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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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slide3

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

slide4

2.1 First-Order Decay Expressions

  • 2.1 (a) Statistical Considerations (1905)

Note: typo “+”

slide5

2.1 First-Order Decay Expressions

  • 2.1 (b) Decay Expressions:
    • (i) N-Expression
slide7

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:

slide8

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

slide9

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.
slide10

2.1 First-Order Decay Expressions

Note: What is N after x half lives?

slide11

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.
slide12

2.1 First-Order Decay Expressions

  • 2.1 (c) (ii) Average/Mean Life:  (common usage in spectroscopy)

slide13

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!

slide14

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

slide15

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:
slide16

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 )
slide17

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 !
slide18

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 .
slide19

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 .
slide20

2.1 First-Order Decay Expressions

  • 2.1 (e) Units of Radioactivity:
slide21

2.2 Multi-Component Decays

  • 2.2 (a) Mixtures of Independently Decay Activities
slide22

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

slide23

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]
slide24

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
slide25

2.2 Multi-Component Decays

  • 2.2 (b) Relationships Among Parent and RA Products
  • N2equation (2.8) and its variations.
slide26

2.2 Multi-Component Decays

  • 2.2 (b) Relationships Among Parent and RA Products
  • N2equation (2.8) and its variations … cont.
slide27

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:
slide28

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 .

slide29

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.
slide30

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.
slide31

2.2 Multi-Component Decays

  • 2.2 (c) Relationships Among Parent and RA Products
    • (2) Secular Equilibrium ( 1 << 2 )
slide32

2.2 Multi-Component Decays

  • 2.2 (c) Relationships Among Parent and RA Products
    • (2) Secular Equilibrium ( 1 << 2 ) … cont.
slide33

2.2 Multi-Component Decays

  • 2.2 (d) Non-Equilibrium Cases
    • (i) If parent is shorter-lived than daughter ( 1 > 2 )
slide34

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 .

slide35

2.2 Multi-Component Decays

  • 2.2 (d) Non-Equilibrium Cases
    • (i) If parent is shorter-lived than daughter ( 1 > 2 ) … cont.
slide36

2.2 Multi-Component Decays

  • 2.2 (d) Non-Equilibrium Cases
    • (ii) If parent is shorter-lived than daughter ( 1 >> 2 )
slide37

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

slide38

2.2 Multi-Component Decays

  • 2.2 (d) Non-Equilibrium Cases
    • (ii) If parent is shorter-lived than daughter ( 1 >> 2 ) … cont.
    • Useful Ratio:
slide39

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 )

slide40

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 .

slide41

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!

slide42

2.2 Multi-Component Decays

  • 2.2 (f) Branching Decays
    • Nuclide decaying via more that one mode.
slide43

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)+ .
slide44

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