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Molecules Structure as Planar Graphs?by referframe
Tags: 2dimensional, graphs, molecular geometry, molecules, organic molecules, planar, planar graphs, structure 
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#1
Oct3013, 11:24 AM

PF Gold
P: 133

I have read that the structure of almost all organic molecules can be represented visually as Planar Graphs, i.e. 2dimensional latticelike structures consisting of nodes (points) connected by lines in which no lines cross. From the perspective of Physical Chemistry, does anybody know why this is so? Some graphs are of such complexity that they required 3 dimensions to guarantee that no lines cross  reference "3 Utilities Puzzle".



#2
Oct3013, 03:19 PM

Sci Advisor
P: 6,077

https://www.google.com/search?q=etha...iw=943&bih=672
It is a simplification for showing on a sheet of paper. The above shows that something as simple as ethane has a three dimensional structure. 


#3
Oct3013, 05:02 PM

PF Gold
P: 133




#4
Oct3013, 09:37 PM

P: 160

Molecules Structure as Planar Graphs?
Because most organic molecules do not have the degree of interconnectedness required to be nonplanar.
Kuratowski's theorem says that every nonplanar graph contains either a topological K_{5} or a topological K_{3,3}. In other words, for an organic molecule to be nonplanar, you have to either have: 1. Five carbon atoms such that there is a chain (of bonds and atoms) from each one to each of the others, with none of these 10 chains sharing any bonds or atoms with any of the others; or: 2. Two sets of atoms of valence 3 or more, such that there is a chain from each atom in the first set to each atom in the second set, with none of these 9 chains sharing any bonds or atoms with any of the others. To satisfy either of these conditions, you need a lot of highorder carbons and a lot of different paths between them. That would require a large and highly interconnected molecule. But it's by no means beyond the realm of possibility, and I have to imagine that some proteins, with their abundant disulfide interconnections, are nonplanar (containing K_{3,3}). It also seems like some people have synthesized nonplanar molecules intentionally: see this article, for example. EDIT: After posting I found this article which discusses the issue further. The author seems to have a bit of confusion between nonplanarity and knottedness (you can have a knotted molecule that has a planar graph), but apart from that the analysis seems good. 


#5
Oct3013, 11:31 PM

Other Sci
Sci Advisor
P: 1,393

Adamantane comes to mind as a molecule for which this does not appear to be true.



#6
Oct3113, 02:07 AM

Sci Advisor
P: 3,593

Of course there are molecules where this is not true, e.g. adamantane.
The problem with this kind of question is which metric to use: All organic molecules with a CAS number? All possible organic molecules with, say, up to 100 C atoms? 


#7
Oct3113, 03:23 AM

Admin
P: 23,594

No idea about explanations. And I am not convinced they are strictly chemical. 


#8
Oct3113, 04:55 AM

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