The half-life of a radionuclide is the time taken for half of the radionuclide's atoms to decay. For the case of one-decay nuclear reactions:
The half-life is related to the decay constant as follows:
A half-life must not be thought of as the time required for exactly half of the entities to decay. Figure 1 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; there are only approximately one-half left because of the random variation in the process. However, with more atoms (the boxes on the right), the overall decay is smoother and less random-looking than with fewer atoms (the boxes on the left), in accordance with the law of large numbers.
The relationship between the half-life and the decay constant shows that highly radioactive substances are quickly spent while those that radiate weakly endure longer. Half-lives of known radionuclides vary widely, from more than 1019 years, such as for the very nearly stable nuclide 209 Bi, to 10−23 seconds for highly unstable ones.
The factor of ln(2) in the above equations results from the fact that the concept of "half-life" is merely a way of selecting a different base other than the natural base e for the lifetime expression. The time constant τ is the e-1-life, the time until only 1/e remains -- about 36.8 percent, rather than the 50 percent in the half-life of a radionuclide. Therefore, τ is longer than t1/2. The following equation can be shown to be valid:
Since radioactive decay is exponential with a constant probability, each process could just as easily be described with a different constant time period that (for example) gave its 1/3-life (how long until only 1/3 is left), or its 1/10-life (how long until only 1/10 is left), and so on. Therefore, the choice of τ and t1/2 for marker-times is only for convenience and for the sake of uploading convention. These marker-times reflect a fundamental principle only in that they show that the same proportion of a given radioactive substance will decay over any time period you choose.
Mathematically, the nth life for the above situation would be found by the same process shown above -- by setting