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Carbon has 4 valence electrons, so it means that it can have a total of four (covalent) bonds, correct?

If that's the case, why can't carbon share four bonds, such as with another carbon atom (to form $\ce{C2}$, for example)?

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    $\begingroup$ The question about fullerenes would probably be best asked as a separate question, with a link back to this one if you really want. $\endgroup$ – chipbuster Aug 13 '15 at 1:54
  • $\begingroup$ Have you heard of Valence Bond Theory (orbital hybridization), VSEPR theory, or MO theory? $\endgroup$ – chipbuster Aug 13 '15 at 1:58
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    $\begingroup$ I have removed the extra fluff outta your question. We really don't need to know about how you learned about the uncertainty principle. The questions should be free-of-noise and stick to the point. We have Chemistry Chat if you want casual conversation with users of the site. $\endgroup$ – M.A.R. Aug 13 '15 at 5:29
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s orbitals have spherical symmetry, and p orbitals have a dumb-bell shape. When these orbitals overlap, then they form the bonding orbitals (and corresponding antibonding orbitals). The p-orbitals are aligned along Cartesian (xyz) axes. If all three p-orbitals overlap, then the s-orbitals can't get 'close enough' to overlap and form a fourth bond. If we start mixing s and p orbitals, the geometry always puts electron density away from the other bonding atom.

Concisely: there are no arrangements which allow overlap of atomic orbitals that create four bonding molecular orbitals with the proper orientation.

C2 as a molecule can exist, it will simply only have 3 bonds plus two unpaired electrons confined to each carbon.

If you want a more elaborate answer we have to unpack the heavy quantum mechanics, and you won't be seeing that until your junior year.

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    $\begingroup$ Side note: $\ce{C2}$ molecule is more interesting than one might think. $\endgroup$ – M.A.R. Aug 13 '15 at 5:28

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