, a quantity that decays exponentially
is one that decreases at a rate proportional to its value. For example, since a radioactive atom
has the same probability to decay at any given time, regardless of how long it already lived, the number of disintegrations in an assembly of such atoms is proportional to their number. Such an exponentially decaying population decreases, in atom per unit of time, three times as fast when there are six million atoms, as it does when there are two millions.
If we call x this quantity, the rate of change dx/dt obeys by definition the differential equation:
where is a positive number called the decay constant
(in units of inverse time). The solution to this equation is the exponential function , whence the name of the associated decay. C
is an arbitrary constant, determined by the initial size of the population.
Examples of Exponential Decays
- Disintegration of unstable particles. See