Radioactive disintegration is a stochastic proces, 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

N0 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 = N0/2 can be placed into eqn 1 to give the following connection between the disintegration constant and the half-life:

 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 1Nucleus 2 Nucleus 3. The initial nucleus is usually referred to as the mother nuclide and the product as the daughter nuclide.

Assume that at the time t = 0, N0 of the mother is N1(t =0), N2(t=0) and N3(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 solutin 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 assuption that t >> T1/2(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    λ2N2 = λ1N1.