Mean lifetime

Mean lifetime
From Wikipedia, the free encyclopedia
Jump to: navigation, search

Given an assembly of elements, the number of which decreases ultimately to zero, the lifetime (also called the mean lifetime) is a certain number that characterizes the rate of reduction ("decay") of the assembly. Specifically, if the individual lifetime of an element of the assembly is the time elapsed between some reference time and the removal of that element from the assembly, the mean lifetime is the arithmetic mean of the individual lifetimes.

Typically, the notion of mean lifetime is used in connection with exponential decay. The remainder of this article confines itself to this particular decay pattern.

[edit] Mean lifetime in exponential decay

The mean lifetime τ of elements in an exponentially decaying assembly is equal to the reciprocal of the decay constant (cf. Exponential decay). Thus, it is the time needed for the assembly to be reduced by a factor of e. It is related to the half-life t1 / 2 by

\tau \cdot \ln 2 = t_{1/2}.\,

Thus the mean lifetime is 44% longer than the half-life, e.g. Polonium-210 has a half-life of 138 days, and a mean lifetime of 200 days.

[edit] Derivation

In exponential decay, the population is governed by the following formula:

N = N_0 e^{-\lambda t} \,

where t is the time, N is the number of elements in the assembly at that time, N0 is the population at the initial reference t = 0, and λ is a parameter characteristic of the decay called the decay constant. The mean lifetime τ is the expected value of the amount of time before an unstable object undergoes a decay. First, we let c be the normalizing factor to convert to a probability space.

1 = \int_{0}^{\infty}c \cdot N_0 e^{-\lambda t}\, dt = c \cdot \frac{N_0}{\lambda}

c = \frac{\lambda}{N_0}.

We see that exponential decay is a scalar multiple of the exponential distribution, which has a well-known expected value. We can compute it here using integration by parts.

\tau = \langle t \rangle = \int_{0}^{\infty} t \cdot c \cdot N_0 e^{-\lambda t}\, dt = \int_{0}^{\infty} \lambda t e^{-\lambda t}\, dt = \frac{1}{\lambda}.

Retrieved from "http://en.wikipedia.org/wiki/Mean_lifetime"

Categories: Exponentials | Physics
Views

* Article
* Discussion
* Edit this page
* History

Personal tools

* Sign in / create account

Navigation

* Main page
* Contents
* Featured content
* Current events
* Random article

interaction

* About Wikipedia
* Community portal
* Recent changes
* Contact Wikipedia
* Donate to Wikipedia
* Help

Search

Toolbox

* What links here
* Related changes
* Upload file
* Special pages
* Printable version
* Permanent link
* Cite this article

In other languages

* Català
* Deutsch
* Español
* Français
* Italiano
* Português
* Русский
* Українська

Powered by MediaWiki
Wikimedia Foundation

* This page was last modified 06:26, 28 May 2007.
* All text is available under the terms of the GNU Free Documentation License. (See Copyrights for details.)
Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a US-registered 501(c)(3) tax-deductible nonprofit charity.
* Privacy policy
* About Wikipedia
* D