Radioactive Isotope Decay-Chain 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 1-gram sample of uranium. How many atoms are decaying each second?

All natural uranium is 99.3% 238U. For now, we'll ignore the remaining 0.7% that's mostly 235U.

238U has a half-life of 4.5 billion years. Let's convert that to seconds:

  (4.5 x 109 years) x (365 days/year) x (24 hours/day) x (60 minutes/hour) x (60 seconds/minute) = 1.4 x 1017 seconds

The get the mean lifetime of a radioactive isotope, often designated τ (Greek letter tau), divide the half-life by the natural logarithm of 2:

mean lifetime  τ  = half-life / ln(2) = (1.4 x 1017 seconds) / 0.69 = 2.0 x 1017 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 1017 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 238U is 238 grams per mole, and mole is 6.0 x 1023 atoms (the Avogadro constant):

  (5.0 x 10-18 gram / second) x (1.0 mole / 238 grams) x (6.0 x 1023 atoms / mole)  =  1.3 x 104 atoms/second 

This is 13,000 atoms/second or 13,000 becquerels (13 kBq)

One-thousandth of a gram (one milligram) of uranium decays at one-thousandth of this rate, or 13 atoms per second.

An 8-milligram sample of uranium decays at a rate of about 100 atoms per second. I chose this amount for my first decay-chain example because it's easy to visualize and discuss this rate.

The 235U and 234U 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.

What's the Difference Between Half-Life and Average (Mean) Lifetime?

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 half-life 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 half-life. You can read more about half-life, mean lifetime, and exponential decay in any algebra II book or the Wikipedia article on half-life.

Equilibrium Concentration of Radium in Uranium-Bearing Rock

In any rock containing uranium, all of the isotopes in the decay chain of 238U accumulate to their respective equilibrium concentrations. Each isotope's concentration is the half-life of the isotope relative to the half-life of 238U, subject to adjustment for branching in the decay chain.

The half-life of uranium-238 is 4.5 x 109 years and the half-life of radium-226 is 1,602 years, and there is no significant branching in the chain from 238U to 226Ra.

Equilibrium Concentration of Astatine in Old Uranium Rock

Astatine-218 is in the decay chain of 238U and has a half-life of 1.5 seconds. However, 218At is produced at a branch in the decay chain. Polonium-218 can decay into either 214Pb with a probability of 99.98% or 218At with a probability of 0.02%.

The number of 218At atoms as a fraction of the number of 238U atoms is the ratio of their respective half-lives multiplied by the probability of decay into 218At from the main branch: 

 (0.0002) x (1.5 seconds) / (1.4 x 1017 seconds)  =  2.1 x 10-21

For a rock sample containing 1.0 gram of uranium, the number of 238U atoms is:

  (1.0 grams uranium) x (mole / 238 grams) x (6 x 1023 atoms / mole)  =  2.5 x 1021 uranium atoms

This ignores the small amount of 235U present in natural uranium.

The number of astatine atoms in the rock is:

  (2.5 x 1021) uranium atoms x (2 x 10-21 astatine atom / uranium atom)  =  5 astatine atoms

The direct parent of astatine-218, polonium-218, 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 polonium-218, only 0.0002 x 13,000  =  2.6 are astatine-218 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 astatine-218 is:
 
 t  =  half-life / 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 219At and 215At appear in the actinium series decay chain. The grandparent isotope is 235U, with a half-life of 7.0 x 108 years. Converting to seconds:

 (7.0 x 108 years) x (365 days/year) x (24 hours/day) x (60 minutes/hour) x (60 seconds/minute)  =  2.2 x 1016 seconds

In natural uranium, the fraction of the uranium that is 235U is 0.007; the remaining 0.993 is the main isotope, 238U.

The half-life of 219At 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 227Ac to 223Fr, and subsequently with a probability of 0.00006 in the decay of 223Fr to 219At. Thus, the equilibrium concentration of 219At compared to 238U, the main isotope of natural uranium, is:

 (0.007) x (0.0138) x (0.00006) x 56 seconds / 2.2 x 1016 seconds  =  1.5 x 10-23

This is less than 1% of the amount of 218At produced in the decay chain of the radium series.

