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Kinetics of Radioactive Decays

Kinetics of Radioactive Decays. Kinetics of First Order Reactions. 2.1 First-Order Decay Expressions. 2.1 (a) Statistical Considerations (1905) Let: p = probability of a particular atom disintegrating in time interval t.

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Kinetics of Radioactive Decays

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  1. Kinetics of Radioactive Decays

  2. Kinetics of First Order Reactions

  3. 2.1 First-Order Decay Expressions • 2.1 (a) Statistical Considerations (1905) • Let: p = probability of a particular atom disintegrating in time interval t. Since this is a pure random event; that is, all decays are independent of past and present information; then each t gives the same probability again. Total time = t = n t

  4. 2.1 First-Order Decay Expressions • 2.1 (a) Statistical Considerations (1905) Note: typo “+”

  5. 2.1 First-Order Decay Expressions • 2.1 (b) Decay Expressions: • (i) N-Expression

  6. 2.1 First-Order Decay Expressions Excel Example

  7. 2.1 First-Order Decay Expressions • 2.1 (b) Decay Expressions: • (ii) A-Expression • Define: A = Activity (counts per second or disintegrations per second) For fixed geometry:

  8. 2.1 First-Order Decay Expressions • 2.1 (b) Decay Expressions: • (ii) A-Expression • Define: A = Activity (counts per second or disintegrations per second) A = c  N Where: c = detection coeff. N A

  9. 2.1 First-Order Decay Expressions • 2.1 (c) Lives • (i) Half-life: t1/2 • Defined as time taken for initial amount ( N or A ) to drop to half of original value.

  10. 2.1 First-Order Decay Expressions Note: What is N after x half lives?

  11. 2.1 First-Order Decay Expressions • 2.1 (c) Lives • (ii) Average/Mean Life:  (common usage in spectroscopy) • Can be found from sums of times of existence of all atoms divided by the total number.

  12. 2.1 First-Order Decay Expressions • 2.1 (c) (ii) Average/Mean Life:  (common usage in spectroscopy) 

  13. 2.1 First-Order Decay Expressions • 2.1 (c) Lives • (iii) Comparing half and average/mean life 1.44 t1/2 Why is  greater than t1/2 by factor of 1.44?  gives equal weighting to those atoms that survives a long time!

  14. 2.1 First-Order Decay Expressions • 2.1 (c) Lives (iii) Comparing half and average/mean life What is the value of N at t =  ? Excel Example

  15. 2.1 First-Order Decay Expressions • 2.1 (d) Decay/Growth Complications • Kinetics can get quite complicated mathematically if products are also radioactive (math/expressions next section) • Examples:

  16. 2.1 First-Order Decay Expressions • 2.1 (e) Units of Radioactivity • Refers to “Activity” • 1 Curie (Ci) = the amount of RA material which produces 3.700x1010 disintegrations per second. • SI unit => 1 Becquerel (Bq) = 1 disintegration per second • Example (1): Compare 1 mCi of 15O ( t1/2 = 2 min ) with 1 mCi of 238U ( t1/2 = 4.5x109 y ) • Use “Specific Activity” = Bq/g ( activity per g of RA material )

  17. 2.1 First-Order Decay Expressions • 2.1 (e) Units of Radioactivity • Rad = quantitative measure of radiation energy absorption (dose) • 1 dose of 1 rad deposits 100 erg/g of material • SI dose unit => gray (Gy) = 1 J/kg; 1 Gy = 100 rad • Roentgen (R) = unit of radiation exposure; • 1 R = 1.61x1012 ion pairs per gram of air. • More Later !

  18. 2.1 First-Order Decay Expressions • 2.1 (e) Units of Radioactivity: • Example (2): Calculate the weight (W) in g of 1.00 mCi of 3Hwith t1/2 = 12.26 y .

  19. 2.1 First-Order Decay Expressions • 2.1 (e) Units of Radioactivity: • Example (3): Calculate W of 1.00 mCi of 14C with t1/2 = 5730 y . • Example (4): Calculate W of 1.00 mCi of 238U with t1/2 = 4.15x109 y .

  20. 2.1 First-Order Decay Expressions • 2.1 (e) Units of Radioactivity:

  21. 2.2 Multi-Component Decays • 2.2 (a) Mixtures of Independently Decay Activities

  22. 2.2 Multi-Component Decays • 2.2 (a) Mixtures of Independently Decay Activities • Resolution of Decay Curves • (i) Binary Mixture ( unknowns 1 , 2 , initial A1 & A2 ) Excel plot

  23. 2.2 Multi-Component Decays • 2.2 (a) Mixtures of Independently Decay Activities • Resolution of Decay Curves • (ii) If 1 & 2 are known but 12(not very different) • (iii) Least Square Analysis ( if only At versus t ) [Multi-parameter fitting software]

  24. 2.2 Multi-Component Decays • 2.2 (b) Relationships Among Parent and RA Products • Consider general case of Parent(N1)/daughter(N2) in which daughter is also RA. • (i) If (2) is stable • (ii) If (2) is RA and (3) is stable

  25. 2.2 Multi-Component Decays • 2.2 (b) Relationships Among Parent and RA Products • N2equation (2.8) and its variations.

