> "There are, after all, only a finite number of elements in the periodic table..."
That's because we've only discovered or figured out how to make a finite number of them. Is there a reason that other (alien) elements can't exist that we've never been exposed to?
Yes, there is a reason. We've thoroughly explored the periodic table through our studies of nuclear processes (lots of government funding for anything nuclear weapons related). It turns out you can create a ton of elements we don't see in nature, but almost none of them are stable: you smash atoms together to make a new one, but the new one flys apart after a <second (and that's actually a long lifetime for these atoms, many have lifetimes of nanoseconds). There is a potential "island of stability" around element 120-130, but even there lifetimes are predicted to be less than a minute.
In addition to the stability argument, there are also energy requirements. It turns out fusion (stars) only gives energy up to iron, then it requires energy to make bigger atoms. In other words, you get energy back out when you split big atoms (I.e. nuclear reactors). In nature, all elements past iron are created only in the spectacular energies of supernovas, which occur in less than a second.
Basically, we have a really good grasp on the elements that can exist. We're only missing the details on a few things that occur on nanosecond and less timescales.
>> In addition to the stability argument, there are also energy requirements. It turns out fusion (stars) only gives energy up to iron, then it requires energy to make bigger atoms.
That's actually a common misunderstanding. If you fuse hydrogen (or even lithium) with iron, you can get a higher numbered element and some excess energy. See the chart on this page:
While iron is at the top of the curve, that just means you won't see iron-iron fusion. There really is no reason the lighter (common) elements can't fuse with the heavier elements.
This is also the idea the LENR (low energy nuclear reaction, formerly known as cold-fusion) guys are considering. If you fuse Hydrogen with Nickel62 to produce Coppper63 you could get some energy out. Notice that Nickel is already heavier than Iron. Some claim to have seen this copper production in hydrogen-nickel cells. The claims are not really relevant - the math supports it as a possibility. Weather it can happen on earth or in a star is open for debate. One key question is how the excess energy would get out as heat, and there are ideas about that.
I for one find it amusing that people don't think something like this is where all the naturally existing heavy elements came from.
Wrong, it really does mean that it takes energy input to fuse iron and hydrogen. Naturally existing heavy elements are the result of supernovas, where a percentage of the energy of the dying star is converted to fusing elements beyond iron.
The structure of elements/atoms is well understood based on their subatomic constituents. Naively, you might think that can you just keep combining increasingly larger numbers of electrons, protons, and neutrons to create new elements. However, the stability of an atom becomes problematic when the size of the nucleus approaches the interaction length of the strong force (i.e. the nucleus is too large for the strong force to hold it together). These elements are unstable and therefor not relevant as far as organic chemistry is concerned.
Furthermore, the formation of elements in the Universe is also a fairly well understood process. For elements lighter than Fe it generally occurs through nuclear fusion in the center of stars. For elements larger than Fe it generally occurs through the r-process and s-process. With these we can model nucleosynthesis extremely well and it gives us a very good idea of the elemental composition of the Universe. That being said, there could be some crazy unknown element out there but it would contradict almost everything know about atomic physics.
To add some visualization, you can have a look at the isotope chart [1] from Wikipedia showing the half-live times of the known isotopes to get the big picture. The distinct area towards the top is called island of stability [2] and contains long-lived but nonetheless unstable elements. A second island of stability is suspected even further up in yet uncharted territory but nobody expects additional stable elements beyond lead.
> Is there a reason that other (alien) elements can't exist that we've never been exposed to?
Yes, because you form new chemical elements by adding protons (and stabilizing neutrons) to the nucleus, and humans have found or synthesized the first 118 of these. The ones that don't occur in naturally on earth are short-lived an unstable.
If there was a lot of exotic elements out there, we'd have seen it with our telescopes. It would point to new physics, for a start, since we don't think very heavy elements are likely to form naturally at all, let alone be stable long enough to be observed in any quantity.
Even if such elements are out there in the universe in tiny quantities, we'd run into a billion carbon-based biospheres before we found any.
To elaborate on the first point, we can identify elements by the spectrum of light they produce. We discovered helium in the sun before we discovered it on Earth.
(This only works if the element is common enough and hot enough to emit light that is seen from Earth.)
Physics is the reason that other elements likely don't exist. One can only arrange protons, neutrons, and electrons in a fixed number of "stable" ways.
Of course, this is today's understanding. We may be wrong. :-)
That's because we've only discovered or figured out how to make a finite number of them. Is there a reason that other (alien) elements can't exist that we've never been exposed to?