Decay Kinetics - UNLV Radiochemistry
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Transcript Decay Kinetics - UNLV Radiochemistry
Radioactive Decay Kinetics
•
Outline
Radioactive decay
kinetics
Basic decay
equations
Utilization of
equations
Mixtures
Equilibrium
Branching
Natural radiation
Dating
3-1
Introduction
• Number of radioactive nuclei that decay in a sample
decreases with time
Exponential decrease
Independent of P, T, mass action
Conditions associated with chemical kinetics
* Electron capture and internal conversion can be
affected by conditions
Specific for isotope
Irreversible
• Decay of given radionuclide is random
Statistical
Evaluate behavior of group
3-2
Basic Decay Equations
• Decay is 1st order
Rate proportional to amount of parent
isotope
Equal to the rate of isotope
disintegration
Proportional to number of radioactive
nuclei
* rate of decay=decay constant*#
radioactive nuclei
Decay constant is average decay
probability per nucleus for a unit time
3-3
Represented by l
Basic decay equations
• The radioactive process is a subatomic change within
the atom
• The probability of disintegration of a particular atom
of a radioactive element in a specific time interval is
independent of its past history and present
circumstances
• The probability of disintegration depends only on the
length of the time interval.
Probability of decay: p=lDt
Probability of not decaying: 1-p=1- lDt
3-4
StatisticsofofRadioactive
Radioactive Decay
Statistics
Decay
1-p=1-lDt=probability that atom will survive Dt
(1-l Dt)n=probability that atom will survive n intervals of t
nDt=t, therefore (1- l Dt)n =(1- l t/n)n
Since limn∞(1+x/n)n=ex, (1- lt/n)n=e-lt, the limiting value.
Considering No atoms, the fraction remaining unchanged
after time t is N/No= e-lt
N=Noe-lt
where l is the decay constant
3-5
Radioactivity as Statistical
Phenomenon
• Binomial Distribution for Radioactive Disintegrations
probability W(m) of obtaining m disintegrations in
time t from No original radioactive atoms
No!
W (m)
p m (1 p) N o m
( N o m )! m!
probability of atom not decaying in time t, 1-p, is
(N/No)=e-lt, where N is number of atoms that survive
in time interval t and No is initial number of atoms
• Time Intervals between Disintegrations
probability of time interval having value between t
3-6
and t+d:
P(t )dt N ole N lt dt
o
• Average Disintegration Rate
W (r )
where 1-p=q
n!
p r q n r
( n r )! r !
np r 0 rW ( r ) r
r n
for radioactive disintegration--if n=No and p=1e-lt--average number M of atoms disintegrating
in time t is M=No(1-e-lt); for small lt, M=Nolt
and disintegration R=M/t=No l, which
corresponds to -dN/dt=lN
• Expected Standard Deviation
s N o (1 e lt )e lt Me lt
Since in counting practice lt is generally small , s M
• M is number of counts
• Relative error = s-1
3-7
Measured Activity
• In practicality, activity (A)
is used instead of the
number of atoms (N).
• A= clt, m
where c is the detection
coefficient
A=AOe-lt
• Units
Curie
3.7E10 decay/s
* 1 g Ra
Becquerel
1 decay/s
3-8
Half Life and decay constant
Half-life is time needed to decrease
nuclides by 50%
Relationship between t1/2 and l
N/No=1/2=e-lt
ln(1/2)=-lt1/2
ln 2= lt1/2
t1/2=(ln 2)/l
3-9
Half lives
• Large variation in half-lives for different
isotopes
Short half-lives can be measured
Evaluate activity over time
* Observation on order of half-life
Long half-lives
Based on decay rate and sample
* Need to know total amount of nuclide
in sample
* A=ln
A is activity, n is number of nuclei
3-10
Exponential Decay
• Average Life () for a radionuclide
found from sum of times of existence of all
atoms divided by initial number of nuclei
1
No
t
1
t dN l
t 0
1/l=1/(ln2/t1/2)=1.443t1/2=
Average life greater than half life by factor
of 1/0.693
during time 1/l activity reduced to 1/e it’s
initial value
3-11
Lifetime
• Total number of nuclei that decay over time
Dose
Atom at a time
• Couple with Heisenberg uncertainty principle
DE Dt≥h/2p
Dt is
with energy in eV
DE≥(4.133E-15 eV s/2p)/ G
G is decay width
* Resonance energy
t1/2=1 sec, 1.44 s, G=4.56E-16 eV
3-12
Width and energy
• Need very short half-lives for
large widths
• Useful in Moessbauer
spectroscopy
Absorption distribution
is centered around
Eg+DE
emission centered Eg
DE .
