Difference between revisions of "Radioactive Disintegration (Introduction to Radiochemistry)"

Revision as of 13:33, 2 July 2012

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.

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

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

The above equation can be solved into the following:

N_t = N₀e ^{- λt}

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 equation 1.1 to give the following connection between the disintegration constant and the half-life:

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

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:

dN₁ = -λN₁dt

dN₂ = λ₁N₁dt - λ₂N₂dt

of equation 1.1 is allready known, see e.q.1.1, while the solution of the number of daughter nuclides is given by:

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