Exponential decay
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 N is the quantity and λ (lambda) is a positive rate called the exponential decay constant, disintegration constant,[1] rate constant,[2] or transformation constant:[3] The solution to this equation (see derivation below) is: where N(t) is the quantity at time t, N0 = N(0) is the initial quantity, that is, the quantity at time t = 0.
If the decaying quantity, N(t), is the number of discrete elements in a certain set, it is possible to compute the average length of time that an element remains in the set.
, relates to the decay rate constant, λ, in the following way: The mean lifetime can be looked at as a "scaling time", because the exponential decay equation can be written in terms of the mean lifetime,
A very similar equation will be seen below, which arises when the base of the exponential is chosen to be 2, rather than e. In that case the scaling time is the "half-life".
The half-life can be written in terms of the decay constant, or the mean lifetime, as: When this expression is inserted for
in the exponential equation above, and ln 2 is absorbed into the base, this equation becomes: Thus, the amount of material left is 2−1 = 1/2 raised to the (whole or fractional) number of half-lives that have passed.
The equation that describes exponential decay is or, by rearranging (applying the technique called separation of variables), Integrating, we have where C is the constant of integration, and hence where the final substitution, N0 = eC, is obtained by evaluating the equation at t = 0, as N0 is defined as being the quantity at t = 0.
This is the form of the equation that is most commonly used to describe exponential decay.
In this case, λ is the eigenvalue of the negative of the differential operator with N(t) as the corresponding eigenfunction.
Given an assembly of elements, the number of which decreases ultimately to zero, the mean lifetime,
, (also called simply the lifetime) is the expected value of the amount of time before an object is removed from the assembly.
Starting from the population formula first let c be the normalizing factor to convert to a probability density function: or, on rearranging, 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.
The total decay rate of the quantity N is given by the sum of the decay routes; thus, in the case of two processes: The solution to this equation is given in the previous section, where the sum of
by a constant factor, the same equation holds in terms of the two corresponding half-lives: where
The term "partial half-life" is misleading, because it cannot be measured as a time interval for which a certain quantity is halved.
In terms of separate decay constants, the total half-life
can be shown to be For a decay by three simultaneous exponential processes the total half-life can be computed as above: In nuclear science and pharmacokinetics, the agent of interest might be situated in a decay chain, where the accumulation is governed by exponential decay of a source agent, while the agent of interest itself decays by means of an exponential process.
In the pharmacology setting, some ingested substances might be absorbed into the body by a process reasonably modeled as exponential decay, or might be deliberately formulated to have such a release profile.
Exponential decay occurs in a wide variety of situations.
Many decay processes that are often treated as exponential, are really only exponential so long as the sample is large and the law of large numbers holds.
For small samples, a more general analysis is necessary, accounting for a Poisson process.
