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What are coplanar atoms? How does this relate to VSEPR theory?

Answer
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Hint: Mixing of atomic orbitals to form a new hybrid orbital with different energies, shapes, etc. suitable for the pairing of electrons to form chemical bonds in valence bond theory is orbital hybridization. The intermixing of hybridization is based on quantum mechanics. Based on the involved orbital types in mixing, the hybridization can be classified as $sp,s{p^2},s{p^3},s{p^3}d,s{p^3}{d^2},s{p^3}{d^3}$.

Complete answer:
 Atoms or groups of an atom which lie on the same plane are known as coplanar. In a compound when all atoms are in the same plane they are called coplanar compounds. Coplanarity in organic compounds is generally observed in unsaturated molecules.
To check the coplanarity, hybridization of the molecule should be known. Like for example taking benzene, in the compound all atoms have $s{p^2}$ hybridization, all are in one plane, that is why it is a co-planar compound.
When a carbon atom is $s{p^2}$ hybridized or $sp$ hybridized all atoms attached to it shall be in a plane, with an even number of a carbon atom, the $\pi $ bonds in the carbon are parallel to each other which enables the hydrogen bonds to be parallel.

Note:
The shape of the molecule from the electron pairs surrounding the central atoms of the molecule is predicted by VSEPR (valence shell electron pair repulsion).
Lone pair-Lone pair repulsions are stronger than lone pair- bond pair repulsions. Lone pairs are confined on the central atom whereas bonded pairs are shared between the two atoms. Lone pair electron occupies more space in the molecule in comparison to the bonding pair electron and this causes greater repulsion in lone pairs of electrons in contrast to lone pair – bond pair repulsion.