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>,
+
<math>-\frac{dN}{dt} = \lambda N \rightarrow \lambda N = A</math> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp; (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></span>  
+
<span class="texhtml">''N''<sub>''t''</sub> = ''N''<sub>0</sub>''e''<sup> - λ''t&nbsp;''</sup> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp;&nbsp; (eqn 2)</span>  
 +
 
 +
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 to give the following connection between the disintegration constant and the half-life:
 +
 
 +
 
 +
 
 +
<math>\lambda = \frac{ln2}{T_{1/2}}</math> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; (eqn 3)
  
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:
 
  
<math>\lambda = \frac{ln2}{T_{1/2}}</math>
 
  
 
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''  
+
''dN<sub>1</sub> = -<span class="texhtml">λ</span>N<sub>1</sub>dt &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; (eqn 4)''  
 +
 
 +
''dN<sub>2</sub> =&nbsp;<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 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp; (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)
  
''dN<sub>2</sub> =&nbsp;<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''
 
  
<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>
 
  
If the half-life of the mother is much less than that of the daughter, equation 1.2 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>  
+
<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> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; (eqn 7)
  
 
<br>  
 
<br>  
  
where&nbsp;<math>(1-e^{-\lambda_{2}t})</math>is the saturation factor and <math>\lambda_{2}- \lambda_{1}\cong \lambda_{2}</math>  
+
where&nbsp;<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 &gt;&gt; T<sub>1/2</sub>(2)&nbsp;''(the observed time is much larger than the daughters half-life)''.''  
 
The above equation can be further reduced by the assuption that ''t &gt;&gt; T<sub>1/2</sub>(2)&nbsp;''(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> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; (eqn 8)
  
 
<br>  
 
<br>  
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   & & N_{1}
 
   & & N_{1}
 
   \end{matrix}   
 
   \end{matrix}   
  </math>
+
  </math> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; (eqn 9)
  
when&nbsp;<math>e^{-\lambda_{2}t} \rightarrow 0</math> the equation above 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&nbsp;<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 15:08, 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.

[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:

Nt = N0e - λ                                                                                                                                                  (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:


[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:

dN1 = -λN1dt                                                                                                            (eqn 4)

dN2λ1N1dt - λ2N2dt                                                                                               (eqn 5)


[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)


If the half-life of the mother is much less than that of the daughter, equation 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 assuption 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] equation 9 is called a secular radioactive equilibrium and can be written as λ2N2 = λ1N1.