Half-life

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.

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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.)

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

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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:

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

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where

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• $N\_0$ 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 N_0. As t approaches infinity, the exponential approaches zero.

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

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:N(t_{1/2}) = N_0cdotrac{1}{2}

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Substituting into the formula above, we have:

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:N_0cdotrac{1}{2} = N_0 e^{-lambda t_{1/2}} ,

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:e^{-lambda t_{1/2}} = rac{1}{2} ,

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:- lambda t_{1/2} = ln rac{1}{2} = - ln{2} ,

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:t_{1/2} = rac{ln 2}{lambda} ,

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

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Half-life: Decay by two or more processes ►