Halflife Information
Number of halflives elapsed 
Fraction remaining 
Percentage remaining  

0  1⁄1  100  
1  1⁄2  50  
2  1⁄4  25  
3  1⁄8  12  .5 
4  1⁄16  6  .25 
5  1⁄32  3  .125 
6  1⁄64  1  .5625 
7  1⁄128  0  .78125 
...  ...  ...  
n  1⁄2^{n}  100⁄2^{n} 
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Halflife (symbol t_{1⁄2}) is the time required for a quantity to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay or how long stable atoms survive. The term is also used more generally to characterize any type of exponential (or, rarely, nonexponential) decay. For example, the medical sciences refer to the biological halflife of drugs and other chemicals in the human body. The converse of halflife (in exponential growth) is doubling time.
The original term, halflife period, dating to Ernest Rutherford's discovery of the principle in 1907, was shortened to halflife in the early 1950s.^{ [1]} Rutherford applied the principle of a radioactive element's halflife to studies of age determination of rocks by measuring the decay period of radium to lead206.
Halflife 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 halflives elapsed.
Probabilistic nature
A halflife often describes the decay of discrete entities, such as radioactive atoms. In that case, it does not work to use the definition that states "halflife is the time required for exactly half of the entities to decay". For example, if there is just one radioactive atom, and its halflife is one second, there will not be "half of an atom" left after one second.
Instead, the halflife is defined in terms of probability: "Halflife is the time required for exactly half of the entities to decay on average". In other words, the probability of a radioactive atom decaying within its halflife is 50%.^{ [2]}
For example, the image on the right is a simulation of many identical atoms undergoing radioactive decay. Note that after one halflife there are not exactly onehalf 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 halflife.
Various simple exercises can demonstrate probabilistic decay, for example involving flipping coins or running a statistical computer program.^{ [3]}^{ [4]}^{ [5]}
Formulas for halflife in exponential decay
An exponential decay can be described by any of the following four equivalent formulas:^{ [6]}^{: 109–112 }
 N_{0} is the initial quantity of the substance that will decay (this quantity may be measured in grams, moles, number of atoms, etc.),
 N(t) is the quantity that still remains and has not yet decayed after a time t,
 t_{1⁄2} is the halflife of the decaying quantity,
 τ is a positive number called the mean lifetime of the decaying quantity,
 λ is a positive number called the decay constant of the decaying quantity.
The three parameters t_{1⁄2}, τ, and λ are directly related in the following way:
Halflife and reaction orders
In chemical kinetics, the value of the halflife depends on the reaction order:
 Zero order kinetics: The rate of this kind of reaction does not depend on the substrate concentration, [A]:The integrated rate law of zero order kinetics is:In order to find the halflife, we have to replace the concentration value for the initial concentration divided by 2:and isolate the time:This t_{1/2} formula indicates that the halflife for a zero order reaction depends on the initial concentration and the rate constant.
 First order kinetics: In first order reactions, the concentration of the reactant will decrease exponentially as time progresses until it reaches zero, and the halflife will be constant, independent of concentration.
The time t_{1/2} for [A] to decrease from [A]_{0} to 1/2 [A]_{0} in a firstorder reaction is given by the following equation:
It can be solved forFor a firstorder reaction, the halflife of a reactant is independent of its initial concentration. Therefore, if the concentration of A at some arbitrary stage of the reaction is [A], then it will have fallen to 1/2 [A] after a further interval of (ln 2)/k. Hence, the halflife of a first order reaction is given as the following:The halflife of a first order reaction is independent of its initial concentration and depends solely on the reaction rate constant, k.  Second order kinetics: In second order reactions, the concentration [A] of the reactant decreases following this formula:
We replace [A] for 1/2 [A]_{0} in order to calculate the halflife of the reactant Aand isolate the time of the halflife (t_{1/2}):This shows that the halflife of second order reactions depends on the initial concentration and rate constant.
Decay by two or more processes
Some quantities decay by two exponentialdecay processes simultaneously. In this case, the actual halflife T_{1⁄2} can be related to the halflives t_{1} and t_{2} that the quantity would have if each of the decay processes acted in isolation:
For three or more processes, the analogous formula is:
Examples
There is a halflife describing any exponentialdecay process. For example:
 As noted above, in radioactive decay the halflife is the length of time after which there is a 50% chance that an atom will have undergone nuclear decay. It varies depending on the atom type and isotope, and is usually determined experimentally. See List of nuclides.
 The current flowing through an RC circuit or RL circuit decays with a halflife of ln(2)RC or ln(2)L/R, respectively. For this example the term half time tends to be used rather than "halflife", but they mean the same thing.
