Editing Exponential decay (section)

== Measuring rates of decay ==

=== Mean lifetime ===
If the decaying quantity, ''N''(''t''), is the number of discrete elements in a certain [[set (mathematics)|set]], it is possible to compute the average length of time that an element remains in the set.  This is called the '''mean lifetime''' (or simply the '''lifetime'''), where the '''exponential [[time constant]]''', <math>\tau</math>, relates to the decay rate constant, &lambda;, in the following way:
:<math>\tau = \frac{1}{\lambda}.</math>

The mean lifetime can be looked at as a "scaling time", because the exponential decay equation can be written in terms of the mean lifetime, <math>\tau</math>, instead of the decay constant, &lambda;:
:<math>N(t) = N_0 e^{-t/\tau}, </math>
and that <math>\tau</math> is the time at which the population of the assembly is reduced to {{Fraction|1|[[e (mathematical constant)|''e'']]}} ≈ 0.367879441 times its initial value. This is equivalent to <math>\log_{2}{e}</math> ≈ 1.442695 half-lives.

For example, if the initial population of the assembly, ''N''(0), is 1000, then the population at time <math>\tau</math>, <math>N(\tau)</math>, is 368.

A very similar equation will be seen below, which arises when the base of the exponential is chosen to be 2, rather than ''e''. In that case the scaling time is the "half-life".

===Half-life===
{{main|Half-life}}

A more intuitive characteristic of exponential decay for many people is the time required for the decaying quantity to fall to one half of its initial value. (If ''N''(''t'') is discrete, then this is the median life-time rather than the mean life-time.) This time is called the ''half-life'', and often denoted by the symbol ''t''<sub>1/2</sub>.  The half-life can be written in terms of the decay constant, or the mean lifetime, as:

:<math>t_{1/2} = \frac{\ln (2)}{\lambda} = \tau \ln (2).</math>

When this expression is inserted for <math>\tau</math> in the exponential equation above, and [[Natural logarithm of 2|ln&nbsp;2]] is absorbed into the base, this equation becomes:

:<math>N(t) = N_0 2^{-t/t_{1/2}}. </math>

Thus, the amount of material left is 2<sup>−1</sup>&nbsp;=&nbsp;1/2 raised to the (whole or fractional) number of half-lives that have passed. Thus, after 3 half-lives there will be 1/2<sup>3</sup>&nbsp;=&nbsp;1/8 of the original material left.

Therefore, the mean lifetime <math>\tau</math> is equal to the half-life divided by the natural log of 2, or:

: <math>\tau = \frac{t_{1/2}}{\ln (2)} \approx 1.4427 \cdot t_{1/2}.</math>

For example, [[polonium-210]] has a half-life of 138 days, and a mean lifetime of 200 days.