How many unique molecules can exist in our universe

I’m not sure if there is an answer to this because who knows how big atoms could become on the periodic table. Isn’t there a potential island of stability among much larger atoms?

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?

What would be a rough estimate and how could you calculate it? Would it be a factorial?

It would be theoretically on a scale like c[sup]O(n[sup]3[/sup])[/sup] where n is how big (in dimension) of a molecule you want to build.

Note that you will hit some limits: Once a large 3-D molecule gets big enough it will basically be a large planet and the heat from gravitational collapse will ruin the molecule. At a larger scale it could start undergoing fusion. After that you’d have to worry about neutron stars collapse or a black hole.

Given such limits you have to start thinking about more linear molecules (not sure if very large 2-D ones don’t automatically collapse as well). But at some modest scale (like maybe hundreds of thousands of miles) you run into issues of gravitational instabilities and all that. Things might break up into modest pieces which collapse, etc.

The OP didn’t say anything about the stability of the molecules. If a molecules exists for a fraction of a second and then collapses, it still counts.

The island of stability is relative, those larger atoms don’t occur naturally. 98% of the 10^80 atoms in the Universe are Hydrogen or helium, so H2 is the only molecule, 10^800.752. The remaining 2%, ~1% is oxygen and ~0.5% is carbon. Organic molecules come in many varieties, so you could probably come up with an estimate based on the permutations of combining all the C/0/H atoms, but my guess is that it would be insignificant compared to the number of H2 molecules.

The potential for an island of stability is being seriously discussed. However, so far no such element has been found, which suggests they are not - or rarely - produced naturally.

What do you mean by unique molecules? How many different ways there are to arrange some or all of the atoms in the universe into a single molecule?

If you are looking for a purely mathematical answer of combinations it will involve huge factorials. It will also depend on how many ways you allow to arrange them. My basic statistic skills fail me here.

If you are looking for a realistic answer, there are two limits to the maximum size of any molecule: First, mass. If your molecule becomes too big, it will collapse into a sun, at which point I would no longer call it a molecule.

Second, abundance: Hydrogen and helium together are estimated (in Wikipedia) to account for 98% of all atoms. Hydrogen will often only bond to a single atom. So, no chains of hydrogen. Helium is even harder to get into any bond at all.

So, to get an estimate for the second case: Calculate how many DNA bases you can make from the atoms that are there. Then use the same kind of calculus as above for combining them. If your DNA string approaches the mass of, say, our sun, it’s probably no longer realistic.

As I understand it, you can have a carbon nanotube of indefinite length or a layer of graphene of indefinite area.

At some point, gravity is going to cause it to collapse in on itself.

I think we need more from the OP on what his definition of “unique molecule” Is? Like, are two different water molecules unique? The fact that he mentions the total number of atoms (and not the types of atoms) suggests yes.

If so, an upper bound for # of unique molecules would be the power set (set of all subsets) of a size 10^80 things. Or, 2^(10^80). Obviously, that’s a very conservative upper bound, since atoms are going to stop interacting chemically in any reasonable way long before you put a sizable fraction of all of them into the vicinity of a single molecule.

ETA: Hrm, actually, power set is not conservative, since there’s still the arrangement in 3 dimensions. So, back to the drawing board.

But if they were stable, they would. But “Island of stability” can mean the atom lasts for a thousandth of a second instead of a billionth like nearby superheavy atoms.

I wouldn’t consider two water molecules unique.

Also while the vast majority of atoms in the universe are hydrogen, using fusion they can be formed into any other element on the period table. That would reduce the number of atoms in the universe, but the scales would be the same (there would be roughly 10x fewer atoms if they were all fused into carbon for example).

What is this limit for diamond?

Diamond is a crystal and we usually don’t count those as molecules.

The question is (like so many others in this message board) poorly specified. Is the total number of different molecules possible what is wanted or something else?

What I’m asking is this.

There are 10^80 atoms in the universe. Mostly hydrogen. But that hydrogen can be fused into larger atoms.

Assume (for whatever reason) you wanted to determine how many different kinds of molecules you could create with up to 10^80 atoms to work with.

So if you create a H2O water molecule, that counts as one. You don’t need to create 10^80 of them. Creating a trillion water molecules doesn’t count as different than creating one.

As others have said, due to size and gravity a molecule will collapse into a black hole if it gets too big. So there is an upper limit on size. Bot other than that, there are a large number of stable atoms to work with (some of the higher atomic mass elements would decay too rapidly).

Anyway, with 10^80 atoms, and about ~100 possible elements, how many possible unique molecules configurations are there?

Would the number be something like 10^200, or would it be vastly, vastly more?

I have no idea what the upper limit on a molecule size is before it becomes a black hole. I assume around 10^60.

So would the calculation be something like a factorial of 10^60 and ~100 (for the number of different elements), something akin to that?

If I understand the question, we’ll want to count all the distinct molecule structures with weights 1, 2, 3, 4, …, N (call these counts c(1), c(2), … c(N)) where N is just large enough that we’ve then used up all the baryons in the universe. So what is c(k) ?

