Editing Half-life

{{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 decay|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 &#91;sic&#93; 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) &= \left(2^{-\frac{t}{t_{1/2}}}\right) N_0 \\
  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===
{{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 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]]