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Kinetics of Radioactive Decays Decay Expressions Half-Life Average Life First-Order Decays Multi- Component Decays Mixtures – Independent Decays Consecutive & Branching Decays Equilibrium Phenomena Non-Equilibrium Decay/Growth Complications

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Kinetics of First Order Reactions

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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

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2.1 First-Order Decay Expressions 2.1 (a) Statistical Considerations (1905) Note: typo +

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

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2.1 First-Order Decay Expressions Excel Example

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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:

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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.

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2.1 First-Order Decay Expressions 2.1 (c) Lives (i) Half-life: t 1/2 Defined as time taken for initial amount ( N or A ) to drop to half of original value.

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

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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.

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

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2.1 First-Order Decay Expressions 2.1 (c) Lives (iii) Comparing half and average/mean life Why is greater than t 1/2 by factor of 1.44? gives equal weighting to those atoms that survives a long time!

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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

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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:

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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 3.700x10 10 disintegrations per second. SI unit => 1 Becquerel (Bq) = 1 disintegration per second Example (1): Compare 1 mCi of 15 O ( t 1/2 = 2 min ) with 1 mCi of 238 U ( t 1/2 = 4.5x10 9 y ) Use Specific Activity = Bq/g ( activity per g of RA material )

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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 = 1.61x10 12 ion pairs per gram of air. More Later !

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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 3 H with t 1/2 = y.

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2.1 First-Order Decay Expressions 2.1 (e) Units of Radioactivity: Example (3): Calculate W of 1.00 mCi of 14 C with t 1/2 = 5730 y. Example (4): Calculate W of 1.00 mCi of 238 U with t 1/2 = 4.15x10 9 y.

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2.1 First-Order Decay Expressions 2.1 (e) Units of Radioactivity: NucleiA (mCi)t 1/2 (y)W (g)Sp. Act. (Bq/q) 3H3H x x C x x U x x x10 4

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

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2.2 Multi-Component Decays 2.2 (a) Mixtures of Independently Decay Activities Resolution of Decay Curves (i) Binary Mixture ( unknowns 1, 2, initial A 1 & A 2 ) Excel plot

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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 A t versus t ) [Multi-parameter fitting software]

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2.2 Multi-Component Decays 2.2 (b) Relationships Among Parent and RA Products Consider general case of Parent(N 1 )/daughter(N 2 ) in which daughter is also RA. (i) If (2) is stable (ii) If (2) is RA and (3) is stable

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2.2 Multi-Component Decays 2.2 (b) Relationships Among Parent and RA Products N 2 equation (2.8) and its variations.

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2.2 Multi-Component Decays 2.2 (b) Relationships Among Parent and RA Products N 2 equation (2.8) and its variations … cont.

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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:

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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.

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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) A 1 + A 2 (starting with pure 1) Will go through a maximum before transient equilibrium is achieved.

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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) A 1 + A 2 (starting with pure 1) Will go through a maximum before transient equilibrium is achieved.

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

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

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

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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 ( dN 2 /dt=0 ), then decays at characteristic 2.

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

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

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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 c 2 2 N 1 o and slope=- 2

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

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2.2 Multi-Component Decays 2.2 (d) Non-Equilibrium Cases (iii) Use of t m for both transit & non-equilibrium analysis Idea: Differentiate original N 2 equation to get maximum ( with N 2 o = 0 )

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2.2 Multi-Component Decays 2.2 (d) Non-Equilibrium Cases (iii) Use of t m for both transit & non-equilibrium analysis Idea: Differentiate original N 2 equation to get maximum ( with N 2 o = 0 ) Note: t m = for secular equilibrium.

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2.2 Multi-Component Decays 2.2 (e) Many Consecutive Decays: (note: previous N 1 & N 2 equations are still valid.) H. Bateman gives the solutions for n numbers for pure N 1 o at t=0. (i.e. N 2 o = N 3 o = N n o = 0) Can also be found for N 2 o, N 3 o, N 4 o … N n o 0. But even more tedious!

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

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2.2 Multi-Component Decays 2.2 (f) Branching Decays Example: 130 Cs has a t 1/2 = 30.0 min and decays by + and - emissions. It is found that for every 2 atoms of 130 Ba in the products there are 55 atoms of 130 Xe. Calculate (t 1/2 ) - and (t 1/2 ) +.

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2.2 Multi-Component Decays 2.2 (f) Branching Decays Example: 130 Cs has a t 1/2 = 30.0 min and decays by + and - emissions. It is found that for every 2 atoms of 130 Ba in the products there are 55 atoms of 130 Xe. Calculate (t 1/2 ) - and (t 1/2 ) +. (t 1/2 ) - = 855 min (t 1/2 ) + = 31.1 min

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Kinetics of Radioactive Decays Decay Expressions Half-Life Average Life First-Order Decays Multi- Component Decays Mixtures – Independent Decays Consecutive & Branching Decays Equilibrium Phenomena Non-Equilibrium Decay/Growth Complications

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