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, rate constant, or transformation constant:

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. This is called the mean lifetime (or simply the lifetime), where the exponential time constant, ${\displaystyle \tau }$, 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, ${\displaystyle \tau }$, instead of the decay constant, λ:

and that ${\displaystyle \tau }$ is the time at which the population of the assembly is reduced to 1⁄e ≈ 0.367879441 times its initial value. This is equivalent to ${\displaystyle \log _{2}{e}}$ ≈ 1.442695 half-lives.

For example, if the initial population of the assembly, N(0), is 1000, then the population at time ${\displaystyle \tau }$, ${\displaystyle N(\tau )}$, is 368.

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".