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# Half-life

This article describes the scientific meaning. For the computer game, see Half-Life.

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.)

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:

$N(t) = N_0 e^{-\lambda t} \,$

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:

$N(t_{1/2}) = N_0\cdot\frac{1}{2}$

Substituting into the formula above, we have:

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

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

## Related topics

Last updated: 02-08-2005 16:27:01
Last updated: 03-18-2005 11:16:12