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

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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. <br>  
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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&nbsp;''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&nbsp;''will be proportional with N, see the formula below.  
  
<math>-\frac{dN}{dt} = \lambda N \rightarrow \lambda N = A</math>,
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{| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;"
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|-
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| <math>-\frac{dN}{dt} = \lambda N \rightarrow \lambda N = A</math>  
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| align="right" | Eqn 1<br>
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|}
  
 
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></span>  
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{| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;"
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|-
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| <math>N_{t}=N_{0}e^{-\lambda t}\,</math>  
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| align="right" | Eqn 2<br>
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|}
  
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.<br> <br> ''N = N<sub>0</sub>/2'' can be placed into equation 1.1 to give the following connection between the disintegration constant and the half-life:  
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<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:&lt;/span&gt;
  
<math>\lambda = \frac{ln2}{T_{1/2}}</math>  
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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;"
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|-
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| <math>\lambda = \frac{ln2}{T_{1/2}}</math>
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| align="right" | Eqn 3<br>
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|}
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<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''  
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{| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;"
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|-
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| <math>dN_{1}=-\lambda N_{1}dt\,</math>
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| align="right" | Eqn 4<br>
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|}
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<br>
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{| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;"
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|-
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| <math>dN_{2}=\lambda{1}N_{1}dt-\lambda_{2}N_{2}dt\,</math>
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| align="right" | Eqn 5<br>
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|}
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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:
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{| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;"
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|-
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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  </math>
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|-
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| <math>N_{2}= \frac{\lambda_{1}}{\lambda_{2} -\lambda_{1}} N_{1}(t) ( 1 - e^{-(\lambda_{2} - \lambda_{1}t)})</math>
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| align="right" | Eqn 6<br>
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|}
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<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;"
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|-
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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>
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| align="right" | Eqn 7<br>
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|}
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<br>
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where&nbsp;<math>(1-e^{-\lambda_{2}t})</math>is the saturation factor and <math>\lambda_{2}- \lambda_{1}\cong \lambda_{2}</math>.
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The above equation can be further reduced by the assumption that ''t &gt;&gt; T<sub>1/2</sub>(2)&nbsp;''(the observed time is much larger than the daughters half-life)''.''  
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{| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;"
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|-
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| <math>N_{2} = \frac{\lambda_{1}}{\lambda_{2}}N_{0}e^{-\lambda_{1}}</math>
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| align="right" | Eqn 8<br>
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|}
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<br>
  
''dN<sub>2</sub> =&nbsp;<span class="texhtml"<sub>1</sub></span>N<sub>1</sub>dt - <span class="texhtml"<sub>2</sub></span>N<sub>2</sub>dt''
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{| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;"
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|-
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| <math>\begin{matrix}& N_{2} = \frac{\lambda_{1}}{\lambda_{2}}
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  & \underbrace{N_{0}e^{-\lambda_{1}}} \\
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  & & N_{1}
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  \end{matrix}</math>  
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| align="right" | Eqn 9<br>
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|}
  
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<br> When&nbsp;<math>e^{-\lambda_{2}t} \rightarrow 0</math> eqn 9 is called a secular radioactive equilibrium and can be written as &nbsp;&nbsp; <span class="texhtml">λ<sub>2</sub>N<sub>2</sub> = λ<sub>1</sub>N<sub>1</sub></span>.
  
<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>
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[[Category:Laboratory_exercise]] [[Category:Radio_chemistry]] [[Category:Nuclear_Properties]] [[Category:Half_life]] [[Category:Natural_activity]] [[Category:Detection]]

Latest revision as of 13:23, 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.

[math]-\frac{dN}{dt} = \lambda N \rightarrow \lambda N = A[/math] Eqn 1

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

The above equation can be solved into the following:

[math]N_{t}=N_{0}e^{-\lambda t}\,[/math] 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>



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

[math]dN_{1}=-\lambda N_{1}dt\,[/math] Eqn 4


[math]dN_{2}=\lambda{1}N_{1}dt-\lambda_{2}N_{2}dt\,[/math] 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:

[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] Eqn 6


If the half-life of the mother is much less than that of the daughter, Eqn 6 can be simplified into:

[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


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 assumption that t >> T1/2(2) (the observed time is much larger than the daughters half-life).

[math]N_{2} = \frac{\lambda_{1}}{\lambda_{2}}N_{0}e^{-\lambda_{1}}[/math] Eqn 8


[math]\begin{matrix}& N_{2} = \frac{\lambda_{1}}{\lambda_{2}} & \underbrace{N_{0}e^{-\lambda_{1}}} \\ & & N_{1} \end{matrix}[/math] Eqn 9


When [math]e^{-\lambda_{2}t} \rightarrow 0[/math] eqn 9 is called a secular radioactive equilibrium and can be written as    λ2N2 = λ1N1.