Difference between revisions of "Radioactive Disintegration (Introduction to Radiochemistry)"
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where <math>(1-e^{-\lambda_{2}t})</math>is the saturation factor and <math>\lambda_{2}- \lambda_{1}\cong \lambda_{2}</math> | 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 assuption that ''t >> T<sub>1/2</sub>(2) (the observed time is much larger than the daughters half-life). | + | The above equation can be further reduced by the assuption that ''t >> T<sub>1/2</sub>(2) ''(the observed time is much larger than the daughters half-life)''.'' |
− | <br> <math>N_{2} = \frac{\lambda_{1}}{\lambda_{2}}N_{0}e^{-\lambda_{1}}</math> | + | <br> <math>N_{2} = \frac{\lambda_{1}}{\lambda_{2}}N_{0}e^{-\lambda_{1}}</math> |
+ | |||
+ | <br> | ||
+ | |||
+ | |||
+ | :<math> | ||
+ | \begin{matrix}& N_{2} = \frac{\lambda_{1}}{\lambda_{2}} | ||
+ | & \underbrace{N_{0}e^{-\lambda_{1}}} \\ | ||
+ | & & N_{1} | ||
+ | \end{matrix} | ||
+ | </math> |
Revision as of 15:29, 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.
,
where λ is the disintegration constant and A is the rate of disintegration.
The above equation can be solved into the following:
Nt = 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:
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, 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
If the half-life of the mother is much less than that of the daughter, equation 1.2 can be simplified into:
where
is the saturation factor andThe 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).