• overlapping part of the peaks
can be changed by changing
the temperature of the source
and/or the absorber.
3-13
Equations
• Nt=Noe-lt
N=number of nuclei, l= decay constant,
t=time
Also works for A (activity) or C (counts)
* At=Aoe-lt, Ct=Coe-lt
• A= lN
• 1/l=1/(ln2/t1/2)=1.443t1/2=
3-14
Half-life calculation
Using Nt=Noe-lt
• For an isotope the initial count rate was 890 Bq.
After 180 minutes the count rate was found to
be 750 Bq
What is the half-life of the isotope
750=890exp(-l*180 min)
750/890=exp(-l*180 min)
ln(750/890)= -l*180 min
-0.171/180 min= -l
9.5E4 min-1 =lln2/t1/2
t1/2=ln2/9.5E-4=729.6 min
3-15
Half-life calculation
A=lN
• A 0.150 g sample of 248Cm has a alpha activity of 0.636 mCi.
What is the half-life of 248Cm?
Find A
* 0.636 E-3 Ci (3.7E10 Bq/Ci)=2.35E7 Bq
Find N
* .150 g x 1 mole/248 g x 6.02E23/mole= 3.64E20
atoms
lA/N= 2.35E7 Bq/3.64E20 atoms=6.46E-14 s-1
* t1/2=ln2/l0.693/6.46E-14 s-1=1.07E13 s
* 1.07E13 s=1.79E11 min=2.99E9 h=1.24E8 d
=3.4E5 a
3-16
Counting
A=lN
• Your gamma detector efficiency at 59 keV is 15.5 %.
What is the expected gamma counts from 75
micromole of 241Am?
Gamma branch is 35.9 % for 241Am
C=(0.155)(0.359)lN
t1/2=432.7 a* (3.16E7 s/a)=1.37E10 s
l=ln2/1.37E10 s=5.08E-11 s-1
N=75E-6 moles *6.02E23/mole=4.52E19 atoms
• C=(0.155)(0.359)5.08E-11 s-1*4.52E19 =1.28E8 Bq
3-17
Decay Scheme
3-18
Specific activity
• Activity of a given amount of radionuclide
Use A=lN
Use of carrier should be included
• SA of 226Ra
1 g 226Ra, t1/2= 1599 a
1 g * 1 mole/226 g * 6.02E23 atoms/mole =
2.66E21 atom = N
t1/2=1599 a *3.16E7 s/a = 5.05E10 s
l=ln2/ 5.05E10 s =1.37E-11 s-1
A= 1.37E-11 s-1 * 2.66E21=3.7E10 Bq
3-19
Specific Activity
• 1 g 244Cm, t1/2=18.1 a
1 g * 1 mole/244 g * 6.02E23 atoms/mole = 2.47E21
atom = N
t1/2=18.1 a *3.16E7 s/a = 5.72E8 s
l=ln2/ 5.72E8 s =1.21E-9 s-1
A= 1.21E-9 s-1 * 2.47E21=2.99E12 Bq
• Generalized equation for 1 g
6.02E23/Isotope mass *2.19E-8/ t1/2 (a)
1.32E16/(Isotope mass* t1/2 (a))
3-20
Isotope
14 C
Specific
Activity
t 1/2 a
SA (Bq/g)
5715
1.65E+11
228 Th
1.91E+00
3.03E+13
232 Th
1.40E+10
4.06E+03
233 U
1.59E+05
3.56E+08
235 U
7.04E+08
7.98E+04
238 U
4.47E+09
1.24E+04
237 Np
2.14E+06
2.60E+07
238 Pu
8.77E+01
6.32E+11
239 Pu
2.40E+04
2.30E+09
242 Pu
3.75E+05
1.45E+08
244 Pu
