Radioactive Isotope DecayChain
Calculations
Here are the calculations used to get
the figures shown in the associated web page
How
Much Astatine Is Present in
a Uranium Ore Rock? You are encouraged to
check these calculations and the assumptions on which they are based.
If you find anything questionable, please let me know! All calculations
have a precision of two significant digits or less, so quantities are
rounded accordingly, and quantities or conditions having an effect of
less than 1%
are ignored.
Decay Rate of Uranium in Atoms per
Second
Suppose you have a 1gram sample of uranium. How many atoms are
decaying each second?
All natural
uranium
is 99.3%
^{238}U. For now, we'll ignore
the
remaining 0.7% that's mostly
^{235}U.
^{238}U has a
halflife
of 4.5 billion years. Let's convert that to seconds:
(4.5 x 10
^{9} years) x (365 days/year) x
(24 hours/day) x (60 minutes/hour) x (60 seconds/minute) = 1.4 x 10
^{17} seconds
The get the
mean
lifetime of a radioactive isotope, often designated τ (Greek letter tau), divide the
halflife by the natural logarithm of 2:
mean lifetime τ = halflife
/
ln(2) = (1.4 x 10
^{17} seconds) / 0.69 = 2.0 x 10
^{17} seconds
The
decay
rate, often designated λ
(Greek letter lambda), is the fraction of the total mass that decays in
one unit of time. It is equal to the inverse of the mean lifetime:
decay rate λ = 1/τ = 1 / (2.0 x 10
^{17} seconds) = (5.0 x 10
^{18} per second)
This is the fraction of the uranium that decays in one second, so a
mass of 1.0 gram of uranium decays at a rate of 5.0 x 10
^{18}
gram per second. To convert this decay rate from grams per second to
atoms per second, you use the fact that the atomic weight of
^{238}U is 238 grams per mole, and mole is
6.0 x 10
^{23}
atoms (the
Avogadro
constant):
(5.0 x 10
^{18} gram / second) x (1.0
mole / 238 grams) x (6.0 x 10
^{23} atoms
/ mole) = 1.3 x 10
^{4}
atoms/second
This is
13,000 atoms/second
or 13,000
becquerels
(13 kBq)
Onethousandth of a gram (one milligram) of uranium decays at
onethousandth of this rate, or 13 atoms per second.
An 8milligram sample of uranium decays at a rate of about
100 atoms
per second. I chose this amount for my first decaychain example
because it's
easy to visualize and discuss this rate.
The
^{235}U and
^{234}U
present in the sample also contribute significantly to the total number
of atoms decaying per second. For details, see
Decay Rate of One Gram of Uranium.
Imagine that you put 100 radioactive atoms in a box and observe them
decay, one by one, until they are all gone. If you add up the lifetimes
of the 100 individual atoms and divide the sum by 100, you get the
mean lifetime, also known as the
average lifetime. On the other
hand, the halflife is the
median
lifetime  half of the atoms live longer and half live shorter
than this time. This is the time at which 50 of the atoms
have decayed and 50 remain in the box.
Why are the average and median lifetimes different? There are always a
few Methuselah atoms that live much longer than
usual, purely by chance. These atoms pull up the
average, but they have no effect on
the
median. Meanwhile, the
shortest time that an atom can live is zero. There are no atoms with
negative lifetimes that could pull the average down. So the average
lifetime is somewhat longer than the median.
If you multiply the average lifetime by the natural logarithm of 2 (the
number 0.6931), you get the halflife. You can read more about
halflife,
mean lifetime, and exponential decay in any algebra II book or the
Wikipedia article on
halflife.
Equilibrium Concentration of Radium in
UraniumBearing Rock
In any rock containing uranium, all of
the isotopes in the decay chain of
^{238}U
accumulate to their respective
equilibrium concentrations. Each isotope's concentration is the
halflife of
the isotope relative to the halflife of
^{238}U,
subject to
adjustment for branching in the decay chain.
The halflife of uranium238 is 4.5 x 10
^{9}
years and the halflife of radium226 is 1,602 years, and there is no
significant branching in the chain from
^{238}U
to
^{226}Ra.
 Atoms of radium as a fraction of the atoms of uranium:
1,602 / (4.5 x 10^{9}) = 3.6
x 10^{7}
 Atoms of uranium as a multiple of the atoms of radium: (4.5
x 10^{9}) / 1,602 = 2.8 x 10^{6} = 2.8 million
Equilibrium Concentration of Astatine in
Old Uranium Rock
Astatine218 is
in the decay chain of
^{238}U and has a
halflife of 1.5 seconds.
However,
^{218}At is produced at a
branch
