[curiouscat]

Can the natural abundance of isotopes of an element be mathematically predicted from the half-lives of these isotopes?
e.g. In nature, uranium is found as uranium-238 (~99.3%), uranium-235 (~0.7%), and uranium-234 (~0.005%).

If not, then is there any way to predict the theoretical natural abundances at all or is empirical measurement the only option?

[Borek]

In some (rare) cases half life of the isotope depends on the compound in which the element is present*, so in general you can't predict the composition not knowing the sample history.

*see http://en.wikipedia.org/wiki/Electron_capture

[curiouscat]

In some (rare) cases half life of the isotope depends on the compound in which the element is present,

That's interesting. I did not know that. Thought half life was a strict isotopic property.

So the compounds ^wAB and ^wAC can have different half lives for ^wA  decay? Do you know an example off-hand?

so in general you can't predict the composition not knowing the sample history.

What about not in a sample specific sense but an average sense. e.g. The way atomic-masses are defined by IUPAC is via some sort of relative natural abundance metric, right?

[Borek]

So the compounds ^wAB and ^wAC can have different half lives for ^wA  decay? Do you know an example off-hand?

See the wiki article. It gives Be as an example.

I remember reading about "huge" differences between molar mass of B coming from different sources on Earth, which suggest similar processes (it was even discussed at CF in the past). Atomic mass of Zn is listed by NIST as 65.38, while molar mass of Co is listed as 58.933195 - I bet that means Zn is much more variable.

What about not in a sample specific sense but an average sense. e.g. The way atomic-masses are defined by IUPAC is via some sort of relative natural abundance metric, right?

I suppose it would be better to ask someone fluent in nuclear synthesis. I can just use common logic. Abundance of ¹⁴C is a function of the solar radiation (which is why there are special tables used for radiocarbon dating) - so atomic mass of carbon is a function of the solar radiation as well (although differences are minute). When there are several stable isotopes their production in stellar processes is most likely a function of the temperature and pressure inside the star (or during supernova explosion) - so the final ratio should be not universal, but rather local, with some variation between even neighboring start systems. Most likely when there is only one stable isotope and all others are relatively short living atomic mass is less variable.

[gippgig]

In the case of ²³⁸U & ²³⁴U it can be since ²³⁴U is only present because it is produced in the decay chain of ²³⁸U. The ratio of the abundance of ²³⁴U to ²³⁸U is the same as the ratio of the half-life of ²³⁴U to the half-life of ²³⁸U. This doesn't work for ²³⁵U (or for just about any other isotope in nature) since ²³⁵U is not a decay product of ²³⁸U.
Note that while it is (generally) not possible to predict the natural abundances from the half-lives, it is possible to predict how the natural abundances have changed over time. For example, 1.7 billion years ago the abundance of ²³⁵U was about 3% (since it has decayed faster than ²³⁸U) so it was possible for a vein of uranium ore to go critical in Oklo in Gabon, Africa.

[curiouscat]

Note that while it is (generally) not possible to predict the natural abundances from the half-lives, it is possible to predict how the natural abundances have changed over time.

How does one predict the natural abundance at an ancient point in time; say 1.7 billion years ago. Is there any other independent strategy other than to measure the current natural abundances and then extrapolate back in time using the known half lives?

[gippgig]

It would theoretically be possible to predict abundances from the mechanisms of nucleosynthesis but there are so many uncertainties & unknowns at this point that I believe we are using isotopic abundances to develop the theory of nucleosynthesis rather than the other way around.

[Enthalpy]

Electron capture is THE case where decay rate can be influenced by traditional means, and beryllium the only example I remember... Explanation being that its 2s orbital makes the chemical bonds and, as a spherical orbital, has some probability density right at the nucleus so sometimes a 2s electron gets swallowed instead of a 1s, so chemistry influences this secondary process, by about 10^-2 in total.

One group believed to have seen a change in decay rate, again at electron capture, due to high presure in sonoluminescence, but the other groups didn't observe it. It is indeed observed in a diamond anvil.

If someone could at will accelerate a lot the decay rate of ⁴⁰K, that would provide us abundent clean energy. Good luck...

