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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.


MEAN LIFETIME IN EXPONENTIAL DECAY


As you will see derived below, 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 t_{1/2} thus:

au \cdot \ln 2 = t_{1/2}


Derivation


For purposes of this derivation, the following terminology and notation (some of it introduced above) will be useful:
  • A large (but decreasing) number of ''elements'' is grouped into an ''assembly''.

  • The ''individual lifetime'' of an element is the time elapsed between some reference time and the removal of that element from the assembly.

  • The ''total lifetime'' (''T'') is the sum of the individual lifetimes.

  • The ''mean lifetime'' (τ) is the arithmetic mean of the individual lifetimes.

  • The ''population'' (''N'') is the number of elements in the assembly at any given time.

  • The ''initial population'' (N_0) is the population at some initial reference time.


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

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

During a Differential period of time ''dt'', the population shrinks slightly. The (positive) number of elements leaving the assembly is equal to the negative of the change in population:

:-dN = N_0\lambda e^{-\lambda t} dt \,

If ''dt'' is taken at some time ''t'', the individual lifetimes of the elements are simply equal to ''t''. These elements' contribution to the total lifetime is thus given by:

:dT = t \cdot -dN = t\cdot N_0\lambda e^{-\lambda t} dt \,

: au = rac{1}{N_0} \int_{0}^{\infty} dT = rac{1}{N_0} \int_{0}^{\infty} N_0\lambda e^{-\lambda t}\, t\, dt

: au = \lambda \int_{0}^{\infty} t\, e^{-\lambda t} dt

Integration By Parts yields:

: au = rac{1}{\lambda}