in the decay chain. Polonium218 can decay into either
^{214}Pb
with a
probability of 99.98% or
^{218}At with
a probability of 0.02%.
The number of
^{218}At atoms as a
fraction of the number of
^{238}U atoms
is the ratio of their respective halflives
multiplied by the probability of decay into
^{218}At
from the main
branch:
(0.0002) x (1.5 seconds) / (1.4 x 10
^{17}
seconds) =
2.1 x 10^{21}
For a rock sample containing 1.0 gram of uranium, the number of
^{238}U
atoms is:
(1.0 grams uranium) x (mole / 238 grams) x (6 x 10
^{23}
atoms / mole) = 2.5 x 10
^{21}
uranium atoms
This ignores the small amount of
^{235}U
present in natural uranium.
The number of astatine atoms in the rock is:
(2.5 x 10
^{21}) uranium atoms x
(2 x 10
^{21} astatine atom / uranium
atom) =
5 astatine
atoms
The direct parent of astatine218, polonium218, is produced at a rate
of
13,000 atoms per second
and
also decays at the same rate, like all isotopes in the main branch. Of
the atoms produced by this decay of
polonium218, only 0.0002 x 13,000 = 2.6 are astatine218
atoms. Therefore, astatine is produced at
a rate of
2.6 atoms per second
and also decays at at the same rate.
The average lifetime of astatine218 is:
t =
halflife /
ln(2)
= (1.5 seconds) / 0.69 =
2.2 seconds
So in a rock containing 1.0 gram of uranium, there are about 5 astatine
atoms in existence at any given time. A few new ones are created and
few decay away every
second. Each astatine atom has an average lifetime of 2.2 seconds. You
can see in the
simulation that the
actual number of atoms varies quite a lot from time to time.
Astatine Isotopes in the Actinium Series
The astatine isotopes
^{219}At and
^{215}At appear in the
actinium
series decay chain. The
grandparent isotope is
^{235}U, with a
halflife of 7.0 x 10
^{8} years.
Converting to seconds:
(7.0 x 10
^{8} years) x (365
days/year) x (24 hours/day) x (60 minutes/hour) x (60
seconds/minute) = 2.2 x 10
^{16}
seconds
In natural uranium, the fraction of the uranium that is
^{235}U
is 0.007; the remaining 0.993 is the main isotope,
^{238}U.
The halflife of
^{219}At is 56
seconds, and it appears in a rare branch of another rare branch of the
decay
chain, having a probability of 0.0138 in the decay of
^{227}Ac
to
^{223}Fr, and subsequently with a
probability of 0.00006 in the decay of
^{223}Fr
to
^{219}At. Thus, the equilibrium
concentration of
^{219}At compared to
^{238}U, the main isotope of natural uranium,
is:
(0.007) x (0.0138) x (0.00006) x 56 seconds / 2.2 x 10
^{16} seconds =
1.5 x 10^{23}
This is less than 1% of the amount of
^{218}At
produced in the decay chain of the
radium
series.
Meanwhile, the halflife of
^{215}At is
0.0001 second, and it appears in a rare branch of the decay chain, with
a probability of 0.0000023 in the decay of
^{215}Po
to
^{215}At. Thus, the equilibrium
concentration of
^{215}At compared to
^{238}U, the main isotope of natural uranium,
is:
(0.007) x (0.0000023) x 0.0001 second / 2.2 x 10
^{16}
seconds =
7.3 x 10^{30}
This amount is truly negligible.
Astatine Isotope in the Thorium Series
In the
thorium
series,
^{216}At is the product of
a minor branch in the decay of
^{216}Po,
with
a probability of 0.00013 and a halflife of 0.00003 second. Therefore,
the equilibrium concentration of
^{216}At
compared to thorium is:
(0.00013) x (0.00003 second) / [ (1.4 x 10
^{10}
years) x (365 days/year) x
(24 hours/day) x (60 minutes/hour) x (60 seconds/minute) = 9 x 10
^{27}
Thus, the resulting amount of
^{216}At
is
negligible.
How Much
Astatine Is Present in Pure Extracted Uranium?
Uranium extracted from its ore is separated chemically from its
decaychain isotopes, so it is pure uranium. Once extracted,
^{238}U starts decaying into its decaychain
isotopes:
^{234}Th,
^{234}Pa,
^{234}U, and so on. The first two
isotopes in
this chain have fairly short halflives, so they accumulate quickly to
their equilibrium
concentrations. However,
^{234}U has a
halflife of a quartermillion years, so it accumulates very slowly
toward its equilibrium concentration and
causes a bottleneck in the chain.
The following table shows the approximate number atoms that exist in
the
radium
series
from
^{238}U to
^{218}At
after 100 years of decay,
starting with 1 gram of pure uranium.
Note (May 2013):
There is an error in this table (not a calculation
error, but an error in the assumptions and starting data). Can you find
the error? I'll fix it later when I have some time. In the meantime, you
can find the correct data here.
Isotope

