## The Decay Constant

The decay constantappears in the equations

(1)

whereoris the number of undecayed atoms at any timeis the activity or number of decays in one second andis 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 perioda fractionof the undecayed atoms at the start of the time period will remain undecayed at the end of the period. We can rewrite (1) withto becomewith dt tending to 0 so thatis the instantaneous rate of change ofThe minus sign is necessary becauseis decreasing sois negative.

To see why the distinction is important, consider an isotope with a very short half life, say

Then from the equation relatingtowe havebut no probability greater than one is possible.