## The Mathematics of Radioactive Decay

The basic relationship that expresses random decay as a random process is sxpressed mathematically as where N is the number of undecayed atoms remaining.

The constant of proportionality is negative since N is decreasing so dN &lt; 0. We may write where is the number of decays in each time period typically taken to be 1 second, and is a positive constant called the decay constant. The proportion of atoms that will decay in a time dt or probability that an individual atom will decay in a time is then We can solve the equation obtaining where N-0 is the number of undecayed atoms initially present. A similar expression exists for the activity We can find the decay constant %lambda in the following way.

Take natural logarithms of obtaining This is the equation of a straight line graph, with on the vertical axis and gradient A graph of N against t is shown below. The number of undecayed atoms fall by a half in each fixed time period called the half life, labelled  We can find the half life by substituting and into  Alternatively we could observe the time taken for the activity to decay by a half, but this is less accurate. 