The process of radioactive decay has a particular mathematical property. The rate of radioactive decayis proportional to the numberof undecayed atoms present. We can write
whereis the decay rate,
is the number of undecayed atoms present
is the decay constant.
We can solve this differential equation using a method called separation of variables:
To fully solve the original differential equation we have to findWhenthe initial number of undecayed atoms. Substituting these into the equationgives Hence
The half lifeis the time taken for half the atoms to decay ie forto fall toWe can find
The half life is a constant for a particular isotope, so the time taken for the number of atoms to fall from tois the same as the time taken to fall fromtoetc. The graph below illustrates this.