Meanwhile, the half-life of 215At 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 215Po to 215At. Thus, the equilibrium concentration of 215At compared to 238U, the main isotope of natural uranium, is:

  (0.007) x (0.0000023) x 0.0001 second / 2.2 x 1016 seconds  =  7.3 x 10-30

This amount is truly negligible.

Astatine Isotope in the Thorium Series

In the thorium series, 216At is the product of a minor branch in the decay of 216Po, with a probability of 0.00013 and a half-life of 0.00003 second. Therefore, the equilibrium concentration of 216At compared to thorium is:

 (0.00013) x (0.00003 second) / [ (1.4 x 1010 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 216At is negligible. 

How Much Astatine Is Present in Pure Extracted Uranium?

Uranium extracted from its ore is separated chemically from its decay-chain isotopes, so it is pure uranium. Once extracted, 238U starts decaying into its decay-chain isotopes: 234Th, 234Pa, 234U, and so on. The first two isotopes in this chain have fairly short half-lives, so they accumulate quickly to their equilibrium concentrations. However, 234U has a half-life of a quarter-million 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 238U to 218At 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
Half-life, centuries
Average life, centuries
Decay Rate, per century
Atoms present
Atoms decayed
Decay product
238U 4.5 x 107 6.5 x 107 1.5 x 10-8 2.5 x 1021 3.8 x 1013 234Th
234Th zero
zero
100 %
small amount
3.8 x 1013 234Pa
234Pa zero
zero
100 %
small amount
3.8 x 1013 234U
234U 2.5 x 103 3.6 x 103 2.8 x 10-4 1.9 x 1013 5.3 x 109 230Th
230Th 7.5 x 102 1.1 x 103 9.2 x 10-4 2.6 x 109 2.4 x 106 226Ra
226Ra 1.6 x 101 2.3 x 101 4.3 x 10-2 1.2 x 106 5.2 x 104 222Ra
222Ra zero
zero
100 %
small amount 5.2 x 104 218Po
218Po zero
zero
100 %
small amount
5.2 x 104
10
214Po 0.9998
218At 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 1-gram 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.

Half-life, centuries -- The half-life of the isotope, in centuries. For several of these isotopes, the half-life 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 half-life 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  238U 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 234U, the average amount is calculated as one-half the total number of 238U atoms produced over the century, because 234U starts at zero and grows in a straight line as 238U 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 over-estimate 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, 218Po decays into two isotopes, 214Po with a probability of 99.98% and 218At with a probability of 0.02%.

No doubt an exact formula,  a year-by-year computer simulation, or a year-by-year 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 over-estimates 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 1027 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 238U and ignore that part that's 235U, for the reasons discussed earlier. That means the total amount of uranium in the Earth's crust is:

 (6.0 x 1027 grams) x (0.0050) x (2.0 x 10-6) = 6 x 1019 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 1019 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 215At, 216At, and 219At existing in the decay chains of 235U and 232Th, 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 241Am. To convert this to a number of atoms:

  (0.30 x 10-6 gram) x (1.0 mole / 241 grams) x (6.0 x 1023 atoms / mole)  =  7.5 x 1014 atoms

The following table shows the approximate number atoms that exist in the neptunium series from 241Am to 217At after 100 years of decay, starting with 0.3 microgram of 241Am. 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
Half-life, centuries
Average life, centuries
Decay Rate, per century
Atoms present
Atoms decayed
Decay product
241Am 4.3 6.3
0.16 7.5 x 1015 1.2 x 1015 237Np
237Np 2.1 x 104 3.1 x 104 3.2 x 10-5 0.6 x 1015 1.9 x 1010 233Pa
233Pa zero
zero
100 %
small amount
1.9 x 1010 233U
233U 1.6 x 103 2.3 x 103 4.3 x 10-4 0.9 x 108 3.9 x 104 229Th
229Th 73 106 9.4 x 10-3 1.9 x 104 180 224Ra
225Ra zero
zero
100 % 180 180
225Ac
225Ac zero
zero
100 %
180 180 221Fr
221Fr zero
zero
100 %
180 180 217At

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 half-life of 217At 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 decay-chain isotopes and branch probabilities are based on the values provided in the decay chain article in Wikipedia. For the 216At decay probability in the throium series, see THEORIA science journal.

I learned about decay-chain 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 half-lives of many isotopes. Amazingly, they did this before the discovery of the atomic nucleus, atomic numbers, the proton, and the neutron.

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2012 Gray Chang