# 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> (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 ''</sup> (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> (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. <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 (eqn 4)'' |

+ | |||

+ | ''dN<sub>2</sub> = <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 (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) | ||

− | |||

− | |||

− | If the half-life of the mother is much less than that of the daughter, equation | + | 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> (eqn 7) |

<br> | <br> | ||

− | where <math>(1-e^{-\lambda_{2}t})</math>is the saturation factor and <math>\lambda_{2}- \lambda_{1}\cong \lambda_{2}</math> | + | 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 >> T<sub>1/2</sub>(2) ''(the observed time is much larger than the daughters half-life)''.'' | The above equation can be further reduced by the assuption that ''t >> T<sub>1/2</sub>(2) ''(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> (eqn 8) |

<br> | <br> | ||

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& & N_{1} | & & N_{1} | ||

\end{matrix} | \end{matrix} | ||

− | </math> | + | </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 <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.

(eqn 1)

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

The above equation can be solved into the following:

*N*_{t} = *N*_{0}*e*^{ - λt } (eqn 2)

N_{0} 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 _{0}/2* can be placed into equation 1 to give the following connection between the disintegration constant and the half-life:

(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*, N_{0 }of the mother is *N _{1}(t =0), N_{2}(t=0) and N_{3}(t=0)*, the change in number of mother- and daughter nuclides can then respectively be described through the following equations:

*dN _{1} = -λN_{1}dt (eqn 4)*

*dN _{2} = λ_{1}N_{1}dt - λ_{2}N_{2}dt (eqn 5)*

(eqn 6)

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

(eqn 7)

where

is the saturation factor and .The above equation can be further reduced by the assuption that *t >> T _{1/2}(2) *(the observed time is much larger than the daughters half-life)

*.*

(eqn 8)

- (eqn 9)

When λ_{2}*N*_{2} = λ_{1}*N*_{1}.