## The Mathematics of Radioactive Decay

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

The constant of proportionality is negative since N is decreasing so dN < 0. We may writewhereis the number of decays in each time periodtypically taken to be 1 second, andis 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 timeis then

We can solve the equationobtainingwhere 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 ofobtainingThis is the equation of a straight line graph, withon 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 substitutingandinto

Alternatively we could observe the time taken for the activity to decay by a half, but this is less accurate.