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Only ⁵⁹Co exists naturally, so the element's mass is more accurate. I had proposed to inject a bit in Fukushima's cooling water and check for ⁶⁰Co to know if neutrons were available, as a proof of ongoing criticality.

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Very few radioactive nuclides a primordial (as old as Earth or more): ⁴⁰K, ²³²Th and varied U. All others result from decays (Po, Rn...) and even more indirect ways (traces of natural Pu, as U fissions spontaneously, produces neutrons absorbed by U which turns to Np then Pu); as different isotopes can (and do) have different parent elements, adn not just different isotopes of one parent element, the isotopic composition of children can depend on the chemical composition of the parents. The geographic origin of uranium ore is traced that way.

One extreme case: one Ar isotope results partly from ⁴⁰K decay, so Ar composition varies accordingly.

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Intense gravitation, over the full potential of a planet, star or galaxy, should segregate elements and isotopes very efficiently, under the sole condition that atoms were gaseous at some time of the object's evolution, and had a mean free path not too small compared with the object's size. I suggested it to a student as a research topic. Maybe some theoretical results will emerge one day. That would rock a few established models.

[curiouscat]

Intense gravitation, over the full potential of a planet, star or galaxy, should segregate elements and isotopes very efficiently, under the sole condition that atoms were gaseous at some time of the object's evolution, and had a mean free path not too small compared with the object's size. I suggested it to a student as a research topic. Maybe some theoretical results will emerge one day. That would rock a few established models.

Can you elaborate on this a bit? I tried thinking about it but didn't get the significance? Also how'd you test that hypothesis?

[Enthalpy]

Take for instance the Sun's gravitation potential at our present distance to it. Earth moves at 30km/s, which means each neutron has -7*10^-19J gravitation+kinetic energy, or 200 times 300K. So temperature does not suffice to mix the isotopes evenly over the Sun's gravitation field, and the elements even less so.

Many more cases exist in astronomy at different scales: galaxies, planets... On Earth, the top atmosphere has slightly more of the lighter isotopes than the bottom, but Earth is a bit too light and the atmosphere too shallow for that. Jupiter would be far better.

At the microscopic scale, you can consider that an elastic collision between atoms tends to redistribute the kinetic energy evenly as a mean result, including between atoms of different mass, but at the same distance from the Sun or any massive object, atoms would need the same speed to stay on the same orbit. So the heavier atom falls down.

Though, this is a dynamic process. Whether gravitation had the opportunity to segregate the elements and isotopes depends on how long they take and how much time they had.

My quick-and-dirty answer is: over the early evolution of a gas clump, we're likely to find an age where density is still small enough that atoms go through the cloud with few collisions; this is when segregation has chances to occur. I didn't model these conditions; an astronomy student on a forum was interested at the project. Maybe this will be the bingo for him.

One example is the formation of a proto-star. Oxygen is known to vanish as the cloud begins to collapse, but reappears as the star is formed. That would fit.

Present models of gas cloud collapse want it to radiate heat, but evaporation (from the gravitation well) of the lighter elements and nuclides could help condensate (fall down) the heavier ones.

If isotope (and element) segregation has been important in our Solar system, it would rock very seriously many models that use isotopic abundance as a proof for the origin of varied objects.

Intergalactic hydrogen clouds have too little deuterium.

So in fact, tests exist already - we just need to check if segregation is a plausible explanation.

Or even... Stars with more elements beyond helium reside in the deeper gravitation wells of galaxies: the bulge, the globular clusters... Commonly explained through the age of the stars. If segregation contributes to explain the proportions, a little bit, or a little bit more... this would lead to re-tune the models a little bit, with many implications.

I suggested to measure the ²H/¹H ratio at Jupiter's big spot, which is said to result from ascending wind. Maybe H₂O or NH₃ rotations and vibrations can be measured in the GHz or IR domains.

Segregation can result in anything between
- Small effect at few exotic objects with no other implication
and
- General mechanism at all scales that needs to re-tune all explanations of elements and isotopes abundances, and every deduction made from them.

Fun, isn't it?
Marc Schaefer, aka Enthalpy