# How many unique molecules can exist in our universe **URL:** \ **Category:** Factual Questions\ **Created:** [November 14, 2018, 7:31pm UTC](https://boards.straightdope.com/t/how-many-unique-molecules-can-exist-in-our-universe/824516 "2018-11-14T19:31:46Z")\ **Posts on this page:** 4\ **Page:** 2
**Author:** ![Wesley\_Clark](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wesley_clark/32/20581_2.png) [@Wesley\_Clark](https://boards.straightdope.com/u/Wesley_Clark)\ **Post date:** [November 16, 2018, 5:02am UTC](https://boards.straightdope.com/t/how-many-unique-molecules-can-exist-in-our-universe/824516/21 "2018-11-16T05:02:17Z")
> [@rat\_avatar](#): > > 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](https://webbook.nist.gov/cgi/cbook.cgi?ID=C7719871). > > 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](https://periodic.lanl.gov/6.shtml) 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!)](http://bigocheatsheet.com/) due to similar issues. 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. > [@septimus](#): > > 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! 🙂 > > However, c(k) ≈ Y[sup]k[/sup] is probably a much likelier law. [Wikipedia implies 3[sup]k[/sup] as](https://en.wikipedia.org/wiki/List_of_straight-chain_alkanes) 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? Math was never my strong suit. What is Z in this calculation? ---
**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\ **Post date:** [November 16, 2018, 6:05am UTC](https://boards.straightdope.com/t/how-many-unique-molecules-can-exist-in-our-universe/824516/22 "2018-11-16T06:05:59Z")
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. > [@Wesley\_Clark](#): > > … What is Z in this calculation? 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! 🙂 ---
**Author:** ![dtilque](https://avatars.discourse-cdn.com/v4/letter/d/d6d6ee/32.png) [@dtilque](https://boards.straightdope.com/u/dtilque)\ **Post date:** [November 16, 2018, 7:25am UTC](https://boards.straightdope.com/t/how-many-unique-molecules-can-exist-in-our-universe/824516/23 "2018-11-16T07:25:18Z")
> [@Napier](#): > > 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 that should be irrelevant for this discussion. > [@](#): > > 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. 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. ---
**Author:** ![HeiLo](https://avatars.discourse-cdn.com/v4/letter/h/c77e96/32.png) [@HeiLo](https://boards.straightdope.com/u/HeiLo)\ **Post date:** [November 16, 2018, 10:10pm UTC](https://boards.straightdope.com/t/how-many-unique-molecules-can-exist-in-our-universe/824516/24 "2018-11-16T22:10:29Z")
> [@ftg](#): > > So a quick upper bound to the number of arrangements is a constant raised to a cubic polynomial due to space being 3-D. I think I’m being obtuse. How does the dimensionality of the space determine the order of this polynomial? [Previous page](https://boards.straightdope.com/t/how-many-unique-molecules-can-exist-in-our-universe/824516.md?page=1)