  26. 2.2 Multi-Component Decays • 2.2 (b) Relationships Among Parent and RA Products • N2equation (2.8) and its variations … cont.

  27. 2.2 Multi-Component Decays • 2.2 (c) Equilibrium Phenomena (Transient & Secular): Parent longer lived • Consider equation (2.8) • (1) Transient Equilibrium ( 1 < 2 ) • (i) When t is large:

  28. 2.2 Multi-Component Decays • 2.2 (c) Equilibrium Phenomena (Transient & Secular): Parent longer lived • Consider equation (2.8) • (1) Transient Equilibrium ( 1 < 2 ) • (ii) for activities Note: Main point is that for transient equilibrium, after some time, both species will decay with 1 .

  29. 2.2 Multi-Component Decays • 2.2 (c) Equilibrium Phenomena (Transient & Secular): Parent longer lived • Consider equation (2.8) • (1) Transient Equilibrium ( 1 < 2 ) • (iii) A1 + A2 (starting with pure 1) • Will go through a maximum before transient equilibrium is achieved.

  30. 2.2 Multi-Component Decays • 2.2 (c) Equilibrium Phenomena (Transient & Secular): Parent longer lived • Consider equation (2.8) • (1) Transient Equilibrium ( 1 < 2 ) • (iii) A1 + A2 (starting with pure 1) • Will go through a maximum before transient equilibrium is achieved.

  31. 2.2 Multi-Component Decays • 2.2 (c) Relationships Among Parent and RA Products • (2) Secular Equilibrium ( 1 << 2 )

  32. 2.2 Multi-Component Decays • 2.2 (c) Relationships Among Parent and RA Products • (2) Secular Equilibrium ( 1 << 2 ) … cont.

  33. 2.2 Multi-Component Decays • 2.2 (d) Non-Equilibrium Cases • (i) If parent is shorter-lived than daughter ( 1 > 2 )

  34. 2.2 Multi-Component Decays • 2.2 (d) Non-Equilibrium Cases • (i) If parent is shorter-lived than daughter ( 1 > 2 ) … cont. Note: If parent is made free of daughter at t=0, then daughter will rise, pass through a maximum ( dN2/dt=0 ), then decays at characteristic 2 .

  35. 2.2 Multi-Component Decays • 2.2 (d) Non-Equilibrium Cases • (i) If parent is shorter-lived than daughter ( 1 > 2 ) … cont.

  36. 2.2 Multi-Component Decays • 2.2 (d) Non-Equilibrium Cases • (ii) If parent is shorter-lived than daughter ( 1 >> 2 )

  37. 2.2 Multi-Component Decays • 2.2 (d) Non-Equilibrium Cases • (ii) If parent is shorter-lived than daughter ( 1 >> 2 ) At large t, extrapolate back to t=0 to get c22N1o and slope=-2

  38. 2.2 Multi-Component Decays • 2.2 (d) Non-Equilibrium Cases • (ii) If parent is shorter-lived than daughter ( 1 >> 2 ) … cont. • Useful Ratio:

  39. 2.2 Multi-Component Decays • 2.2 (d) Non-Equilibrium Cases • (iii) Use of tm for both transit & non-equilibrium analysis Idea: Differentiate original N2 equation to get maximum ( with N2o = 0 )

  40. 2.2 Multi-Component Decays • 2.2 (d) Non-Equilibrium Cases • (iii) Use of tm for both transit & non-equilibrium analysis Idea: Differentiate original N2 equation to get maximum ( with N2o = 0 ) Note: tm =  for secular equilibrium .

  41. 2.2 Multi-Component Decays • 2.2 (e) Many Consecutive Decays: (note: previous N1 & N2 equations are still valid.) H. Bateman gives the solutions for n numbers for pure N1o at t=0. (i.e. N2o = N3o = Nno = 0) Can also be found for N2o , N3o , N4o … Nno 0 . But even more tedious!

  42. 2.2 Multi-Component Decays • 2.2 (f) Branching Decays • Nuclide decaying via more that one mode.

  43. 2.2 Multi-Component Decays • 2.2 (f) Branching Decays • Example: 130Cs has a t1/2 = 30.0 min and decays by + and - emissions. It is found that for every 2 atoms of 130Ba in the products there are 55 atoms of 130Xe. Calculate (t1/2)- and (t1/2)+ .

  44. 2.2 Multi-Component Decays • 2.2 (f) Branching Decays • Example: 130Cs has a t1/2 = 30.0 min and decays by + and - emissions. It is found that for every 2 atoms of 130Ba in the products there are 55 atoms of 130Xe. Calculate (t1/2)- and (t1/2)+ . (t1/2)-= 855 min (t1/2)+ = 31.1 min

  45. Kinetics of Radioactive Decays

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