(adsbygoogle = window.adsbygoogle || []).push({}); Definition/Summary

The half-life, [itex]t_{1/2}[/itex], of an inverse exponential process (an exponential decay) is the time taken for the amount to reduce by one-half. It is constant.

Processes with a half-life include radioactive decay, first-order chemical reactions, and current flowing through an RC electrical circuit.

The half-life divided by the (natural) logarithm of 2 is the mean lifetime, [itex]{\tau}[/itex]. It is the time taken for the amount to reduce by a factor e (ie 2.718...). It is the inverse of the decay constant, [itex]{\lambda}[/itex], also referred to as the decay rate, or probability per unit time of decay.

Equations

Inverse exponential process (exponential decay) with decay constant [itex]\lambda[/itex]:

[tex]A = A_0e^{-\lambda t}[/tex]

Mean lifetime:

[tex]\tau\ =\ \frac{1}{\lambda} \ =\ \frac{t_{1/2}}{\log 2}[/tex]

where [itex]\log[/itex] denotes the natural logarithm.

Half-life:

[tex]t_{1/2}\ =\ \frac{log2}{\lambda} \ = \ \tau\ \log 2 [/tex]

For decay of the same population by two or more simultaneous inverse exponential processes with decay constants [itex]\lambda_1,\cdots,\lambda_n[/itex]:

[tex]\lambda\ =\ \lambda_1\ +\ \cdots\ +\ \lambda_n[/tex]

[tex]\frac{1}{\tau}\ =\ \frac{1}{\tau_1}\ +\ \cdots\ +\ \frac{1}{\tau_n}[/tex]

[tex]\frac{1}{t_{1/2}}\ =\ \frac{1}{\left(t_1\right)_{1/2}}\ +\ \cdots\ +\ \frac{1}{\left(t_n\right)_{1/2}}[/tex]

Extended explanation

Radioactive decay:

The quantity which reduces is theexpectation valueof the quantity of radioactive material.

RC circuits:

The flow of current discharged from a capacitor through a resistor (an RC circuit) is an inverse exponential process with mean lifetime (time constant) equal to the resistance times the capacitance: [itex]\frac{1}{\lambda}\ =\ \tau\ =\ RC[/itex].

Other meanings:

Technically, a half-life could be defined foranyprocess, at each stage of that process, but it would not be constant …

it is only for an inverse exponential process that the half-life is the same at each stage …

and so it is only for an inverse exponential process that a half-lifefor a processcan be defined.

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# What is half-life

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads - half life | Date |
---|---|

A Prove half step scheme is TVD | Jan 26, 2016 |

Challenge 22: Half Again As Big | Nov 10, 2014 |

Hard half life equation | Jul 7, 2014 |

Raising to half power = PRINCIPAL square root? | Mar 17, 2014 |

How do you calculate daily dose of a med based on half life | Mar 16, 2007 |

**Physics Forums - The Fusion of Science and Community**