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 | + | 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 ''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. | ||
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− | The | + | 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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− | If the half-life of the mother is much less than that of the daughter, | + | 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 <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 | + | The above equation can be further reduced by the assumption that ''t >> T<sub>1/2</sub>(2) ''(the observed time is much larger than the daughters half-life)''.'' |
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<br> When <math>e^{-\lambda_{2}t} \rightarrow 0</math> eqn 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>. | <br> When <math>e^{-\lambda_{2}t} \rightarrow 0</math> eqn 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>. | ||
− | [[Category:Laboratory_exercise]][[Category:Radio_chemistry]][[Category:Nuclear_Properties]][[Category:Half_life]][[Category:Natural_activity]][[Category:Detection]] | + | [[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.
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 |
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:
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, 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:
Eqn 4 |
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:
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 assumption that t >> T1/2(2) (the observed time is much larger than the daughters half-life).
Eqn 8 |
Eqn 9 |
When eqn 9 is called a secular radioactive equilibrium and can be written as λ2N2 = λ1N1.