Chemistry - Carbon with 5 bonds?

Solution 1:

Carbon cannot have more then 4 double-electron bonds in reasonable conditions. However, in can form a bond with 5 or 6 atoms, like $\ce{Fe6C}$ fragment, where iron atoms form octahedron around the carbon atom. However, the sum of orders of 6 $\ce{C-Fe}$ bonds will be still 4.

The situation is different if we consider exited states. Indeed, it is possible for hight excited state of carbon to be able to form 5 or 6 bonds. The resulting structure, however, will die quickly.

Noble gases are able to form molecules in excited states, or, to be precise, their molecules are stable only in excited state. This kind of molecules is known as excimer (excited dimer).

Solution 2:

The hypervalent carbon do exist. Look at this article: Kin-ya Akiba, et. al., J. Am. Chem. Soc., 2005, 127 (16), 5893–5901.

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X-ray measurements confirmed the 10-C-5 structure.

Solution 3:

Carbon can make 5 bonds on methanium, $\ce{CH5+}$. It is a kind of carbonium ion and superacid. Methanium can be produced in the laboratory as a dilute and low-temperature gas.

Reaction: $$\ce{CH4 + H+ -> CH5+}$$

Solution 4:

There are stable crystal structures where one could notionally see carbon atoms with more than 4 'bonds'. One example that comes to mind is a carborane unit; e.g. see

For example. But of course, such 'bonds' are a notional thing really and don't reflect the true nature of the underlying electronic structure.