Half-life (symbolt½) is the time required for a quantity (of substance) to reduce to half of its initial value. The term is commonly used innuclear physics to describe how quickly unstableatoms undergoradioactive decay or how long stable atoms survive. The term is also used more generally to characterize any type ofexponential (or, rarely,non-exponential) decay. For example, the medical sciences refer to thebiological half-life of drugs and other chemicals in the human body. The converse of half-life (in exponential growth) isdoubling time.
The original term,half-life period, dating toErnest Rutherford's discovery of the principle in 1907, was shortened tohalf-life in the early 1950s.[1] Rutherford applied the principle of a radioactiveelement's half-life in studies of age determination of rocks by measuring the decay period ofradium tolead-206.
Half-life is constant over the lifetime of an exponentially decaying quantity, and it is acharacteristic 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.
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 thelaw 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 willnot be "half of an atom" left after one second.
Instead, the half-life is defined in terms ofprobability: "Half-life is the time required for exactly half of the entities to decayon average". In other words, theprobability of a radioactive atom decaying within its half-life is 50%.[2]
For example, the accompanying image is a simulation of many identical atoms undergoing radioactive decay. Note that after one half-life there are notexactly one-half of the atoms remaining, onlyapproximately, because of the random variation in the process. Nevertheless, when there are many identical atoms decaying (right boxes), thelaw of large numbers suggests that it is avery 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 statisticalcomputer program.[3][4][5]
In order to find the half-life, we have to replace the concentration value for the initial concentration divided by 2:and isolate the time:Thist½ formula indicates that the half-life for a zero order reaction depends on the initial concentration and the rate constant.
In first order reactions, the rate of reaction will be proportional to the concentration of the reactant. Thus the concentration will decrease exponentially.as time progresses until it reaches zero, and the half-life will be constant, independent of concentration.
The timet½ for[A] to decrease from[A]0 to1/2[A]0 in a first-order reaction is given by the following equation:It can be solved forFor a first-order reaction, the half-life of a reactant is independent of its initial concentration. Therefore, if the concentration ofA at some arbitrary stage of the reaction is[A], then it will have fallen to1/2[A] after a further interval of Hence, the half-life of a first order reaction is given as the following:
The half-life of a first order reaction is independent of its initial concentration and depends solely on the reaction rate constant,k.
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[A] of the reactant decreases following this formula:
We replace[A] for1/2[A]0 in order to calculate the half-life of the reactantAand isolate the time of the half-life (t½):This shows that the half-life of second order reactions depends on the initial concentration andrate constant.
Some quantities decay by two exponential-decay processes simultaneously. In this case, the actual half-lifeT½ can be related to the half-livest1 andt2 that the quantity would have if each of the decay processes acted in isolation:
There is a half-life describing any exponential-decay process. For example:
As noted above, inradioactive decay the half-life is the length of time after which there is a 50% chance that an atom will have undergonenuclear decay. It varies depending on the atom type andisotope, and is usually determined experimentally. SeeList of nuclides.
The current flowing through anRC circuit orRL circuit decays with a half-life ofln(2)RC orln(2)L/R, respectively. For this example the termhalf time tends to be used rather than "half-life", but they mean the same thing.
In achemical reaction, 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 isln(2)/λ, whereλ (also denoted ask) is thereaction rate constant.
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 asbiological 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.[7]
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 inblood 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 intissues, activemetabolites, andreceptor interactions.[8]
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,[9] though this can be altered by behavior and other conditions. The biological half-life ofcaesium in human beings is between one and four months.
Inepidemiology, 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 modeledexponentially.[12][13]
^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.
