Kinetics of Radioactive Decays Decay Expressions Half-Life Average Life First-Order Decays Multi- Component Decays Mixtures – Independent Decays Consecutive

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

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.

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

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 “+”

2.1 First-Order Decay Expressions 2.1 (b) Decay Expressions:

(i) N-Expression𝑅𝑎𝑡𝑒𝑜𝑓 𝐷𝑒𝑐𝑎𝑦=−

𝑑𝑁𝑑𝑡

𝑁=𝑁 𝑜 ∙𝑒−𝜆 ∙𝑡

2.1 First-Order Decay Expressions

Excel Example


(ii) A-Expression Define: A = Activity (counts per second or disintegrations per second)

For fixed geometry: ffEArcps

24


(ii) A-Expression Define: A = Activity (counts per second or disintegrations per second)

A

N A = c NWhere: c = detection coeff.

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


Note: What is N after x half lives?

x

oN

N

2

1


(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.

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


(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!

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

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:

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 )

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 !

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

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 .


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

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


Resolution of Decay Curves (i) Binary Mixture ( unknowns 1 , 2 , initial A1 & A2 )

totot eAeAA 21

21

tott eAA

22)(

Excel plot


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

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

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

2.2 Multi-Component Decays 2.2 (b) Relationships Among Parent and RA Products N2 equation (2.8) and its variations … cont.


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

2.2 Multi-Component Decays 2.2 (c) Equilibrium Phenomena (Transient & Secular): Parent longer lived


(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 .

2.2 Multi-Component Decays 2.2 (c) Equilibrium Phenomena (Transient & Secular): Parent longer lived


(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

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

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

2.2 Multi-Component Decays 2.2 (c) Relationships Among Parent and RA Products

(2) Secular Equilibrium ( 1 << 2 ) … cont.

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


(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


(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


(ii) If parent is shorter-lived than daughter ( 1 >> 2 )

t largeat 21112

1222

2

too

t eNNcA

tot eNcA 2

122


(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


(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


(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


(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


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!

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

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)+ .

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

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

Documents

Kinetics of Radioactive Decays Decay Expressions Half-Life Average Life First-Order Decays Multi- Component Decays Mixtures – Independent Decays Consecutive