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Exponential decay is a scalar multiple of the exponential distribution (i.e. the individual lifetime of each object is exponentially distributed), which has a well-known expected value. We can compute it here using integration by parts .
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.
In exponential decay, such as of a radioactive isotope, the time constant can be interpreted as the mean lifetime. The half-life THL or T1/2 is related to the exponential decay constant by
The doubling time is a characteristic unit (a natural unit of scale) for the exponential growth equation, and its converse for exponential decay is the half-life .
Biological half-life ( elimination half-life, pharmacological half-life) is the time taken for concentration of a biological substance (such as a medication) to decrease from its maximum concentration ( C max) to half of C max in the blood plasma.
Its decay product, 99 Tc, has a relatively long half-life (211,000 years) and emits little radiation. The short physical half-life of 99m Tc and its biological half-life of 1 day with its other favourable properties allows scanning procedures to collect data rapidly and keep total patient radiation exposure low.
Derivation of equations that describe the time course of change for a system with zero- order input and first-order elimination are presented in the articles Exponential decay and Biological half-life, and in scientific literature. [1] [7]
Particle decay is a Poisson process, and hence the probability that a particle survives for time t before decaying (the survival function) is given by an exponential distribution whose time constant depends on the particle's velocity: