### Lecture 6

```Lecture 6 Radioactive Isotopes
Definitions and types of decay
Derivation of decay equations
Half lives and mean lives
Secular Equilibrium
E & H Chpt 5
The chart of the nuclides - decay
Q. 230Th90 How many protons / neutrons?
Full Chart of the Nuclides
Valley of Stability
1:1 line
For 230Th N/P = 1.55
Definitions and Units
Daughter – The product of decay
Decay Chain – A series of sequential decays
from one initial parent
Decay is independent of chemistry and T and P.
Decay is only a property of the nucleus (see Chart of Nuclides)
Types of Decay
DP
DN
-2
+1
-2
-1
Alpha
Beta
a
b
He2+
e-
Gamma
g
“packets of excess energy”
Measurements
DAtomic Wt.
-4
0
(n → P+ + e-)
The chart of the nuclides – decay pathways
b decay X
X
a decay
Mathematical Formulation of Decay
Decay Activity (A) = decays per time
(e.g. minutes (dpm) or second (dps))
A=lN
l = decay constant (t-1)
N = # of atoms or concentration (atoms l-1)
Remember 1 mol = 6.02 x 1023 atoms
Units:
Becquerel (Bq) = 1 dps (the official SI unit)
Curie (Ci) = 3.7 x 1010 Bq = Activity of 1 gram of 226Ra
Named after Pierre Curie
See this link for the history:
http://www.orau.org/ptp/articlesstories/thecurie.htm
Decay Equations (essential math lessons)
Decay is proportional to the # of atoms present (first order)
dN/dt = - lN = AN
where
N=
l=
the number of atoms of the radioactive substance present at time t
the first order decay constant (time-1)
The number of parent atoms at any time t can be calculated as follows.
The decay equation can be rearranged and integrated over a time interval.
N dN
t
N dN
N
=
ln N = ln N - ln N o
 N o N  – l 0 dt
No N
No
ò
ò
where No is the number of parent atoms present at time zero. Integration leads to
ln
or
N
 – lt
No
N  N oe
lt
or
A  A e
lt
Decay Curve
Both N and A decrease exponentially
Half Life
The half life is defined as the time required for half of the
atoms initially present to decay.
N
Al
A 1
=
=
=
N o Ao l Ao 2
After one half life:
From the decay equation
1
– ln   = l t1/2
2
ln (2) = l t1/2
0.693 = l t1/2
so
t1 / 2 
0 .6 9 3
l
Math note:
-ln(1/2) = - (ln 1 – ln 2)
= - ( 0 – ln 2)
= + ln2 = 0.693
Mean Life = Average Life of an Atom
1
t=
No
ò
t=¥
t=0
t dN
= 1 / l  (1/0.693) t1/2
t = 1.44 t1/2
Q. Why is the mean life longer than the half life?
Isotopes used
in Oceanography
U-Th series are shown on the next
page. These tracers have a range
of chemistries and half lives.
Very useful for applications in
oceanography.
transient
Two forms of Helium
3He
2
from beta decay of 3H1 (called
tritium) and primordial from the mantle
3H = 3He + b
1
2
4He
2
the product of alpha decay from
many elements (especially in U-Th series)
How would you expect their distributions
to vary in the ocean?
Example distributions of 3He
from mid-ocean ridge crest
John Lupton (NOAA) et al (various)
Q. Why is the inside of the earth hot?
Q. What is the age of the earth? 6000 years or 4.5 x 109 years
238U
decay products in the ocean
109 y
105 y
24 d
Q. What controls the concentration of 238U in SW?
104 y
1600 y
3d
22 y
U – conservative
Th – particle reactive
Ra – intermediate (like Ca)
Rn = conservative
Pb – particle reactive
Parent-Daughter Relationships
Stable Daughter (B)
A → B e.g. 14C → 15N (stable)
Production of Daughter = Decay of Parent
dN B
dt
 l A N A  l A N A ,o e
 l At
2-box model
Example:
14C
15N
→
(stable)
t1/2 = 5730 years
l = 0.693 / t1/2
A
lA
B
2-box model
A → B →
lA
lB
source
sink
dN B
A
 l A N A  lB N B
dt
lA
mass balance for B
solution:
NB 
AB 
l B ( N A ,0 )
(
lB  lA
l B ( A A ,0 )
lB  l A
(
e
e
 l At
 l At
e
–e
 lB t
 lB t
)
)
solution after assuming NB = 0 at t = 0
B
lB
Three Limiting Cases
1) t1/2(A) > t1/2(B) or lA < lB
2) t1/2(A) = t1/2(B) or lA = lB
3) t1/2(A) < t1/2(B) or lA > lB
one important example:
e.g. 226Ra → 222Rn
1600yrs 3.8 days
Case #1: long half life of parent = small decay constant of parent
AB 
l B ( A A ,0 )
lB  lA
AB
AA
(e
 l At
) l
lB
B
 lA
AA
 l B /( l B  l A )
AA
AB
1
SECULAR EQUILIBRIUM
Activity of daughter
equals activity of
parent!
Are concentrations also equal???
Q. Are concentrations also equal???
AA l A N A
=
=1
AB lB N B
l A N A = lB N B
N A lB
=
N B lA
Example: 226Ra and 222Rn
Secular equilibrium (hypothetical)
t1/2 daughter = 0.8 hr
t1/2 parent = 
Total Activity
(parent+daughter)
Parent
doesn’t change
★
Activity of parent
and daughter equal at
secular equilibrium
daughter
Activity
(log scale)
★
! Daughter grows
in with half life of
the daughter!
t1/2
time (hr)
Example:
Grow in of 222Rn
from 226Ra
Another way to plot
After 5 half lives
activity of daughter =
95% of activity of parent
Example: Rate of grow in
Assume we have a really big wind storm over the ocean so that all the inert gas
222Rn is stripped out of the surface ocean by gas exchange. The activity of the parent
of 222Rn, 226Ra, is not affected by the wind.
Then the wind stops and 222Rn starts to increase (grows in) due to decay.
Q. How many half lives will it take for the activity of 222Rn to equal 50% (and then 95%)
of the 226Ra present?
Answer: Use the following equation to calculate the activity A at time t
(
AA,t = AA,0 1- e
-0.693t/t1/2
)
Why are some zones high (red)?
There is considerable exposure due to artificially produced sources!
Possibly largest contributor is tobacco which
contains radioactive 210Po which emits 5.3 MeV a particles
with an half life of T1/2=138.4days.
Was Litvinenko (a former Russian spy) killed by 210Po?? A case study of 210Po
Toxicity of Polonium 210
Weight-for-weight, polonium's toxicity is around 106 times greater than
hydrogen cyanide (50 ng for Po-210 vs 50 mg for hydrogen cyanide).
The main hazard is its intense radioactivity (as an alpha emitter), which makes it very
difficult to handle safely - one gram of Po will self-heat to a temperature of around 500°C.
It is also chemically toxic (with poisoning effects analogous with tellurium).
Even in microgram amounts, handling 210Po is extremely dangerous, requiring
specialized equipment and strict handling procedures. Alpha particles emitted by
polonium will damage organic tissue easily if polonium is ingested, inhaled, or absorbed
(though they do not penetrate the epidermis and hence are not hazardous if the polonium
is outside the body).
Acute effects
The lethal dose (LD50) for acute radiation exposure is generally about 4.5 Sv. (Sv = Sievert
which is a unit of dose equivalent). The committed effective dose equivalent 210Po
is 0.51 µSv/Bq if ingested, and 2.5 µSv/Bq if inhaled. Since 210Po has an activity of
166 TBq per gram (1 gram produces 166×1012 decays per second),
a fatal 4-Sv dose can be caused by ingesting 8.8 MBq (238 microcurie),
about 50 nanograms (ng), or inhaling 1.8 MBq (48 microcurie), about 10 ng.
One gram of 210Po could thus in theory poison 100 million people of which 50 million
would die (LD50).
Body burden limit
The maximum allowable body burden for ingested polonium is only 1,100 Bq
(0.03 microcurie), which is equivalent to a particle weighing only 6.8 picograms.
The maximum permissible concentration for airborne soluble polonium compounds is
about 10 Bq/m3 (2.7 × 10-10 µCi/cm3). The biological half-life of polonium in
humans is 30 to 50 days. The target organs for polonium in humans are the spleen
and liver. As the spleen (150 g) and the liver (1.3 to 3 kg) are much smaller than the
rest of the body, if the polonium is concentrated in these vital organs, it is a greater
threat to life than the dose which would be suffered (on average) by the whole body
if it were spread evenly throughout the body, in the same way as cesium or tritium.
Notably, the murder of Alexander Litvinenko in 2006 was announced as due to
210Po poisoning. Generally, 210Po is most lethal when it is ingested. Litvinenko was
probably the first person ever to die of the acute α-radiation effects of 210Po , although
Irene Joliot-Curie was actually the first person ever to die from the radiation effects of
polonium (due to a single intake) in the late 1950s. It is reasonable to assume that
many people have died as a result of lung cancer caused by the alpha emission of
polonium present in their lungs, either as a radon daughter or from tobacco smoke.
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