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.

[math]-\frac{dN}{dt} = \lambda N \rightarrow \lambda N = A[/math] Eqn 1

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

The above equation can be solved into the following:

[math]N_{t}=N_{0}e^{-\lambda t}\,[/math] 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:</span>

[math]\lambda = \frac{ln2}{T_{1/2}}[/math] 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[math]\rightarrow[/math]Nucleus 2 [math]\rightarrow[/math]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:

[math]dN_{1}=-\lambda N_{1}dt\,[/math] Eqn 4

[math]dN_{2}=\lambda{1}N_{1}dt-\lambda_{2}N_{2}dt\,[/math] 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:

[math]N_{2} = \frac{\lambda_{1}}{\lambda_{2} -\lambda_{1}} N_{0} (e^{-\lambda_{1} t} -e^{-\lambda_{2}t}) \rightarrow [/math]
[math]N_{2}= \frac{\lambda_{1}}{\lambda_{2} -\lambda_{1}} N_{1}(t) ( 1 - e^{-(\lambda_{2} - \lambda_{1}t)})[/math] Eqn 6

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

[math]N_{2} = \frac{\lambda_{1}}{\lambda_{2}} N_{0} (e^{-\lambda_{1} t} -e^{-\lambda_{2}t}) \rightarrow N_{2} \frac{\lambda_{1}}{\lambda_{2}}N_{1}(t) (1-e^{-\lambda_{2}t}) [/math] Eqn 7

where [math](1-e^{-\lambda_{2}t})[/math]is the saturation factor and [math]\lambda_{2}- \lambda_{1}\cong \lambda_{2}[/math].

The above equation can be further reduced by the assumption that t >> T1/2(2) (the observed time is much larger than the daughters half-life).

[math]N_{2} = \frac{\lambda_{1}}{\lambda_{2}}N_{0}e^{-\lambda_{1}}[/math] Eqn 8

[math]\begin{matrix}& N_{2} = \frac{\lambda_{1}}{\lambda_{2}} & \underbrace{N_{0}e^{-\lambda_{1}}} \\ & & N_{1} \end{matrix}[/math] Eqn 9

When [math]e^{-\lambda_{2}t} \rightarrow 0[/math] eqn 9 is called a secular radioactive equilibrium and can be written as    λ2N2 = λ1N1.