# Half-life

(Redirected from 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.

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 # of
Half-lives
Percent of quantity
remaining
0 100%
1 50
2 25
3 12.5
4 6.25
5 3.125
6 1.5625
7 0.78125%

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:

[itex]N(t) = N_0 e^{-\lambda t} \,[itex]

where

• [itex]N_0[itex] 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 [itex]N_0[itex]. As t approaches infinity, the exponential approaches zero.

In particular, there is a time [itex]t_{1/2} \,[itex] such that:

[itex]N(t_{1/2}) = N_0\cdot\frac{1}{2} [itex]

Substituting into the formula above, we have:

[itex]N_0\cdot\frac{1}{2} = N_0 e^{-\lambda t_{1/2}} \,[itex]
[itex]e^{-\lambda t_{1/2}} = \frac{1}{2} \,[itex]
[itex]- \lambda t_{1/2} = \ln \frac{1}{2} = - \ln{2} \,[itex]
[itex]t_{1/2} = \frac{\ln 2}{\lambda} \,[itex]

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 occuring, and thus there is also a different half-life associated with each process.

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

[itex]N(t) = N_0 e^{-\lambda _1 t} e^{-\lambda _2 t} = N_0 e^{-(\lambda _1 + \lambda _2) t}[itex]

In a fashion similar to the previous section, we can calculate the new total half-life [itex]T _{1/2} \,[itex] and we'll find it to be

[itex]T_{1/2} = \frac{\ln 2}{\lambda _1 + \lambda _2} \,[itex]

or, in terms of the two half-lives

[itex]T_{1/2} = \frac{t _1 t _2}{t _1 + t_2} \,[itex]

Where [itex]t _1 \,[itex] is the half-life of the first process, and [itex]t _2 \,[itex] is the half life of the second process.

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