Difference between revisions of "Radioactive Disintegration (Introduction to Radiochemistry)"
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− | Radioactive disintegration is a stochastic | + | Return to [[Introduction to Radiochemistry - Counting statistics|Main]] |
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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. <br> | ||
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> | + | {| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;" |
+ | |- | ||
+ | | <math>-\frac{dN}{dt} = \lambda N \rightarrow \lambda N = A</math> | ||
+ | | align="right" | Eqn 1<br> | ||
+ | |} | ||
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 | + | {| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;" |
+ | |- | ||
+ | | <math>N_{t}=N_{0}e^{-\lambda t}\,</math> | ||
+ | | align="right" | Eqn 2<br> | ||
+ | |} | ||
+ | |||
+ | <br> 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 eqn 1 to give the following connection between the disintegration constant and the half-life:</span> | ||
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+ | <br> | ||
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+ | <br> | ||
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+ | {| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;" | ||
+ | |- | ||
+ | | <math>\lambda = \frac{ln2}{T_{1/2}}</math> | ||
+ | | align="right" | Eqn 3<br> | ||
+ | |} | ||
+ | |||
+ | <br> | ||
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+ | 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> | ||
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+ | {| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;" | ||
+ | |- | ||
+ | | <math>dN_{1}=-\lambda N_{1}dt\,</math> | ||
+ | | align="right" | Eqn 4<br> | ||
+ | |} | ||
+ | |||
+ | <br> | ||
+ | |||
+ | {| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;" | ||
+ | |- | ||
+ | | <math>dN_{2}=\lambda{1}N_{1}dt-\lambda_{2}N_{2}dt\,</math> | ||
+ | | align="right" | Eqn 5<br> | ||
+ | |} | ||
+ | |||
+ | 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: | ||
+ | |||
+ | {| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;" | ||
+ | |- | ||
+ | | <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> | ||
+ | |} | ||
+ | |||
+ | <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: | ||
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+ | {| 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>\ | + | where <math>(1-e^{-\lambda_{2}t})</math>is the saturation factor and <math>\lambda_{2}- \lambda_{1}\cong \lambda_{2}</math>. |
− | The | + | The above equation can be further reduced by the assumption 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}}</math> | ||
+ | | align="right" | Eqn 8<br> | ||
+ | |} | ||
− | + | <br> | |
− | + | {| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;" | |
+ | |- | ||
+ | | <math>\begin{matrix}& N_{2} = \frac{\lambda_{1}}{\lambda_{2}} | ||
+ | & \underbrace{N_{0}e^{-\lambda_{1}}} \\ | ||
+ | & & N_{1} | ||
+ | \end{matrix}</math> | ||
+ | | align="right" | Eqn 9<br> | ||
+ | |} | ||
− | + | <br> When <math>e^{-\lambda_{2}t} \rightarrow 0</math> eqn 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>. | |
− | + | [[Category:Laboratory_exercise]] [[Category:Radio_chemistry]] [[Category:Nuclear_Properties]] [[Category:Half_life]] [[Category:Natural_activity]] [[Category:Detection]] |
Latest revision as of 14:23, 3 July 2012
Return to Main
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.
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:</span>
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 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:
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 assumption that t >> T1/2(2) (the observed time is much larger than the daughters half-life).
Eqn 8 |
Eqn 9 |
When eqn 9 is called a secular radioactive equilibrium and can be written as λ2N2 = λ1N1.