Half-life

: For other uses, see Half-life (disambiguation).

The half-life of a radioactive substance is the time required for half of a sample to undergo Radioactive decay. The term also has pharmaceutical and other uses.

More generally, for a quantity subject to Exponential decay, the half-life is the time required for the quantity to fall to half of its initial value. (This article is a narrow discussion of half-life. For phenomena where half-life is applied, see "Related topics" below.)

After # ofHalf-lives \ Percent of quantityremaining
0 100%
1 50%
2 25%
3 12.5%
4 6.25%
5 3.125%
6 1.5625%
7 0.78125%
... ...
N
100%
–––––––––––
2^N
... ...

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

Quantities subject to exponential decay are commonly denoted by the symbol N. (This convention suggests a decaying number of discrete items. 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:

: N (t) = N₀ e^{- λ t}

where

N₀ is the initial value of N (at t=0)
λ is a positive constant (the decay constant).

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

In particular, there is a time t_1/2 such that:

N (t_1/2 ) = N₀ •

1
––
2

Substituting into the formula above, we have:

N₀ •

1
––
2

= N₀ e^{- λ t_{1 / 2}}

e^{- λ t_{1 / 2}} =

1
––
2

- λ t_1/2 = ln

1
––
2

= - ln2

t_1/2 =

ln 2
–––––
λ

Thus the half-life is 69.3% of the Mean lifetime.

Decay by two or more processes

A radioactive element may decay via two or more different processes. These processes may have different probabilities of occurring, and thus there is also a different half-life associated with each process.

As an example, for two decay modes, the amount of substance left after time t is given by

: N (t) = N₀ e^{- λ₁ t} e^{- λ₂ t} = N₀ e^{- ( λ₁ + λ₂ ) t}

In a fashion similar to the previous section, we can calculate the new total half-life T_1/2 and we'll find it to be

T_1/2 =

ln 2
–––––––––
λ₁ + λ₂

or, in terms of the two half-lives

T_1/2 =

t₁ t₂
–––––––––
t₁ + t₂

Where t₁ is the half-life of the first process, and t₂ is the half life of the second process.

Half-life

Decay by two or more processes

See also