The decay constant
appears in the equations
(1)
where
or
is the number of undecayed atoms at any time
is the activity or number of decays in one second and
is the number of undecayed atoms at time
Many elementary textbooks explain thatis the probability of a single atom decaying in one second. This is false, and equation (1) is also misleading – it is only theoretically true in the limit of infinite half life,
The actual meaning ofis: in each time period
a fraction
of the undecayed atoms at the start of the time period will remain undecayed at the end of the period. We can rewrite (1) with
to become
with dt tending to 0 so that
is the instantaneous rate of change of
The minus sign is necessary becauseis decreasing so
is negative.
To see why the distinction is important, consider an isotope with a very short half life, say
Then from the equation relatingto
we have
but no probability greater than one is possible.