If we assumed, incorrectly, that c(k) is approximated by some c(k) ≈ k[sup]Z-1[/sup] then ∑[sup]N[/sup] c(k) ≈ N[sup]Z[/sup]. This leads to N ≈ 10[sup]87/Z[/sup] where 10[sup]87[/sup] is the number of baryons. If Z is 10, then our largest molecules will be smaller than a human DNA molecule … so we needn’t worry about gravitational collapse! :slight_smile:

However, c(k) ≈ Y[sup]k[/sup] is probably a much likelier law. Wikipedia implies 3[sup]k[/sup] as a good approximation to the counts for simple paraffins. If we stick with c(k) ≈ 3[sup]k[/sup] and approximate ∑[sup]N[/sup] c(k) as 3[sup]N[/sup], then the largest molecule we’ll need is “ridiculously small.” We’ll end up with 10[sup]84[/sup] distinct molecules, or thereabouts.

The Wikipedia table counts stereo-isomers as distinct. That’s how they end up with over a quadrillion isomers of octatriacontane (C38H78) instead of a mere 9 trillion. Is this OK, OP?

Exercise: What are the three structural isomers of 3,4-dimethyl hexane?

Unless you are willing to consider macromolecules where intermolecular forces and not covalent bonds create the structure like proteins the upper limit for a single molecule is going to be ~ 900 g/mol

As an example of non-macromolecules one of the largest is tetrahexacontane which is C64H130.

If you accept macromolecules your answer is going to be close to infinity.

Polymers, proteins and other macromolecules assembly permutations grow not just exponentially but factorially and even just carbon and hydrogen will have a massive number of possible combinations.

Consider just the group 14 hydrides, which tetrahexacontane is a member of. Almost any combination of XnH2n+2 will work. Note C64H130 where n=64

Propane C3H8
Butane C4H10

Add one oxygen and you get

Propanol C3H8O

Have 10-16 carbon atoms? that is what we call kerosene

There are around 10 million defined organic compounds but there is an indefinitely large number of such compounds in theory.

Note that Carbon has seven isotopes (ignoring particular behaviors) which will cause this count to grow even more if you care about those permutations too.

Combinatorics is not an area I have spent much time in due to practical limits of permutations in computing. The time complexity of computer permutation function is O(n!) due to similar issues.

This is mostly irrelevant to the question. There’s plenty of non-hydrogen atoms around to make all the possible molecules. That was what was confusing me. Even if we could make some super-heavy atoms that are stable, they won’t contribute much to the total. The organic molecules are going to make up the bulk of the total. And while some of those will have metallic atoms incorporated, adding a handful more metalic elements won’t increase the total by much.

And forget about black holes. Molecules will get unstable long before you get to that size, or even the size of a smallish asteroid.

At any rate, I found someone asked the same question in another forum: here. (That wasn’t you, was it?) Note that they closed the question for being too broad.

Note: there is no factorial here! Just a simpler exponential.

Atoms of an isotope don’t have labels. Take H-O-H. You can swap the two hydrogen atoms and the molecule stays the same. So what unique atom is at a given position doesn’t matter, only what type of atom it is.

There are a constant number of isotopes. So a quick upper bound to the number of arrangements is a constant raised to a cubic polynomial due to space being 3-D.

The number of permutations of n distinct objects is n factorial, by definition.

While Combinations are not factorial and are involved, the shear number of permutations will make the additional Combinations a smaller set of a numbers that will be effectively non-finite due to the growth of permutations

It doesn’t involve swapping H-O-H, but placing an O atom at difference points in middle of a large carbon chain will produce different unique substances. It doesn’t take many of these types of cases for the number of permutations to approach infinity.

Why discuss permutations at all???

Swapping one atom with a different atom makes a change. But there only a constant number of different isotopes. You can only make a simple exponential number of arrangements of objects of length n in a line for a constant number of sets of items to choose from.

So if you had 2 isotopes to choose from to make a linear molecule of length n there are 2^n possible options. n! doesn’t figure into it. (And note that n! is bigger than 2^n so it’s a poor bound for this case.)

More isotopes means you replace the “2” with a bigger number. 3-D means the exponent is a cubic function. Just having a 3-D array of carbon and silicon atoms gives you the option of producing a lower bound that looks like this.

(Well, until it becomes sentient and starts to take over the Universe. ;))

As perhaps a quasi-related question, are there realistic limitations on what molecules we can expect to find in nature, in even vanishingly tiny amounts? It’s very probable for example that fluorinated organic molecules have never existed outside of human guided synthesis.

I think people are using “universe” here to refer merely to the observable universe. The whole thing is thought to be infinite in size.

But I’m sure a molecule could get way bigger than an asteroid and still be stable. Lots of polymers get cross-linked and the whole block becomes one big molecule. It isn’t a crystal, and all the atoms are bound together, that is, bound to other atoms in one big network. I think some kind of degeneracy sets in for stellar masses that don’t have internally generated heat to support them from collapse, but certainly planets can stay molecular and stable. Somewhere between big Jupiters and big stars, that’s where the limit ought to be.