 Halflife

This article is about the scientific and mathematical term. For other uses, see Halflife (disambiguation).
Number of
halflives
elapsedFraction
remainingPercentage
remaining0 ^{1}/_{1} 100 1 ^{1}/_{2} 50 2 ^{1}/_{4} 25 3 ^{1}/_{8} 12 .5 4 ^{1}/_{16} 6 .25 5 ^{1}/_{32} 3 .125 6 ^{1}/_{64} 1 .563 7 ^{1}/_{128} 0 .781 ... ... ... n 1/(2^{n}) 100/(2^{n}) Halflife, abbreviated t_{½}, is the period of time it takes for the amount of a substance undergoing decay to decrease by half. The name was originally used to describe a characteristic of unstable atoms (radioactive decay), but it may apply to any quantity which follows a setrate decay.
The original term, dating to 1907, was "halflife period", which was later shortened to "halflife" in the early 1950s.^{[1]}
Halflives are used to describe quantities undergoing exponential decay—for example, radioactive decay—where the halflife is constant over the whole life of the decay, and is a characteristic unit (a natural unit of scale) for the exponential decay equation. However, a halflife can also be defined for nonexponential decay processes, although in these cases the halflife varies throughout the decay process. For a general introduction and description of exponential decay, see the article exponential decay. For a general introduction and description of nonexponential decay, see the article rate law. Corresponding to sediments in environmental processes, if the halflife is greater than the residence time, then the radioactive nuclide will have enough time to significantly alter the concentration. The converse of halflife is doubling time.
The table on the right shows the reduction of a quantity in terms of the number of halflives elapsed.
Contents
Probabilistic nature of halflife
A halflife describes the decay of discrete entities, such as radioactive atoms. In that case, it does not work to use the definition "halflife is the time required for exactly half of the entities to decay". For example, if there is just one radioactive atom with a halflife of 1 second, there will not be "half of an atom" left after 1 second. There will be either zero atoms left or one atom left, depending on whether or not the atom happens to decay.
Instead, the halflife is defined in terms of probability. It is the time when the expected value of the number of entities that have decayed is equal to half the original number. For example, one can start with a single radioactive atom, wait its halflife, and measure whether or not it decays in that period of time. Perhaps it will and perhaps it will not. But if this experiment is repeated again and again, it will be seen that  on average  it decays within the halflife 50% of the time.
In some experiments (such as the synthesis of a superheavy element), there is in fact only one radioactive atom produced at a time, with its lifetime individually measured. In this case, statistical analysis is required to infer the halflife. In other cases, a very large number of identical radioactive atoms decay in the timerange measured. In this case, the law of large numbers ensures that the number of atoms that actually decay is essentially equal to the number of atoms that are expected to decay. In other words, with a large enough number of decaying atoms, the probabilistic aspects of the process can be ignored.
There are various simple exercises that demonstrate probabilistic decay, for example involving flipping coins or running a computer program.^{[2]}^{[3]}^{[4]} 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, due to random variation in the process. However, with more atoms (right boxes), the overall decay is smoother and less random than with fewer atoms (left boxes), in accordance with the law of large numbers.
Formulas for halflife in exponential decay
Main article: Exponential decayAn exponential decay process can be described by any of the following three equivalent formulas:
where

 N_{0} is the initial quantity of the substance that will decay (this quantity may be measured in grams, moles, number of atoms, etc.),
 N(t) is the quantity that still remains and has not yet decayed after a time t,
 t_{1 / 2} is the 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 all directly related in the following way:
where ln(2) is the natural logarithm of 2 (approximately 0.693).

Click "show" to see a detailed derivation of the relationship between halflife, decay time, and decay constant. Start with the three equations  N(t) = N_{0}e ^{− t / τ}
 N(t) = N_{0}e ^{− λt}
We want to find a relationship between t_{1 / 2}, τ, and λ, such that these three equations describe exactly the same exponential decay process. Comparing the equations, we find the following condition:
Next, we'll take the natural logarithm of each of these quantities.
Using the properties of logarithms, this simplifies to the following:
Since the natural logarithm of e is 1, we get:
Canceling the factor of t and plugging in , the eventual result is:
By plugging in and manipulating these relationships, we get all of the following equivalent descriptions of exponential decay, in terms of the halflife:
Regardless of how it's written, we can plug into the formula to get
 N(0) = N_{0} as expected (this is the definition of "initial quantity")
 as expected (this is the definition of halflife)
 , i.e. amount approaches zero as t approaches infinity as expected (the longer we wait, the less remains).
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:
For a proof of these formulas, see Decay by two or more processes.
Examples

