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

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Return to [[Introduction to Radiochemistry - Counting statistics|Main]]  
 
Return to [[Introduction to Radiochemistry - Counting statistics|Main]]  
  
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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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.  
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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:  
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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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<br>  
 
<br>  
  
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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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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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)''.''  
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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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<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>.  
 
<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>.  
  
[[Category:Laboratory_exercise]][[Category:Radio_chemistry]][[Category:Nuclear_Properties]][[Category:Half_life]][[Category:Natural_activity]][[Category:Detection]]
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[[Category:Laboratory_exercise]] [[Category:Radio_chemistry]] [[Category:Nuclear_Properties]] [[Category:Half_life]] [[Category:Natural_activity]] [[Category:Detection]]

Revision as of 11:23, 3 July 2012

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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.

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 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.