kinetics of radioactive decays decay expressions half-life average life first-order decays multi-...
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Kinetics of Radioactive Decays
Decay Expressions
Half-LifeAverage Life
First-Order Decays
Multi-Component
Decays
Mixtures – Independent
Decays
Consecutive & Branching
Decays
Equilibrium Phenomena
Non-Equilibrium
Decay/GrowthComplications
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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)
𝑁=𝑁 𝑜 ∙𝑒−𝜆 ∙𝑡
n
n
t
1 remaining atoms of Prob. Note: typo “+”
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2.1 First-Order Decay Expressions 2.1 (b) Decay Expressions:
(i) N-Expression𝑅𝑎𝑡𝑒𝑜𝑓 𝐷𝑒𝑐𝑎𝑦=−
𝑑𝑁𝑑𝑡
𝑁=𝑁 𝑜 ∙𝑒−𝜆 ∙𝑡
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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: ffEArcps
24
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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
N A = c NWhere: c = detection coeff.
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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.
𝑡1 /2=ln 2𝜆
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2.1 First-Order Decay Expressions
Note: What is N after x half lives?
x
oN
N
2
1
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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
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!
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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 3.700x1010 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 )
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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 = 1.61x1012 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 3H with t1/2 = 12.26 y .
L
MW
W
t
2lnA
1/2
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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 14C with t1/2 = 5730 y .
Example (4): Calculate W of 1.00 mCi of 238U with t1/2 = 4.15x109 y .
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2.1 First-Order Decay Expressions
2.1 (e) Units of Radioactivity:
Nuclei A (mCi) t1/2 (y) W (g) Sp. Act. (Bq/q)
3H 1.00 12.26 1.03x10-7 3.59x1014
14C 1.00 5730 2.24x10-4 1.65x1011
238U 1.00 4.51x109 3.00x103 1.23x104
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2.2 Multi-Component Decays 2.2 (a) Mixtures of Independently Decay Activities
tn
i
oit
ieAA
1
ii
n
iit NcA
1
to eNcN
dt
dNA
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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 A1 & A2 )
totot eAeAA 21
21
tott eAA
22)(
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 At versus t ) [Multi-parameter fitting software]
totot eAeAA 21
21
totot eAeAA 21
21
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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
32121 NNN
22112
11111
dt
dN :Daughter
dt
dN- :Parent 1
NN
eNNN to
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2.2 Multi-Component Decays
2.2 (b) Relationships Among Parent and RA Products N2 equation (2.8) and its variations.
)8.2()( 22121
12
12
totto eNeeNN
)7.2(011122
2 to eNNdt
dN
22112
dt
dNNN
to eNN 111
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2.2 Multi-Component Decays 2.2 (b) Relationships Among Parent and RA Products N2 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:
)8.2()( 22121
12
12
totto eNeeNN
(2.10) )(
1
12
2
1
N
N
to eNN
11
12
12
to eNN 111
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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
)8.2()( 22121
12
12
totto eNeeNN
(2.11b) )(
2
12
2
1
A
A
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) A1 + A2 (starting with pure 1)
Will go through a maximum before transient equilibrium is achieved.
)8.2()( 22121
12
12
totto eNeeNN
22211121 NcNcAAAt
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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) A1 + A2 (starting with pure 1) Will go through a maximum before transient equilibrium is achieved.
to
to
tot
eNcA
eNcA
cceNA
1
1
1
112
1222
1111
12
22111
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2.2 Multi-Component Decays 2.2 (c) Relationships Among Parent and RA Products
(2) Secular Equilibrium ( 1 << 2 )1
12
2
1 )(
N
N
22
121
2
1 )(
c
c
A
A
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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 )
)8.2()( 22121
12
12
totto eNeeNN
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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.
)8.2()( 22121
12
12
totto eNeeNN
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 .
)14.2(112
122
2
oot NNeN
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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.
)14.2(112
122
2
oot NNeN
to
too
eNcA
eNNcA
NcA
1
2
1111
112
12222
2222
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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 )
t largeat 21112
1222
2
too
t eNNcA
tot eNcA 2
122
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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 )
tot eNcA 2
122
At large t, extrapolate back to t=0 to get c22N1
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:
12/1
22/1
2
1
22
11
122
111
t
t
c
c
c
c
Nc
Nco
o
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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 )
1
2
12
1
2
ln1
:with
12
m
t
t
e m
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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 .
)8.2()( 22121
12
12
totto eNeeNN
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2.2 Multi-Component Decays
2.2 (e) Many Consecutive Decays: (note: previous N1 & N2 equations are still valid.) etc4321
4321 NNNN
done! becan but tedious,& longVery
(2.8)equation into Subdt
dN :Nfor Now 3322
33 NN
H. Bateman gives the solutions for n numbers for pure N1o at t=0. (i.e. N2
o = N3o = Nn
o = 0)
... etc
...
...C
...
...C :where
N :Solutions
122321
1212
111312
1211
1
o
n
n
o
n
n
n
i
tin
N
N
eC n
Can also be found for N2o , N3
o , N4o … Nn
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.
(2.17) CBt
toAA
CBeNN
(2.18) 2ln
2/1CB
At
CBtttt
2/12/12/1
111
(2.19) C
B
tC
B
N
N
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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)+ .
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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
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Kinetics of Radioactive Decays
Decay Expressions
Half-LifeAverage Life
First-Order Decays
Multi-Component
Decays
Mixtures – Independent
Decays
Consecutive & Branching
Decays
Equilibrium Phenomena
Non-Equilibrium
Decay/GrowthComplications