class="mw-redirect" title="Medical radiography">Medical radiography</a></li>
<li><a href="/wiki/Radiation_protection" title="Radiation protection">Radiation protection</a></li>
<li><a href="/wiki/Radiation_therapy" title="Radiation therapy">Radiation therapy</a></li>
<li><a href="/wiki/Radiation_damage" title="Radiation damage">Radiation damage</a></li>
<li><a href="/wiki/Radioactivity_in_the_life_sciences" title="Radioactivity in the life sciences">Radioactivity in the life sciences</a></li>
<li><a href="/wiki/Radioactive_contamination" title="Radioactive contamination">Radioactive contamination</a></li>
<li><a href="/wiki/Radiobiology" title="Radiobiology">Radiobiology</a></li>
<li><a href="/wiki/Sievert" title="Sievert">Biological dose units and quantities</a></li>
<li><a href="/wiki/Wireless_device_radiation_and_health" title="Wireless device radiation and health">Wireless device radiation and health</a></li>
<li><a href="/wiki/Wireless_electronic_devices_and_health" class="mw-redirect" title="Wireless electronic devices and health">Wireless electronic devices and health</a></li>
<li><a href="/wiki/Heat_transfer" title="Heat transfer">Radiation heat-transfer</a></li>
<li><a href="/wiki/Linear_energy_transfer" title="Linear energy transfer">Linear energy transfer</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Radiation incidents</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/wiki/List_of_civilian_radiation_accidents" title="List of civilian radiation accidents">List of civilian radiation accidents</a></li>
<li><a href="/wiki/1996_San_Juan_de_Dios_radiotherapy_accident" title="1996 San Juan de Dios radiotherapy accident">1996 Costa Rica accident</a></li>
<li><a href="/wiki/Goi%C3%A2nia_accident" title="Goiânia accident">1987 Goiânia accident</a></li>
<li><a href="/wiki/1984_Moroccan_radiation_accident" title="1984 Moroccan radiation accident">1984 Moroccan accident</a></li>
<li><a href="/wiki/1990_Clinic_of_Zaragoza_radiotherapy_accident" class="mw-redirect" title="1990 Clinic of Zaragoza radiotherapy accident">1990 Zaragoza accident</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related articles</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a class="mw-selflink selflink">Half-life</a></li>
<li><a href="/wiki/Nuclear_physics" title="Nuclear physics">Nuclear physics</a></li>
<li><a href="/wiki/Radioactive_source" title="Radioactive source">Radioactive source</a></li>
<li><a href="/wiki/Radiation_hardening" title="Radiation hardening">Radiation hardening</a></li>
<li><a href="/wiki/Havana_syndrome" title="Havana syndrome">Havana syndrome</a></li></ul>
</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><div role="note" class="hatnote navigation-not-searchable selfref">See also the categories <a href="/wiki/Category:Radiation_effects" title="Category:Radiation effects">Radiation effects</a>, <a href="/wiki/Category:Radioactivity" title="Category:Radioactivity">Radioactivity</a>, <a href="/wiki/Category:Radiobiology" title="Category:Radiobiology">Radiobiology</a>, and <a href="/wiki/Category:Radiation_protection" title="Category:Radiation protection">Radiation protection</a></div></div></td></tr></tbody></table></div></div>' |
