Difference between revisions of "Radioactive Disintegration (Introduction to Radiochemistry)"
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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. | 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) | + | <math>-\frac{dN}{dt} = \lambda N \rightarrow \lambda N = A</math> (eqn 1) |
where <span class="texhtml">λ</span> is the disintegration constant and A is the rate of disintegration. | where <span class="texhtml">λ</span> is the disintegration constant and A is the rate of disintegration. | ||
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The above equation can be solved into the following: | The above equation can be solved into the following: | ||
− | <span class="texhtml">''N''<sub>''t''</sub> = ''N''<sub>0</sub>''e''<sup> - λ''t ''</sup> (eqn 2)</span> | + | <span class="texhtml">''N''<sub>''t''</sub> = ''N''<sub>0</sub>''e''<sup> - λ''t ''</sup> (eqn 2) |
+ | N<sub>0</sub> 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 = N<sub>0</sub>/2'' can be placed into equation 1 to give the following connection between the disintegration constant and the half-life: | ||
+ | </span> | ||
− | + | <br> | |
− | |||
− | |||
− | |||
− | |||
+ | <math>\lambda = \frac{ln2}{T_{1/2}}</math> (eqn 3) | ||
+ | <br> | ||
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. <br> | 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. <br> | ||
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Assume that at the time ''t = 0'', N<sub>0 </sub>of the mother is ''N<sub>1</sub>(t =0), N<sub>2</sub>(t=0) and N<sub>3</sub>(t=0)'', the change in number of mother- and daughter nuclides can then respectively be described through the following equations:<br> | Assume that at the time ''t = 0'', N<sub>0 </sub>of the mother is ''N<sub>1</sub>(t =0), N<sub>2</sub>(t=0) and N<sub>3</sub>(t=0)'', the change in number of mother- and daughter nuclides can then respectively be described through the following equations:<br> | ||
− | ''dN<sub>1</sub> = -<span class="texhtml">λ</span>N<sub>1</sub>dt (eqn 4) | + | ''dN<sub>1</sub> = -<span class="texhtml">λ</span>N<sub>1</sub>dt'' (eqn 4) |
− | ''dN<sub>2</sub> = <span class="texhtml">λ<sub>1</sub></span>N<sub>1</sub>dt - <span class="texhtml">λ<sub>2</sub></span>N<sub>2</sub>dt (eqn 5) | + | ''dN<sub>2</sub> = <span class="texhtml">λ<sub>1</sub></span>N<sub>1</sub>dt - <span class="texhtml">λ<sub>2</sub></span>N<sub>2</sub>dt'' (eqn 5) |
<br> <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>(eqn 6) | <br> <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>(eqn 6) | ||
− | + | <br> | |
If the half-life of the mother is much less than that of the daughter, equation 6 can be simplified into: | If the half-life of the mother is much less than that of the daughter, equation 6 can be simplified into: | ||
− | <br> <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) | + | <br> <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) |
<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)''.'' | ||
− | <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) |
<br> | <br> | ||
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& & N_{1} | & & N_{1} | ||
\end{matrix} | \end{matrix} | ||
− | </math> (eqn 9) | + | </math> (eqn 9) |
When <math>e^{-\lambda_{2}t} \rightarrow 0</math> equation 9 is called a secular radioactive equilibrium and can be written as <span class="texhtml">λ<sub>2</sub>''N''<sub>2</sub> = λ<sub>1</sub>''N''<sub>1</sub></span>. | When <math>e^{-\lambda_{2}t} \rightarrow 0</math> equation 9 is called a secular radioactive equilibrium and can be written as <span class="texhtml">λ<sub>2</sub>''N''<sub>2</sub> = λ<sub>1</sub>''N''<sub>1</sub></span>. |
Revision as of 16:11, 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.
(eqn 1)
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 (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 equation 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:
dN1 = -λN1dt (eqn 4)
dN2 = λ1N1dt - λ2N2dt (eqn 5)
(eqn 6)
If the half-life of the mother is much less than that of the daughter, equation 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 λ2N2 = λ1N1.
equation 9 is called a secular radioactive equilibrium and can be written as