# Exponential Decay (with half-life)

## Description

Half-life is the amount of time required for the amount of something to fall to half its initial value. The term is very commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay, but it is also used more generally for discussing any type of exponential decay.

The original term, dating to Ernest Rutherford’s discovery of the principle in 1907, was “half-life period”, which was shortened to “half-life” in the early 1950s. Rutherford applied the principle of a radioactive element’s half-life to studies of age determination of rocks by measuring the decay period of radium to lead-206.

Half-life is used to describe a quantity undergoing exponential decay, and is constant over the lifetime of the decaying quantity. It is a characteristic unit for the exponential decay equation. The term “half-life” may generically be used to refer to any period of time in which a quantity falls by half, even if the decay is not exponential. The table on the right shows the reduction of a quantity in the number of half-lives elapsed. For a general introduction and description of exponential decay, see exponential decay. For a general introduction and description of non-exponential decay, see rate law. The converse of half-life is doubling time.

A half-life usually describes the decay of discrete entities, such as radioactive atoms. In that case, it does not work to use the definition “half-life is the time required for exactly half of the entities to decay”. For example, if there are 3 radioactive atoms with a half-life of one second, there will not be “1.5 atoms” left after one second.

Instead, the half-life is defined in terms of probability: “Half-life is the time required for exactly half of the entities to decay on average”. In other words, the probability of a radioactive atom decaying within its half-life is 50%.

For example, the image on the right is a simulation of many identical atoms undergoing radioactive decay. Note that after one half-life there are not exactly one-half of the atoms remaining, only approximately, because of the random variation in the process. Nevertheless, when there are many identical atoms decaying (right boxes), the law of large numbers suggests that it is a very good approximation to say that half of the atoms remain after one half-life.

There are various simple exercises that demonstrate probabilistic decay, for example involving flipping coins or running a statistical computer program.

An exponential decay process can be described with respect to half-life as shown here.

Related formulas## Variables

N(t) | quantity that still remains and has not yet decayed after a time t (dimensionless) |

N_{0} | initial quantity of the substance that will decay (this quantity may be measured in grams, moles, number of atoms, etc (dimensionless) |

t | positive number called the mean lifetime of the decaying quantity (sec) |

t_{1}/2 | half-life of the decaying quantity (sec) |