Halflife
If a sample of material decays at a certain rate over time, its halflife is defined as the time it takes for the sample to decay to half its initial amount. This concept originated when studying the exponential decay of radioactive isotopes, but it is applied to other phenomena as well, including those described by nonexponential decay. An exponential decay process, as exemplified by the decay of radioactive isotopes, for example, is simply one in which the number of atoms disintegrating per unit time is proportional to the total number of radioactive atoms present.
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In the case of radioactive decay, each different radioactive isotope has a particular halflife that is unaffected by changes in the physical or chemical conditions of the surroundings. This property is the basis for radiometric dating of rocks and fossils. In pharmacology, the halflife of a drug (in a biological system) is the time it takes for the drug to lose half its pharmacologic activity.
Example of radioactive decay
Carbon14 (^{14}C) is a radioactive isotope that decays to produce the isotope nitrogen14 (^{14}N). The halflife of ^{14}C is about 5,730 years. This means that if one starts with 10 grams of ^{14}C, then 5 grams of the isotope will remain after 5,730 years, 2.5 grams will remain after another 5,730 years, and so forth.
Calculation of halflife
Number of halflives elapsed 
Fraction remaining 
As power of 2 

0  1/1  1 / 2^{0} 
1  1/2  1 / 2^{1} 
2  1/4  1 / 2^{2} 
3  1/8  1 / 2^{3} 
4  1/16  1 / 2^{4} 
5  1/32  1 / 2^{5} 
6  1/64  1 / 2^{6} 
7  1/128  1 / 2^{7} 
...  ...  
N  1 / 2^{N}  1 / 2^{N} 
The table at right shows the reduction of the quantity in terms of the number of halflives elapsed.
It can be shown that, for exponential decay, the halflife t_{1 / 2} obeys the following relation:
where

 ln(2) is the natural logarithm of 2, and
 λ, called the decay constant, is a positive constant used to describe the rate of exponential decay.
In addition, the halflife is related to the mean lifetime τ by the following relation:
The constant λ can represent various specific physical quantities, depending on the process being described.
 In firstorder chemical reactions, λ is the reaction rate constant.
 In pharmacology (specifically pharmacokinetics), the halflife of a drug is defined as the time it takes for a substance (drug, radioactive nuclide, or other) to lose half of its pharmacologic, physiologic, or radiologic activity.^{[1]}
 For electronic filters such as an RC circuit (resistorcapacitor circuit) or an RL circuit (resistorinductor circuit), λ is the reciprocal of the circuit's time constant τ, which is the same as the mean lifetime mentioned above. For simple RC or RL circuits, λ equals 1 / RC or R / L, respectively. The symbol τ is related to the circuit's cutoff frequency f_{c} by
 or, equivalently, .
Experimental determination
The halflife of a process can be readily determined by experiment. Some methods do not require advance knowledge of the law governing the decay rate, whether it follows an exponential or other pattern of decay.
Most appropriate to validate the concept of halflife for radioactive decay, in particular when dealing with a small number of atoms, is to perform experiments and correct computer simulations. Validation of physicsmath models consists of comparing the model's behavior with experimental observations of real physical systems or valid simulations (physical and/or computer simulations).^{[2]}
When studying radioactive decay, the exponential model does not apply for a small number of atoms (or a small number of atoms is not within the domain of validity of the formula or equation or table). Some model simulations use pennies or pieces of candy.^{[3]}^{[4]} A similar experiment is performed with isotopes that have very short halflives.^{[5]}
Decay by two or more processes
Some quantities decay by two processes simultaneously. In a manner similar to that mentioned above, one can calculate the new total halflife (T_{1 / 2}) as follows:
or, in terms of the two halflives t_{1} and t_{2}
that is, half their harmonic mean.
Derivation
Quantities that are subject to exponential decay are commonly denoted by the symbol N. (This convention suggests a decaying number of discrete items, an interpretation that is valid in many, but not all, cases of exponential decay.) If the quantity is denoted by the symbol N, the value of N at a time t is given by the formula:
where N_{0} is the initial value of N (at t = 0)
When t = 0, the exponential is equal to 1, and N(t) is equal to N_{0}. As t approaches infinity, the exponential approaches zero. In particular, there is a time such that
Substituting into the formula above, we have
See also
 Exponential decay
 Mean lifetime
 Radioactive decay
 Rate equation
Notes
 ↑ Taken from Medical Subject Headings. Year introduced: 1974 (1971).
 ↑ See link to test the behavior of the last remaining atoms of a radioactive sample. Retrieved October 19, 2007.
 ↑ Radioactive Decay Model Retrieved October 19, 2007.
 ↑ Radioactive Decay Retrieved October 19, 2007.
 ↑ For example, see Figure 5 in the link. See how to write a computer program that simulates radioactive decay including the required randomness in the link and experience the behavior of the last atoms. Retrieved October 19, 2007.
References
 Emsley, John. 2001. Nature's Building Blocks: An A to Z Guide to the Elements. Oxford: Oxford University Press. ISBN 0198503407
 Magill, Joseph, and Jean Galy. 2004. Radioactivity Radionuclides Radiation. Berlin: Springer. ISBN 3540211160
 Brown, G.I. 2002. Invisible Rays: A History of Radioactivity. Sutton Publishing. ISBN 0750926678
External links
All links retrieved January 26, 2014.
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