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.

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:

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>

Half-life

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