Radioactive Isotope Decay-Chain Calculations

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.

• Atoms of radium as a fraction of the atoms of uranium:  1,602 / (4.5 x 10⁹)  =  3.6 x 10^-7
  
  
• Atoms of uranium as a multiple of the atoms of radium:  (4.5 x 10⁹) / 1,602  =  2.8 x 10⁶  =  2.8 million
  
  

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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