8.00E+07
6.76E+05
241 Am
4.33E+02
1.27E+11
243 Am
7.37E+03
7.37E+09
244 Cm
1.81E+01
2.99E+12
248 Cm
3.48E+05
1.53E+08
3-21
10
10
10
10
13
14
10
3
10
2
10
1
10
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10
-7
y = m2 /M0
C
12
11
Value
Error
m2
5.7831e+13
1.3169e+11
Chisq
R
6.7326e+22
0.99996
NA
NA
10
10
10
10
10
10
10
9
8
SA (Ci/g)
SA (Bq/g)
10
14
7
6
5
4
1000
1
100
10
4
10
6
t 1/2 (a)
10
8
10
10
3-22
Specific Activity
• Activity/mole
N=6.02E23
• SA (Bq/mole) of 129I, t1/2=1.57E7 a
t1/2=1.57E7 a *3.16E7 s/a = 4.96E14 s
l=ln2/ 4.96E14 s =1.397E-15 s-1
A= 1.397E-15 s-1 *6.02E23=8.41E8 Bq
• Generalized equation
SA (Bq/mole)=1.32E16/t1/2 (a)
3-23
Specific Activity
Isotope
SA
(Bq/mole)
t 1/2 a
3
H
12.3
1.07E+15
14
C
5715
2.31E+12
22
Na
2.6
5.08E+15
55
Fe
2.73
4.84E+15
228
Th
1.91E+00
6.91E+15
232
Th
1.40E+10
9.43E+05
233
U
1.59E+05
8.30E+10
235
U
7.04E+08
1.88E+07
238
U
4.47E+09
2.95E+06
237
Np
2.14E+06
6.17E+09
238
Pu
8.77E+01
1.51E+14
239
Pu
2.40E+04
5.50E+11
242
Pu
3.75E+05
3.52E+10
244
Pu
8.00E+07
1.65E+08
241
Am
4.33E+02
3.05E+13
243
Am
7.37E+03
1.79E+12
244
Cm
1.81E+01
7.29E+14
3-24
10
15
SA (Bq/mole)
SA (Bq/mole)
10
13
10
11
10
9
y = m2 /M0
10
7
10
5
1
Value
Error
m2
1.3204e+16
1.9321e+12
Chisq
R
3.5919e+25
1
NA
NA
100
10
4
10
t 1/2 (a)
6
10
8
10
10
3-25
SA with carrier
• 1E6 Bq of 152Eu is added to 1 mmole Eu.
Specific activity of Eu (Bq/g)
Need to find g Eu
1E-3 mole *151.96 g/mole = 1.52E-1 g
=1E6 Bq/1.52E-1 g =6.58E6 Bq/g
* =1E9 Bq/mole
• What is SA after 5 years
t1/2=13.54 a
= 6.58E6*exp((-ln2/13.54)*5)=
* 5.09E6 Bq/g
3-26
Lifetime
• Atom at a time chemistry
• 261Rf lifetime
Find the lifetime for an atom of 261Rf
t1/2 = 65 s
=1.443t1/2
=93 s
• Determines time for experiment
• Method for determining half-life
3-27
Mixtures of radionuclides
• Composite decay
Sum of all decay particles
Not distinguished by energy
• Mixtures of Independently Decaying Activities
if two radioactive species mixed together,
observed total activity is sum of two separate
activities:
A=A1+A2=c1l1N1+c2l2N2
any complex decay curve may be analyzed into
its components
Graphic analysis of data is possible
3-28
Can determine initial
concentration and half-life
of each radionuclide
3-29
l=0.554
t1/2=1.25 hr
y = m1*exp(-m2*M0)+m3*exp(-m...
