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Half-life (symbol ) 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 Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies t ...
to describe how quickly unstable
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, ...
s undergo
radioactive decay Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is consid ...
or how long stable atoms survive. The term is also used more generally to characterize any type of
exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value *Expo ...
(or, rarely, non-exponential) decay. For example, the medical sciences refer to the
biological half-life Biological half-life (also known as elimination half-life, pharmacologic half-life) is the time taken for concentration of a biological substance (such as a medication) to decrease from its maximum concentration ( Cmax) to half of Cmax in the bl ...
of drugs and other chemicals in the human body. The converse of half-life (in exponential growth) is
doubling time The doubling time is the time it takes for a population to double in size/value. It is applied to population growth, inflation, resource extraction, consumption of goods, compound interest, the volume of malignant tumours, and many other things ...
. The original term, ''half-life period'', dating to
Ernest Rutherford Ernest Rutherford, 1st Baron Rutherford of Nelson, (30 August 1871 – 19 October 1937) was a New Zealand physicist who came to be known as the father of nuclear physics. ''Encyclopædia Britannica'' considers him to be the greatest ...
's discovery of the principle in 1907, was shortened to ''half-life'' in the early 1950s. Rutherford applied the principle of a radioactive element's half-life in studies of age determination of rocks by measuring the decay period of
radium Radium is a chemical element with the symbol Ra and atomic number 88. It is the sixth element in group 2 of the periodic table, also known as the alkaline earth metals. Pure radium is silvery-white, but it readily reacts with nitrogen (rather ...
to
lead-206 Lead (82Pb) has four stable isotopes: 204Pb, 206Pb, 207Pb, 208Pb. Lead-204 is entirely a primordial nuclide and is not a radiogenic nuclide. The three isotopes lead-206, lead-207, and lead-208 represent the ends of three decay chains: the urani ...
. 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

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 Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
: "Half-life 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 half-life is 50%. For example, the image on the right 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 In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...
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 A computer program is a sequence or set of instructions in a programming language for a computer to execute. Computer programs are one component of software, which also includes documentation and other intangible components. A computer prog ...
.


Formulas for half-life in exponential decay

An exponential decay can be described by any of the following four equivalent formulas:\begin N(t) &= N_0 \left(\frac \right)^ \\ N(t) &= N_0 2^ \\ N(t) &= N_0 e^ \\ N(t) &= N_0 e^ \end where * is the initial quantity of the substance that will decay (this quantity may be measured in grams, moles, number of atoms, etc.), * is the quantity that still remains and has not yet decayed after a time , * is the half-life of the decaying quantity, * is a
positive number In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
called the
mean lifetime A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and ( lambda) is a positive rat ...
of the decaying quantity, * is a positive number called the
decay constant A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and ( lambda) is a positive rat ...
of the decaying quantity. The three parameters , , and are directly related in the following way:t_ = \frac = \tau \ln(2)where is the
natural logarithm of 2 The decimal value of the natural logarithm of 2 is approximately :\ln 2 \approx 0.693\,147\,180\,559\,945\,309\,417\,232\,121\,458. The logarithm of 2 in other bases is obtained with the formula :\log_b 2 = \frac. The common logarithm in particu ...
(approximately 0.693).


Half-life and reaction orders

In
chemical kinetics Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with chemical thermodynamics, which deals with the direction in wh ...
, the value of the half-life depends on the
reaction order In chemistry, the rate law or rate equation for a reaction is an equation that links the initial or forward reaction rate with the concentrations or pressures of the reactants and constant parameters (normally rate coefficients and partial reactio ...
:


Decay by two or more processes

Some quantities decay by two exponential-decay processes simultaneously. In this case, the actual half-life can be related to the half-lives and that the quantity would have if each of the decay processes acted in isolation: \frac = \frac + \frac For three or more processes, the analogous formula is: \frac = \frac + \frac + \frac + \cdots For a proof of these formulas, see Exponential decay § Decay by two or more processes.


