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

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{| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;"
 
{| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;"
 
|-
 
|-
| <math>N_{t}=N_{0}e^{-\lambdat}\,</math>
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| <math>N_{t}=N_{0}e^{-\lambda t}\,</math>
 
| align="right" | Eqn 2<br>
 
| align="right" | Eqn 2<br>
 
|}  
 
|}  
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{| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;"
 
{| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;"
 
|-
 
|-
| <math>dN_{1}=-\lambdaN_{1}dt\,</math>
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| <math>dN_{1}=-\lambda N_{1}dt\,</math>
 
| align="right" | Eqn 4<br>
 
| align="right" | Eqn 4<br>
 
|}  
 
|}  
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{| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;"
 
{| 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 N_{2} = \frac{\lambda_{1}}{\lambda_{2} -\lambda_{1}} N_{1}(t) ( 1 - e^{-(\lambda_{2} - \lambda_{1}t)})</math>
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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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|-
 +
|<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:  
  
<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; &nbsp; &nbsp; (eqn 7)
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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>  
 
<br>  
  
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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)''.''  
  
 +
{| 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> &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 8)  
 
<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;&nbsp; (eqn 8)  
  

Revision as of 09:58, 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.

[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:



[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 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:

[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 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} t} -e^{-\lambda_{2}t}) \rightarrow N_{2} \frac{\lambda_{1}}{\lambda_{2}}N_{1}(t) (1-e^{-\lambda_{2}t}) [/math] Eqn 7


[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.