total Bq
10
4
Value
Error
m1
10000
0.00065416
m2
m3
0.55452
2000
5.3036e-08
0.00069206
m4
0.066906
3.3669e-08
Chisq 3.7138e-07
NA
R
1
l=0.067
t1/2=10.4 hr
NA
1000
100
0
5
10
15
T (hr)
20
25
3-30
Parent – daughter decay
• Isotope can decay into
radioactive isotope
Uranium decay series
Lower energy
Different properties
A
Z
Spin
Parity
• For a decay parent -> daughter
Rate of daughter
formation dependent upon
parent decay ratedaughter decay rate
3-31
Parent - daughter
• For the system 1 decays into 2
dN 2
l1 N1 l2 N 2
dt
• Rearranging gives dN2 + l2 N2dt l1N1dt
• Solve and substitute for N1 using N1t=N1oe-lt
dN 2 + l2 N 2 dt l1 N1o e l1t dt
Linear 1st order differential equation
Solve by integrating factors
• Multiply by el2t
el2t dN 2 + l2 N 2el2t dt l1 N1o e( l2 l1 )t dt
d ( N 2el2t ) l1 N1o e( l2 l1 )t dt
3-32
Parent-daughter
• Integrate from t 0->t
t
t
( l2 l1 ) t
l
N
e
l2 t
1 1o
N
e
0 2
0 l2 l1
N 2e
l2 t
N 2o
l1
l2 l1
N1o (e ( l2 l1 )t 1)
• Multiply by e-l2t and solve for N2
N 2 (t )
l1
l2 l1
N1o (e l1t e l2t ) + N 2o e l2t
Growth of daughter from parent
Decay of initial daughter
3-33
Parent daughter
• Can solve equation for activity from A=lN
l1l2
A2
N1o (e l t e l t ) + A2 o e l t
l2 l1
• Find maximum daughter activity based on
dN/dt=0
l1e l t l2e l t
l2
• Solve for t
ln( )
l1
t
(l2 l1 )
1
2
1
2
2
• For 99mTc (t1/2=6.01 h) from 99Mo (2.75 d)
lTc=2.8 d-1, lMo=0.25 d-1
0.94 d
3-34
Half life relationships
• No daughter decay
No activity of daughter
Number of daughter atoms due to parent decay
• No Equilibrium
N 2 N1o (1 e l1t )
if parent is shorter-lived than daughter
(l1l2), no equilibrium attained at any time
daughter reaches maximum activity when
l1N1=l2N2
All parents decay, then decay is based on
daughter
3-35
Half life relationships
• Transient equilibrium
Parent half life greater than 10 x daughter
half life
(l1 < l2)
• Parent daughter ratio becomes constant over
time
As t goes toward infinity
e l2t e l1t ; N 2 o e l2t
0
N2
l1
l2 l1
N1o e
l1t
N1 N1o e
l 1t
N2
l1
N1 l2 3-36
l1
3-37
Half life relationship
• Secular equilibrium
Parent much longer half-life than daughter
1E4 times greater
(l1 << l2)
Parent activity does not measurably
decrease in many daughter half-lives
N2
l1
N 2 l1
N1 l2 l1
N1 l2
N 2l2 N1l1
A2 A1
3-38
3-39
Many Decays
dN3
l 2N2 l3N3
dt
• Bateman solution
• Only parent present at time 0
Nn C1e
C1
l1 t
+ C2 e
l 2t
+ Cn e
l nt
l1l 2 .....l (n1)
(l 2 l1 )(l 3 l1 )...(l n l1 )
C2
N1o
l1l 2 .....l(n1)
(l1 l 2 )(l3 l2 )...(ln l2 )
N1o
3-40
Branching decay
• Branching Decay
partial decay constants must be considered
N
A has only one half life l l ; 1
i 1
i
t1/ 2
N
t
i
i 1 1/ 2
if decay chain branches and two branches are
later rejoined, the two branches are treated as
separate chains
production of common member beyond
branch point is sum of numbers of
atoms formed by the two paths
• Branching ratio is based on relative constants
li/lt
1
3-41
Branching Decay
• For a branching decay of alpha and beta
At=Aa+Ab
A=lN, so
* ltN =laN+lbN; lt=la+lb
1=Aa /At +Ab /At ; 1=la /lt +lb /lt
• Consider 212Bi, what is the half life for each decay mode?
Alpha branch 36 %, beta branch 64 %
t1/2=60.55 min
lt=0.0114; 0.36=la /lt; 0.36=la /0.0114; la=0.0041
t1/2 alpha = 169 min
lt=la+lb; 0.0114=0.0041+lb; 0.0073=lb
t1/2 beta = 95.0 min
3-42
3-43
Cross Sections
The probability of a nuclear process is generally
expressed in terms of a cross section s that has the
dimensions of an area.