Examples

There is a half-life describing any exponential-decay process. For example: *As noted above, in
radioactive decay Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is consid ...
the half-life 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 Isotopes are two or more types of atoms that have the same atomic number (number of protons in their nuclei) and position in the periodic table (and hence belong to the same chemical element), and that differ in nucleon numbers (mass numbers ...
, and is usually determined experimentally. See
List of nuclides This list of nuclides shows observed nuclides that either are stable or, if radioactive, have half-lives longer than one hour. This represents isotopes of the first 105 elements, except for elements 87 (francium) and 102 (nobelium). At least 3,300 ...
. *The current flowing through an
RC circuit A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC ci ...
or
RL circuit A resistor–inductor circuit (RL circuit), or RL filter or RL network, is an electric circuit composed of resistors and inductors driven by a voltage or current source. A first-order RL circuit is composed of one resistor and one inductor, either ...
decays with a half-life of or , respectively. For this example the term
half time In several team sports, matches are played in two halves. Half-time (also written halftime or half time) is the name given to the interval between the two halves of the match. Typically, after half-time, teams swap ends of the field of play in or ...
tends to be used rather than "half-life", but they mean the same thing. *In a
chemical reaction A chemical reaction is a process that leads to the chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the positions of electrons in the forming and breakin ...
, 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 , where (also denoted as ) is the
reaction rate constant In chemical kinetics a reaction rate constant or reaction rate coefficient, ''k'', quantifies the rate and direction of a chemical reaction. For a reaction between reactants A and B to form product C the reaction rate is often found to have the f ...
.


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 Biological half-life (also known as elimination half-life, pharmacologic half-life) is the time taken for concentration of a biological substance (such as a medication) to decrease from its maximum concentration ( Cmax) to half of Cmax in the bl ...
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.


In biology and pharmacology

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 Blood plasma is a light amber-colored liquid component of blood in which blood cells are absent, but contains proteins and other constituents of whole blood in suspension. It makes up about 55% of the body's total blood volume. It is the intr ...
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 tissues, active
metabolite In biochemistry, a metabolite is an intermediate or end product of metabolism. The term is usually used for small molecules. Metabolites have various functions, including fuel, structure, signaling, stimulatory and inhibitory effects on enzymes, c ...
s, and receptor interactions. 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, though this can be altered by behavior and other conditions. The biological half-life of
caesium Caesium ( IUPAC spelling) (or cesium in American English) is a chemical element with the symbol Cs and atomic number 55. It is a soft, silvery-golden alkali metal with a melting point of , which makes it one of only five elemental metals that a ...
in human beings is between one and four months. The concept of a half-life has also been utilized for
pesticide Pesticides are substances that are meant to control pests. This includes herbicide, insecticide, nematicide, molluscicide, piscicide, avicide, rodenticide, bactericide, insect repellent, animal repellent, microbicide, fungicide, and lam ...
s in
plant Plants are predominantly photosynthetic eukaryotes of the kingdom Plantae. Historically, the plant kingdom encompassed all living things that were not animals, and included algae and fungi; however, all current definitions of Plantae ex ...
s, and certain authors maintain that pesticide risk and impact assessment models rely on and are sensitive to information describing dissipation from plants. In
epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and determinants of health and disease conditions in a defined population. It is a cornerstone of public health, and shapes policy decisions and eviden ...
, 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
exponentially Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value *Expo ...
.


See also

* Half time (physics) *
List of radioactive nuclides by half-life This is a list of radioactive nuclides (sometimes also called isotopes), ordered by half-life from shortest to longest, in seconds, minutes, hours, days, and years. Current methods make it difficult to measure half-lives between approximately 10� ...
*
Mean lifetime A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and ( lambda) is a positive rat ...
*
Median lethal dose In toxicology, the median lethal dose, LD50 (abbreviation for "lethal dose, 50%"), LC50 (lethal concentration, 50%) or LCt50 is a toxic unit that measures the lethal dose of a toxin, radiation, or pathogen. The value of LD50 for a substance is t ...


References


External links

*https://www.calculator.net/half-life-calculator.html Comprehensive half-life calculator
Welcome to Nucleonica
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wiki: Decay Engine
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Bucknell.edu
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 {{DEFAULTSORT:Half-Life Chemical kinetics Radioactivity Nuclear fission Temporal exponentials