I was considering biomolecules and polymers, but assume they use the 900g/mol as the cutoff for a molecule.
What kind of number would they have?
The reason I ask is because assume the universe expands forever and humans become some kind of godlike species in the far future, but we are stuck in this universe. So the matter we have access to can be rearranged in a virtually infinite number of ways, but at the end of the day its still a finite number.
Math was never my strong suit. What is Z in this calculation?
First off: I misunderstood OP’s question. “Anyway, there are about 10^80 atoms in the universe. Is there a rough estimate to how many unique molecules could be built out of that?” I thought the idea was to assemble molecules, all with chemical structures distinct from each other, out of those atoms and stop when we’d used up all 10^80 atoms. The key question becomes: How large of molecules will we need to use up these 10^80 atoms? I concluded that we wouldn’t need molecules larger than 200 atoms or thereabouts, assuming we’re allowed to fuse most of the “useless” Helium atoms into more useful atoms like Carbon. The final answer would end up as something like (10[sup]80[/sup] ÷ 200). (To add further confusion, I substituted 10^87 — an estimate from another recent thread — for OP’s 10^80.)
Others interpreted the question quite differently: After building one molecule, we’re allowed to disassemble it and reuse those atoms over and over. Now a key question is: How large are the largest molecules we can build? Is a chromosome a single molecule? Some of them have as many 7 billion atoms each. Structures (like polymer threads or crystals) much larger than that can be viewed as single molecules. The final answer would end up as something like (e[sup]10[sup]20[/sup][/sup]) where 10[sup]20[/sup] is just an arbitrary estimate of the maximum molecule size.
Despite that the final answers are wildly different, in either interpretation the key question is: *What combinatorial law c(k) approximates the number of distinct molecules with k atoms? * (Or, k might be treated as the molecule’s atomic weight. This is a detail we can worry about later!)
The most obvious guesses for c(k) are
Power law: c(k) ≈ k[sup]Z[/sup]
Exponential: c(k) ≈ Y[sup]k[/sup]
Here Z and Y are just parameters we will eventually want to estimate. The second (exponential) formula is more correct, but I mentioned the power law possibility anyway.
It is the (unknown parameter) if c(k) follows a power law. It’s doomed to remain unknown since c(k) does NOT follow a power law!
But that should be irrelevant for this discussion.
I won’t say it’s impossible that some molecules could be that large, perhaps a vastly expanded buckyball or something. But most will collapse from their own weight long before they even get to Earth sized. Thus my guess of a small asteroid being the upper bound in size.
