# Radioactive Disintegration (Introduction to Radiochemistry)

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Radioactive disintegration is a stochastic process, which means a random process, that can be described statistically. In this task you will learn about the secular radioactive equilibrium, and how any measure of a radioactive source is stated with uncertainty.

In a sample with N radioactive atoms of a particular nuclide, the number of atoms that disintegrates with the time *dt *will be proportional with N, see the formula below.

Eqn 1 |

where λ is the disintegration constant and A is the rate of disintegration.

The above equation can be solved into the following:

Eqn 2 |

N_{0} is the number of atoms of the nuclide at hand present at t = 0. The time past when half of the nuclides has disintegrated is called the half-life. N = N_{0}/2 can be placed into eqn 1 to give the following connection between the disintegration constant and the half-life:</span>

Eqn 3 |

The half-life is a characteristic value for each radioactive nuclide. A radioactive nuclide will often disintegrate into a product that is radioactive as well: Nucleus 1

Assume that at the time *t = 0*, N_{0 }of the mother is *N _{1}(t =0), N_{2}(t=0) and N_{3}(t=0)*, the change in number of mother- and daughter nuclides can then respectively be described through the following equations:

Eqn 4 |

Eqn 5 |

The solution of Eqn 4 is already known it is the expression in Eqn 2 while the solution for the numbers of daughter nuclides are given with:

Eqn 6 |

If the half-life of the mother is much less than that of the daughter, Eqn 6 can be simplified into:

Eqn 7 |

where

is the saturation factor and .The above equation can be further reduced by the assumption that *t >> T _{1/2}(2) *(the observed time is much larger than the daughters half-life)

*.*

Eqn 8 |

Eqn 9 |

When eqn 9 is called a secular radioactive equilibrium and can be written as λ_{2}N_{2} = λ_{1}N_{1}.