Template:Short description Template:About

Number of
half-lives
elapsed
Fraction
remaining
Percentage
remaining
0 Template:Frac 100
1 Template:1/2 50
2 Template:1/4 25
3 Template:Frac 12 .5
4 Template:Frac 6 .25
5 Template:Frac 3 .125
6 Template:Frac 1 .5625
7 Template:Frac 0 .78125
Template:Mvar Template:Frac Template:Frac

Template:E (mathematical constant)

Half-life (symbol Template:Math) 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 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, 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 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 natureEdit

File:Halflife-sim.gif
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 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 on average". In other words, the probability of a radioactive atom decaying within its half-life is 50%.<ref name=PTFP>Template:Cite book</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>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Formulas for half-life in exponential decayEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} An exponential decay can be described by any of the following four equivalent formulas:<ref name=ln(2)/>Template:Rp<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> where

The three parameters Template:Math, Template:Mvar, and Template:Mvar are directly related in the following way:<math display="block">t_{1/2} = \frac{\ln (2)}{\lambda} = \tau \ln(2)</math>where Template:Math is the natural logarithm of 2 (approximately 0.693).<ref name="ln(2)">Template:Cite book</ref>Template:Rp

Half-life and reaction ordersEdit

In chemical kinetics, the value of the half-life depends on the reaction order:

Zero order kineticsEdit

The rate of this kind of reaction does not depend on the substrate concentration, Template:Math. 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 Template:Math formula indicates that the half-life for a zero order reaction depends on the initial concentration and the rate constant.

First order kineticsEdit

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 Template:Math for Template:Math to decrease from Template:Math to Template:Math 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 Template:Math at some arbitrary stage of the reaction is Template:Math, then it will have fallen to Template:Math after a further interval of Template:Tmath Hence, the half-life of a first order reaction is given as the following:

<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, Template:Mvar.

Second order kineticsEdit

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 Template:Math of the reactant decreases following this formula:

<math display="block" chem>\frac{1}{[\ce A]} = kt + \frac{1}{[\ce A]_0}</math>We replace Template:Math for Template:Math in order to calculate the half-life of the reactant Template:Math <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 (Template:Math):<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 rate constant.

Decay by two or more processesEdit

Some quantities decay by two exponential-decay processes simultaneously. In this case, the actual half-life Template:Math can be related to the half-lives Template:Math and Template:Math 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.

ExamplesEdit

Template:Further There is a half-life describing any exponential-decay process. For example:

In non-exponential decayEdit

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>Template:Cite book</ref>

In biology and pharmacologyEdit

Template:See also 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 tissues, active metabolites, and receptor interactions.<ref name="SCM">Template:Cite book</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>Template:Cite book</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 pesticides in plants,<ref name=tebuau>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> and certain authors maintain that pesticide risk and impact assessment models rely on and are sensitive to information describing dissipation from plants.<ref name=acs>Template:Cite journal</ref>

In 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 exponentially.<ref name = "Balkew">Template:Cite thesis</ref><ref name = "Ireland">Template:Cite book</ref>

See alsoEdit

ReferencesEdit

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External linksEdit

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