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

Line 3: | Line 3: | ||

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> | + | {| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;" |

− | + | |- | |

+ | | <math>-\frac{dN}{dt} = \lambda N \rightarrow \lambda N = A</math> | ||

+ | | align="right" | Eqn 1<br> | ||

+ | |} | ||

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. | ||

The above equation can be solved into the following: | The above equation can be solved into the following: | ||

− | + | {| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;" | |

+ | |- | ||

+ | | <math>N_{t}=N_{0}e^{-\lambdat}\,</math> | ||

+ | | align="right" | Eqn 2<br> | ||

+ | |} | ||

<span class="texhtml"> | <span class="texhtml"> | ||

Line 18: | Line 25: | ||

<br> | <br> | ||

− | + | {| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;" | |

− | <math>\lambda = \frac{ln2}{T_{1/2}}</math> | + | |- |

− | + | | <math>\lambda = \frac{ln2}{T_{1/2}}</math> | |

+ | | align="right" | Eqn 3<br> | ||

+ | |} | ||

<br> | <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> | ||

− | 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> |

− | + | ||

− | + | {| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;" | |

+ | |- | ||

+ | | <math>dN_{1}=-\lambdaN_{1}dt\,</math> | ||

+ | | align="right" | Eqn 4<br> | ||

+ | |} | ||

+ | <br> | ||

+ | {| cellspacing="0" cellpadding="0" border="0" style="width: 597px; height: 31px;" | ||

+ | |- | ||

+ | | <math>dN_{2}=\lambda{1}N_{1}dt-\lambda_{2}N_{2}dt\,</math> | ||

+ | | align="right" | Eqn 5<br> | ||

+ | |} | ||

− | + | 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: | |

− | + | {| 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> | ||

+ | | align="right" | Eqn 6<br> | ||

+ | |} | ||

<br> | <br> |

## Revision as of 10:53, 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.

Eqn 1 |

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

The above equation can be solved into the following:

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 eqn 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:

Eqn 4 |

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:

Eqn 6 |

If the half-life of the mother is much less than that of the daughter, eqn 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}.**