• Originates from simple picture that probability for reaction
between nucleus and impinging particle is proportional to the
cross-sectional target area presented by the nucleus
doesn’t hold for charged particles that have to
overcome Coulomb barriers or for slow neutrons
• Total cross section for collision with fast particle is never
greater than twice the geometrical cross-sectional area of the
nucleus
10-24 cm2=1 barn
3-44
For a beam of particles striking a thin target--one in which
the beam is attenuated only infinitesimally--the cross
section for a particular process is defined:
Ri Inxs i
When a sample is embedded in a uniform flux of particles
incident on it from all direction, such as in a nuclear
reactor, the cross section is defined:
Ri Ns i
Ri= # of processes of type under consideration occurring in the target
per unit time
I= # of incident particles per unit
=flux of particles/cm2/sec
time
N=number of nuclei contained in
n= # of nuclei/cm3
sample
3-45
x=target thickness (cm)
Production of radionuclides
• N1=N0s
s=cross section
=neutron flux
• To full consider produced nuclei
N1=N0s/l1 (1-exp-(l1t))
t=time of irradiation
(1-exp-(l1t)) gives maximum level percent
half life
%
1
50
2
75
3
87.5
4
93.75
5
96.875
3-46
Natural Radionuclides
• 70 naturally occurring radioactive isotopes
Mainly decay from actinides
Tritium
14C
40K
• 70 kg “reference man,”
4400 Bq of 40K
3600 Bq of 14C
• US diet
1 pCi/day of 238U, 226Ra, and 210Po
• air
~ 0.15 pCi/L of 222Rn
• earth’s crust
~10 ppm and ~4 ppm of the radioelements Th and U.
• interior heat budget of the planet Earth is dominated by the
contributions from the radioactive decay of U, Th, and K
3-47
Environmental radionuclides
• primordial nuclides that have survived since the time
the elements were formed
t1/2>1E9 a
Decay products of these long lived nuclides
40K., 87Rn, 238U, 235U, 232Th
• cosmogenic are shorter lived nuclides formed
continuously by the interaction of comic rays with
matter
3H., 14C, 7Be
14N(n, 1H )14C (slow n)
14N(n, 3H )12C (fast n)
• anthropogenic are nuclides introduced into the
environment by the activities of man
Actinides and fission products
3-48
14C and 3H
Dating
• Radioactive decay as clock
Based on Nt=Noe-lt
Solve for t
Nt
No
ln
ln
No
Nt
t
l
l
• N0 and Nt are the number of radionuclides present at
times t=0 and t=t
Nt from A = λN
• t the age of the object
Need to determine No
For decay of parent P to daughter D total
number of nuclei is constant
D(t ) + P(t ) Po
3-49
Dating
Dt
t ln( 1 + )
l
Pt
1
• Pt=Poe-lt
• Measuring ratio of daughter to parent atoms
no daughter atoms present at t=0
that they are all due to the parent decay
none have been lost during time t
• A mineral has a 206Pb/238U =0.4. What is the
age of the mineral?
t=(1/(ln2/4.5E9))ln(1+0.4)
2.2E9 years
3-50
Dating
•
14C
dating
Based on constant formation of 14C
No longer uptakes C upon organism
14
death
Ceq
1
)
• 227 Bq 14C /kgC at equilibrium t l ln( 14C
sample
• What is the age of a wooden sample with 0.15
Bq/g C?
t=(1/(ln2/5730 a))*ln(0.227/0.15)=3420 a
3-51
Dating
• Determine when Oklo reactor operated
Today 0.7 % 235U
Reactor 3.5 % 235U
Compare 235U/238U (Ur) ratios and use Nt=Noe-lt
e -l235t
U r (t) U r (o) -l238t U r (o)e (- l235t + l238t )
e
U r (t)
ln
t (-l235 + l238 )
U r (o)
U r (t)
ln
U r (o)
t
(-l235 + l238 )
7.25E - 3
ln
3.63E - 2
t
1.94 E 9 years
(-9.85E - 10 + 1.55E 10)
3-52
Questions
• Make excel sheets to calculate
Mass or mole to activity
Calculate specific activity
Concentration and volume to activity
Determine activity for counting
Parent to progeny
Daughter and granddaughter
* i.e., 239U to 239Np to 239Pu
3-53