half-life

In radioactivity, the time taken for a group of atoms to decay to half their original number; symbol T^½, units s (second), also minutes and years. It varies from seconds to thousands of years, depending on the atomic species. The half-life of plutonium-239 is 24 400 years; for helium-6 it is 0·8 seconds. The term also applies to the decay of excited atoms by the emission of light.

The half-life of a quantity subject to exponential decay is the time required for the quantity to decay to half of its initial value.

Number of half-lives elapsed	Fraction remaining	As power of 2
0	1/1	1 / 20
1	1/2	1 / 21
2	1/4	1 / 22
3	1/8	1 / 23
4	1/16	1 / 24
5	1/32	1 / 25
6	1/64	1 / 26
7	1/128	1 / 27
...
N	1 / 2	1 / 2

The table at right shows the reduction of the quantity in terms of the number of half-lives elapsed.

It can be shown that, for exponential decay, the half-life t1 / 2 obeys this relation:

where

ln(2) is the natural logarithm of 2, and λ is the decay constant, a positive constant used to describe the rate of exponential decay. In a fashion similar to the previous section, we can calculate the new total half-life T1 / 2 and we'll find it to be:

or, in terms of the two half-lives

where t1 is the half-life of the first process, and t2 is the half life of the second process. This interpretation is valid in many, but not all, cases of exponential decay.) If the quantity is denoted by the symbol N, the value of N at a time t is given by the formula:

where N0 is the initial value of N (at t = 0)

When t = 0, the exponential is equal to 1, and N(t) is equal to N0.

In radioactive decay, the exponential model does not apply for a small number of atoms (or a small number of atoms is not within the domain of validity of the formula or equation or table).