- 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:
- <math>N(t) = N_0 e^{-\lambda t} \,<math>
where
- <math>N_0<math> 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 <math>N_0<math>. As t approaches infinity, the exponential approaches zero.
In particular, there is a time <math>t_{1/2} \,<math> such that:
- <math>N(t_{1/2}) = N_0\cdot\frac{1}{2} <math>
Substituting into the formula above, we have:
- <math>N_0\cdot\frac{1}{2} = N_0 e^{-\lambda t_{1/2}} \,<math>
- <math>e^{-\lambda t_{1/2}} = \frac{1}{2} \,<math>
- <math>- \lambda t_{1/2} = \ln \frac{1}{2} = - \ln{2} \,<math>
- <math>t_{1/2} = \frac{\ln 2}{\lambda} \,<math>
Thus the half-life is 69.3% of the mean lifetime.
Related topics
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