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Identifying pi and sigma bonds from Lewis Structures

ex.

I can see how you use the shapes (trigonal planar, tetrahedral, ect) around an atom to get the hybridization, but how do you figure out what kind of bonds there are (parts b and c)?

Sigma bonds are pair of electrons shared in the plane between the atoms. eg in CH4, there are 4 sigma bonds. In C2H6 there are 7 sigma bonds ( six are C-H bonds) and one is a C-C bond. Sigma bonds are what you regard a single covalent bond to be.

In ethene, there are 4 sigma bonds between C and H as there are four C-H bonds. There is also another sigma bond between C-C, making 5 sigma bonds in total. Now what about the extra bond between the C atoms that makes a double bond?. This is not a sigma bond as the electrons shared in it are not between the two atoms but are shared above and below the sigma bond in two orbitals each containing 1 electron and together making ONE pi bond. See attached diagram (Click on them to enlarge them) . So ethene has 5 sigma bonds and one pi bond.

So for the total number of sigma bonds in your structure, count all of the single bonds as being sigma bonds. Add another sigma bond for each double bond, and finally add another pi bond for each double bond. For the total of pi bonds (which is 3) there will be 6 electrons

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That was fairly simple, thanks.