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What's the chance that there might be undiscovered chemical elements in the Solar System - either on planets or around the Sun or on asteroids of the Oort-cloud?

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If you mean elements with atomic number higher that 115-ish then of course there is a chance, so far nothing prohibits atoms from having as many protons - though stability is an issue. But I don't see the point in asking 'What is the chance..?' –  harogaston Jul 5 at 18:59
    
Likelihood is "maybe": Superheavy Element 117 Points to Fabled “Island of Stability” on Periodic Table scientificamerican.com/article/… –  Wayfaring Stranger Jul 9 at 21:56

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As far as elements (e.g. on the periodic table) go, I would say the odds are very slim. We already discovered or produced all the elements of the Periodic Table up to atomic number 112 at least. As the number increases, the half lives of the elements generally decreases, and is very short for elements above 102. If this trend holds true as the number increases, practically all the "undiscovered" elements should have turned into the lower known atomic number elements.

However, there is hope. There is a theorized "island of stability" where a narrow range of yet to be discovered high atomic number elements may be stable: http://en.wikipedia.org/wiki/Island_of_stability I would say there is a slight chance this element could be discovered in the solar system.

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Elements in the "island of stability" are expected to be relatively stable, compared to their neighbors. Quoting the Wikipedia article, "Specifically, they are expected to have radioactive decay half-lives of minutes or days, with "some optimists" expecting half-lives of millions of years." Even with half lives in the millions of years, there could still have gone through hundreds or thousands of half-lives over the history of the Solar System. Unless the optimists are underestimating their stability, there should be practically nothing left of them. –  Keith Thompson Jul 7 at 23:57

Further to the answer of @Jonathan, the thing that distinguishes one chemical element from another is the number of protons in the nucleus, which in turn determines the number of orbital electrons in the uncharged atom.

But we already know the element that corresponds to any given number of protons between 1 and 112; that's the atomic number. And you can't have a fraction of a proton. The only room for possible new elements is on the end.

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Another way to look at this question is to consider how elements are produced. The elements with larger atomic numbers (i.e.: 26 (iron) or so) on the periodic table are primarily produced during supernovae explosions. Based on a lot of findings in stellar physics and nuclear physics in the past half century, it's unlikely that a transfermionic element (an element with 92 or more protons) can be produced in that process. Further, these elements tend to decay with half lives measured in hours or minutes (or less), so even if they were produced in a supernova, they are long since gone.

As @Jonathan pointed out, there is some potential for such elements due to the so-called island of stability, but they are still likely to be highly unstable, with very short decay times.

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Iron is produced around the core of big stars by fusing silicon. The elements produced in supernovae are the ones heavier than iron. Just pointing :) –  Joan.bdm Jul 3 at 7:22
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Good point. However, some elements, europium for instance, are produced in the corona during a star's time on the main sequence. –  Ben Jul 22 at 2:00
    
Didn't know that! I guess the millions Kelvin is the reason for that. Thanks Ben! –  Joan.bdm Jul 22 at 6:18

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