Half-life

Half-life (symbol t½) 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.

[1] 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.

The accompanying table shows the reduction of a quantity as a function of the number of half-lives elapsed.

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".

In other words, the probability 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 not exactly one-half of the atoms remaining, only approximately, because of the random variation in the process.

Various simple exercises can demonstrate probabilistic decay, for example involving flipping coins or running a statistical computer program.

[3][4][5] An exponential decay can be described by any of the following four equivalent formulas:[6]: 109–112

where The three parameters t½, τ, and λ are directly related in the following way:

This t½ formula indicates that the half-life for a zero order reaction depends on the initial concentration and the rate constant.

as time progresses until it reaches zero, and the half-life will be constant, independent of concentration.

The time t½ for [A] to decrease from [A]0 to ⁠1/2⁠[A]0 in a first-order reaction is given by the following equation:

For a first-order reaction, the half-life of a reactant is independent of its initial concentration.

By integrating this rate, it can be shown that the concentration [A] of the reactant decreases following this formula:

This shows that the half-life of second order reactions depends on the initial concentration and rate constant.

In this case, the actual half-life T½ can be related to the half-lives t1 and t2 that the quantity would have if each of the decay processes acted in isolation:

For example: 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).

[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 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.

[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.

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,[10] and certain authors maintain that pesticide risk and impact assessment models rely on and are sensitive to information describing dissipation from plants.

[11] 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.