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Graphite has hexagonal parallel planes. In a hexagonal structure the carbon atom has three bonds. Since the valancy of carbon is 4 is the valency satisfied in graphite?

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Just as much as in benzene or any other $\mathrm{sp^2}$ carbon case. The concept of valency doesn't mean that the given atom has to form bonds with the given number of neighbors, but refer to a simple way of counting electrons and formed bonds.

In organic chemistry, we often find carbon forming 4 (single) bonds to neighbors. For example you can find four $\ce{C-H}$ bonds from the central carbon in $\ce{CH4}$ or one $\ce{C-C}$ and three $\ce{C-H}$ bonds from each carbon in $\ce{C2H6}$.

However, in a benzene or ethene $(\ce{H2C=CH2})$ molecule there is only three neighbors of each carbons. In ethene the $\ce{C=C}$ is a double bond (unlike $\ce{C-C}$ in ethylene) which is formed by using two-two valence electrons of the carbons. When you count valence, you count these valence electrons, therefore a double bond counts twice.

The situation of benzene is a little trickier, as it does not have a double or a single bond between the carbons. If the carbons would form a single bond, there would be a extra dangling electron remaining on each of them. In reality these electrons are delocalized (shared) over the whole ring. If you count the valence electrons of a carbon, you should count this delocalized electron, too. You can understand graphite along this line: three bonds to the three neighbors plus the remaining one valence electron delocalized over the ring system (the whole graphite sheet) are together gives back the four valence electrons.

You can find more on hybridization in the other answer.

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In the hexagonal graphite structure the carbon atoms are $\ce{sp^2}$ hybridized, just like in benzene. In graphite, this p-orbital is used for bonding just as it is in benzene, resulting in an extended pi system in the graphite structure. Graphite is really just a large number of benzene rings annelated together to form a continuous, ring structure. Just as carbon satisfies its valence requirement of 4 in benzene (3 sigma bonds and 1 pi bond), carbon's valency requirements are satisfied the same way in graphite.

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    $\begingroup$ With regards to the physical appearance of both elements : if you notice, pure, mined Graphite is infact a bunch of flat layers (slice) combined together. These slices are fragile. And if you look at diamond, very well known hard build structure. Naturally formed diamonds. $\endgroup$ – bonCodigo Aug 20 '14 at 15:53

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