 In a chemical reaction, the halflife of a species is the time it takes for the concentration of that substance to fall to half of its initial value. In a firstorder reaction the halflife of the reactant is ln(2)/λ, where λ (also denoted as k) is the reaction rate constant.
In nonexponential decay
The term "halflife" 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 halflife discussed below). In a decay process that is not even close to exponential, the halflife will change dramatically while the decay is happening. In this situation it is generally uncommon to talk about halflife in the first place, but sometimes people will describe the decay in terms of its "first halflife", "second halflife", etc., where the first halflife is defined as the time required for decay from the initial value to 50%, the second halflife is from 50% to 25%, and so on.^{ [7]}
In biology and pharmacology
A biological halflife or elimination halflife is the time it takes for a substance (drug, radioactive nuclide, or other) to lose onehalf of its pharmacologic, physiologic, or radiological activity. In a medical context, the halflife may also describe the time that it takes for the concentration of a substance in blood plasma to reach onehalf of its steadystate value (the "plasma halflife").
The relationship between the biological and plasma halflives of a substance can be complex, due to factors including accumulation in tissues, active metabolites, and receptor interactions.^{ [8]}
While a radioactive isotope decays almost perfectly according to socalled "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 halflife of water in a human being is about 9 to 10 days,^{ [9]} though this can be altered by behavior and other conditions. The biological halflife of caesium in human beings is between one and four months.
The concept of a halflife has also been utilized for pesticides in plants,^{ [10]} and certain authors maintain that pesticide risk and impact assessment models rely on and are sensitive to information describing dissipation from plants.^{ [11]}
In epidemiology, the concept of halflife can refer to the length of time for the number of incident cases in a disease outbreak to drop by half, particularly if the dynamics of the outbreak can be modeled exponentially.^{ [12]}^{ [13]}
See also
References
 ^ John Ayto, 20th Century Words (1989), Cambridge University Press.
 ^ Muller, Richard A. (April 12, 2010). Physics and Technology for Future Presidents. Princeton University Press. pp. 128–129. ISBN 9780691135045.
 ^ Chivers, Sidney (March 16, 2003). "Re: What happens during halflifes [sic] when there is only one atom left?". MADSCI.org.
 ^ "RadioactiveDecay Model". Exploratorium.edu. Retrieved 20120425.

^ Wallin, John (September 1996).
"Assignment #2: Data, Simulations, and Analytic Science in Decay". Astro.GLU.edu. Archived from the original on 20110929.
{{ cite web}}
: CS1 maint: unfit URL ( link)  ^ ^{a} ^{b} Rösch, Frank (September 12, 2014). Nuclear and Radiochemistry: Introduction. Vol. 1. Walter de Gruyter. ISBN 9783110221916.
 ^ Jonathan Crowe; Tony Bradshaw (2014). Chemistry for the Biosciences: The Essential Concepts. p. 568. ISBN 9780199662883.
 ^ Lin VW; Cardenas DD (2003). Spinal cord medicine. Demos Medical Publishing, LLC. p. 251. ISBN 9781888799613.
 ^ Pang, XiaoFeng (2014). Water: Molecular Structure and Properties. New Jersey: World Scientific. p. 451. ISBN 9789814440424.
 ^ Australian Pesticides and Veterinary Medicines Authority (31 March 2015). "Tebufenozide in the product Mimic 700 WP Insecticide, Mimic 240 SC Insecticide". Australian Government. Retrieved 30 April 2018.
 ^ Fantke, Peter; Gillespie, Brenda W.; Juraske, Ronnie; Jolliet, Olivier (11 July 2014). "Estimating HalfLives for Pesticide Dissipation from Plants". Environmental Science & Technology. 48 (15): 8588–8602. Bibcode: 2014EnST...48.8588F. doi: 10.1021/es500434p. PMID 24968074.
 ^ Balkew, Teshome Mogessie (December 2010). The SIR Model When S(t) is a MultiExponential Function (Thesis). East Tennessee State University.
 ^ Ireland, MW, ed. (1928). The Medical Department of the United States Army in the World War, vol. IX: Communicable and Other Diseases. Washington: U.S.: U.S. Government Printing Office. pp. 116–7.
External links
 https://www.calculator.net/halflifecalculator.html Comprehensive halflife calculator
 Welcome to Nucleonica, Nucleonica.net (archived 2017)
 wiki: Decay Engine, Nucleonica.net (archived 2016)
 System Dynamics – Time Constants, Bucknell.edu
 Researchers Nikhef and UvA measure slowest radioactive decay ever: Xe124 with 18 billion trillion years
 https://academo.org/demos/radioactivedecaysimulator/ Interactive radioactive decay simulator demonstrating how halflife is related to the rate of decay