Halflife, centuries

Average life, centuries

Decay Rate, per century

Atoms present

Atoms decayed

Decay product

^{238}U 
4.5 x 10^{7} 
6.5 x 10^{7} 
1.5 x 10^{8} 
2.5 x 10^{21} 
3.8 x 10^{13} 
^{234}Th 
^{234}Th 
zero

zero

100 %

small amount

3.8 x 10^{13} 
^{234}Pa 
^{234}Pa 
zero

zero

100 %

small amount

3.8 x 10^{13} 
^{234}U 
^{234}U 
2.5 x 10^{3} 
3.6 x 10^{3} 
2.8 x 10^{4} 
1.9 x 10^{13} 
5.3 x 10^{9} 
^{230}Th 
^{230}Th 
7.5 x 10^{2} 
1.1 x 10^{3} 
9.2 x 10^{4} 
2.6 x 10^{9} 
2.4 x 10^{6} 
^{226}Ra 
^{226}Ra 
1.6 x 10^{1} 
2.3 x 10^{1} 
4.3 x 10^{2} 
1.2 x 10^{6} 
5.2 x 10^{4} 
^{222}Ra 
^{222}Ra 
zero

zero

100 %

small amount 
5.2 x 10^{4} 
^{218}Po 
^{218}Po 
zero

zero

100 %

small amount

5.2 x 10^{4}
10

^{214}Po
0.9998
^{218}At 0.0002

So one gram of freshly extracted, pure uranium will produce
10 atoms of
astatine in the first century, mostly in the latter part of
the
century, after the isotopes in the decay chain have had time to
accumulate. So you can't expect a 1gram sample of uranium metal, or a
similar amount of uranium compounds or Fiesta ware, to produce
any astatine in the first decade,
not even
one atom. Even
if, by chance, an astatine atom is created, it exists for only about 2
seconds.
Here are the details about the contents of the table.
Isotope  Lists the elements
in the decay chain leading up to astatine.
Halflife, centuries  The
halflife of the isotope, in centuries. For several of these isotopes,
the halflife is so short, it is essentially zero on a century time
scale. These isotopes are treated as decaying instantly.
Average life, centuries  The
average life, in centuries, for each atom of the isotope, calculated as
the halflife divided by the natural logarithm of 2.
Decay Rate, per century
 The fraction of the atoms that decay in a century, calculated
as the inverse of the average life.
Atoms present  The
average number of atoms present
during the century of decay, starting with 1 gram of
^{238}U in the first row. This number becomes
more difficult to calculate as you go down the chain because the number
of
atoms is not a constant, but grows from zero as the isotope starts to
accumulate over the century. For
^{234}U,
the average amount is calculated as onehalf the total number of
^{238}U atoms produced over the century,
because
^{234}U starts at zero and
grows
in a straight line as
^{238}U feeds the
supply at a constant rate, and a negligible fraction decays into the
next isotope in the chain during that time. I used this same estimation
method at
subsequent steps in the chain, even though it is an overestimate of
the true average amount, because the lower isotopes in the chain
accumulate more slowly
at first, not in a straight line.
Atoms decayed  The
number of atoms of the isotope that decay into the next isotope in the
chain over the course of a century, calculated as the
Atoms present multiplied by the
Decay rate, per century.
Decay product  The next
isotope in the chain produced by decay, which is the isotope that
begins the next row. In the case of the last row,
^{218}Po
decays into two isotopes,
^{214}Po with
a probability of 99.98% and
^{218}At
with a probability of 0.02%.
No doubt an
exact
formula, a yearbyyear computer simulation, or a
yearbyyear spreadsheet could be used to calculate the amount of
astatine more
accurately. This is left as an exercise to the reader. (Please send me
your results if you do it!) The foregoing
table overestimates the production of astatine, so I can safely say
that
no more than 10 atoms of
astatine
are produced by 1 gram of uranium in a century.
Total Astatine Content in the
Earth's Crust 
Less Than One Gram
The total amount of astatine in the Earth's crust is reported in
various books and websites as less than one ounce, probably less than
one ounce, about an ounce, less than 30 grams, or less than 28 grams.
This statement has been quoted and copied from once source
to another.
I
believe the "less than one ounce" figure originated when someone did
a rough calculation and found an answer that was
much less than one ounce, but
knowing that the calculation was approximate, reported the amount as
"less than one ounce," a nice round figure, making a statement that was
sure to be true. Another possibility was that someone calculated the
amount of astitine in the whole Earth, including the core and mantle,
based on the assumption of 2 parts per million of uranium throughout.
For the total amount of astatine in the Earth's crust, I got an answer
that's about 1/300 of one ounce. Here are my basic
assumptions and calculations.
The mass of the Earth is 6.0 x 10
^{27}
g. The crust makes up 0.005 of the total mass of the Earth. The
concentration of uranium in the Earth's crust is 2.0 x 10
^{6}.
We'll assume that this is all
^{238}U
and ignore that part that's
^{235}U,
for the
reasons
discussed earlier. That means the total amount of uranium in the
Earth's crust is:
(6.0 x 10
^{27} grams) x (0.0050) x
(2.0 x 10
^{6}) = 6 x 10
^{19}
grams uranium
We saw
earlier in this page that under
equilibrium conditions, the concentration of astatine atoms as a
fraction of uranium atoms is 2 x 10
^{21}.
So the total number of astatine atoms in the Earth's crust is:
(6 x 10
^{19} grams U) x (1 mole U / 238
grams) x (2 x 10
^{21} mole At / mole U)
x (218 grams At / mole At) =
0.1
gram At
This is the amount of astatine in the Earth's crust at
any given time.
There are additional, but much smaller, amounts of
^{215}At,
^{216}At,
and
^{219}At existing in the decay
chains of
^{235}U and
^{232}Th,
as explained earlier in this page.
How Much Astatine Is Present in an
Ionization Smoke Detector?
An ionization smoke detector contains 0.3 microgram of
^{241}Am.
To convert this to a number of atoms:
(0.30 x 10
^{6} gram) x (1.0
mole / 241 grams) x (6.0 x 10
^{23} atoms
/ mole) = 7.5 x 10
^{14}
atoms
The following table shows the approximate number atoms that exist in
the
neptunium
series from
^{241}Am to
^{217}At after 100 years of decay, starting
with 0.3 microgram of
^{241}Am. For an
explanation of the table entries, see the description of the table
above, under
How
Much Astatine Is Present in Pure Extracted Uranium?
Isotope