Main article: Exponential decayApplications and examples
There is a halflife describing any exponentialdecay process. For example:
 The current flowing through an RC circuit or RL circuit decays with a halflife of RCln(2) or ln(2)L / R, respectively.
 In a firstorder chemical reaction, the halflife of the reactant is ln(2) / λ, where λ is the reaction rate constant.
 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.
Halflife in nonexponential decay
Main article: Rate equationThe decay of many physical quantities is not exponential—for example, the evaporation of water from a puddle, or (often) the chemical reaction of a molecule. In such cases, the halflife is defined the same way as before: as the time elapsed before half of the original quantity has decayed. However, unlike in an exponential decay, the halflife depends on the initial quantity, and the prospective halflife will change over time as the quantity decays.
As an example, the radioactive decay of carbon14 is exponential with a halflife of 5730 years. A quantity of carbon14 will decay to half of its original amount after 5730 years, regardless of how big or small the original quantity was. After another 5730 years, onequarter of the original will remain. On the other hand, the time it will take a puddle to halfevaporate depends on how deep the puddle is. Perhaps a puddle of a certain size will evaporate down to half its original volume in one day. But on the second day, there is no reason to expect that onequarter of the puddle will remain; in fact, it will probably be much less than that. This is an example where the halflife reduces as time goes on. (In other nonexponential decays, it can increase instead.)
The decay of a mixture of two or more materials which each decay exponentially, but with different halflives, is not exponential. Mathematically, the sum of two exponential functions is not a single exponential function. A common example of such a situation is the waste of nuclear power stations, which is a mix of substances with vastly different halflives. Consider a sample containing a rapidly decaying element A, with a halflife of 1 second, and a slowly decaying element B, with a halflife of one year. After a few seconds, almost all atoms of the element A have decayed after repeated halving of the initial total number of atoms; but very few of the atoms of element B will have decayed yet as only a tiny fraction of a halflife has elapsed. Thus, the mixture taken as a whole does not decay by halves.
Halflife in biology and pharmacology
Main article: Biological halflifeA biological halflife or elimination halflife is the time it takes for a substance (drug, radioactive nuclide, or other) to lose half of its pharmacologic, physiologic, or radiological activity. In a medical context, halflife may also describe the time it takes for the blood plasma concentration of a substance to halve ("plasma halflife") its steadystate. 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.^{[5]}
While a radioactive isotope decays perfectly according to first order kinetics where the rate constant is fixed, the elimination of a substance from a living organism follows more complex kinetics.
For example, the biological halflife of water in a human is about 7 to 14 days, though this can be altered by behavior. The biological halflife of caesium in humans is between one and four months. This can be shortened by feeding the person Prussian blue, which acts as a solid ion exchanger which absorbs the caesium while releasing potassium ions.
See also
 List of isotopes by halflife
 Mean lifetime
References
 ^ John Ayto, "20th Century Words" (1989), Cambridge University Press.
 ^ MADSCI.org
 ^ Exploratorium.edu
 ^ Astro.GLU.edu
 ^ Lin VW; Cardenas DD (2003). Spinal cord medicine. Demos Medical Publishing, LLC. p. 251. ISBN 1888799617. http://books.google.co.uk/books?id=3anl3G4No_oC&pg=PA251&lpg=PA251.
External links
 Nucleonica.net, Nuclear Science Portal
 Nucleonica.net, wiki: Decay Engine
 Bucknell.edu, System Dynamics  Time Constants
 Subotex.com, HalfLife elimination of drugs in blood plasma  Simple Charting Tool
Radiation (Physics & Health) Main articles Electromagnetic radiation
and healthRadiation therapy · Radiation poisoning · Radioactivity in the life sciences · List of civilian radiation accidents
Health physics · Laser safety · Lasers and aviation safety · Mobile phone radiation and health · Wireless electronic devices and healthRelated articles See also categories: Radiation effects, Radioactivity, and Radiobiology.Categories: Radioactivity
 Exponentials
 Chemical kinetics

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