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In several pictures that I have seen of buckminsterfullerene and graphite (such as below), each carbon is bonded to only 3 other carbons, so it only forms 3 covalent bonds. enter image description here enter image description here

I asked my teacher about this and he says that each carbon only make 3 bonds and has a lone pair. These lone pairs are de-localised hence, that is why buckminsterfullerene is able to conduct electricity.

However, I am doubtful about this as how can a lone pair on an atom be de-localised. Would the electrons be in their orbitals, and not be able to freely move around the compound. I try searching the internet for any information, but there is no mention on how many bonds each carbon atoms makes.

I am guessing that there is alternating double bonds between each carbon so that each carbon has 4 bonds. It is these pi bonds which along the compound that makes it a hyper-conjugated compound. This would explain why buckminsterfullerene and graphite are able to conduct electricity. Am I correct?

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Have you ever seen a picture of graphite, to begin with? It shares the same feature, i.e., that each carbon is bonded to exactly 3 other carbons. You may call it "alternating double bonds", but this is not quite correct either, because in fact all bonds are equal. Think of it as a resonance structure, much like benzene. This is the reason for graphite conductivity, too.

The idea about that lone pair is plain wrong. Carbon has but 4 valence electrons; once 3 of them are spent for sigma bonds, how could it have a lone pair?

Pure buckminsterfullerene does not conduct electricity (strictly speaking, it is a semiconductor, not a conductor like any metal). Some of its derivatives do, but that's another story.

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    $\begingroup$ Thanks. So, if graphite and buckminsterfullerene both have the same structure, why is only graphite able to conduct electricity, but not buckminsterfullerene? $\endgroup$ – Nanoputian Sep 14 '15 at 7:46
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    $\begingroup$ Because they don't have the same structure. Look again at that picture of graphite. Look at them layers, they are infinite. It's a huge system of conjugated pi bonds, from one end to another. That's what conducts electricity. With fullerene, it's just a system of such bonds within one molecule. Supposedly it could conduct electricity from one side of molecule to another, but that's about it. $\endgroup$ – Ivan Neretin Sep 14 '15 at 7:55
  • $\begingroup$ Fun fact: Graphite (at least an ideally layered) is only a conductor in two dimensions. $\endgroup$ – Martin - マーチン Sep 14 '15 at 12:36
  • $\begingroup$ Yes, of course. And nanotubes (at least some of them) are 1D conductors. $\endgroup$ – Ivan Neretin Sep 14 '15 at 13:44
  • $\begingroup$ Fun fact: Calculations gave me alternating single and double bonds along the six-membered rings and the five-membered rings of fullerene were exclusively single-bonds. $\endgroup$ – Jan Sep 19 '15 at 17:40

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