Difference between revisions of "Radioactive Disintegration (Introduction to Radiochemistry)"
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− | | <math>N_{t}=N_{0}e^{-\ | + | | <math>N_{t}=N_{0}e^{-\lambda t}\,</math> |
| align="right" | Eqn 2<br> | | align="right" | Eqn 2<br> | ||
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− | | <math>dN_{1}=-\ | + | | <math>dN_{1}=-\lambda N_{1}dt\,</math> |
| align="right" | Eqn 4<br> | | align="right" | Eqn 4<br> | ||
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− | | <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> | + | | <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> | ||
| align="right" | Eqn 6<br> | | align="right" | Eqn 6<br> | ||
|} | |} | ||
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If the half-life of the mother is much less than that of the daughter, eqn 6 can be simplified into: | If the half-life of the mother is much less than that of the daughter, eqn 6 can be simplified into: | ||
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− | + | |- | |
+ | | <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> | ||
+ | | align="right" | Eqn 7<br> | ||
+ | |} | ||
<br> | <br> | ||
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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)''.'' | ||
+ | {| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;" | ||
+ | |- | ||
+ | | <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> | ||
+ | | align="right" | Eqn 7<br> | ||
+ | |} | ||
<br> <math>N_{2} = \frac{\lambda_{1}}{\lambda_{2}}N_{0}e^{-\lambda_{1}}</math> (eqn 8) | <br> <math>N_{2} = \frac{\lambda_{1}}{\lambda_{2}}N_{0}e^{-\lambda_{1}}</math> (eqn 8) | ||
Revision as of 10:58, 3 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.
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 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:
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 7 |
(eqn 8)
- (eqn 9)
When λ2N2 = λ1N1.
eqn 9 is called a secular radioactive equilibrium and can be written as