Halflife, centuries

Average life, centuries

Decay Rate, per century

Atoms present

Atoms decayed

Decay product

^{241}Am 
4.3 
6.3

0.16 
7.5 x 10^{15} 
1.2 x 10^{15} 
^{237}Np 
^{237}Np 
2.1 x 10^{4}

3.1 x 10^{4}

3.2 x 10^{5}

0.6 x 10^{15}

1.9 x 10^{10} 
^{233}Pa 
^{233}Pa 
zero

zero

100 %

small amount

1.9 x 10^{10} 
^{233}U 
^{233}U 
1.6 x 10^{3} 
2.3 x 10^{3} 
4.3 x 10^{4} 
0.9 x 10^{8} 
3.9 x 10^{4} 
^{229}Th 
^{229}Th 
73 
106 
9.4 x 10^{3} 
1.9 x 10^{4} 
180 
^{224}Ra 
^{225}Ra 
zero

zero

100 % 
180 
180

^{225}Ac 
^{225}Ac 
zero

zero

100 %

180 
180 
^{221}Fr 
^{221}Fr 
zero

zero

100 %

180 
180 
^{217}At 
So the americium in a smoke detector produces no more than
180 atoms of
astatine in the first century, mostly in the latter part of
the
century, after the isotopes in the decay chain have had time to
accumulate. The halflife of
^{217}At
is 0.032 second. Therefore, you'll have an astatine atom in
existence for a total of only a few seconds over the course of 100
years.
Information Sources
The decaychain isotopes and branch probabilities are based on the
values provided in the
decay chain article
in Wikipedia. For the
^{216}At decay
probability in the throium series, see
THEORIA
science journal.
I learned about decaychain accumulation and equilibrium from the book
Interpretation of Radium,
by Frederick Soddy, published in 1909. Soddy and his colleagues
discovered transmutation, isotopes, decay chains, and endless energy;
and calculated decay energies, particle velocities, and halflives of
many isotopes. Amazingly, they did this
before the discovery of the atomic
nucleus, atomic numbers, the proton, and the neutron.