(Polar molecules, Non-polar molecules, etc.)
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Coplanar just means that the atoms in the molecule are on the same flat surface when visualized. An even number of carbon atoms in the molecule given leads to coplanar hydrogen atoms because the terminal C-C pi bonds are parallel to each other, and this can only occur with an even x value.
I believe you are referring to question #19 on the Sapling homework for weeks #7 - 8. Just for starters, coplanar means that the atoms in question lie on the same plane as one another and thus, are not perpendicular. I like to think of it as being parallel. This way of thinking ties into understanding the solution because for even numbers of carbon atoms, the end pi bonds that form are parallel to each other and thus, the hydrogen atoms are parallel to one another. Hence, they lie on the same plane. Meanwhile for odd numbers of carbon atoms, the pi bonds that form are perpendicular to each other, meaning the hydrogen atoms are perpendicular to one another. Therefore, they are not on the same plane. In general, to determine coplanarity, you must observe the bonding and hybridization taking place within the molecule. I do not think they are necessarily important to molecular structure, however, they are likely to be crucial to a molecule's properties.
When things are coplanar it means it lies in the same plane. There is an x, y, and z plan. So, if something were coplanar all the molecules would be in the same plane. If you picture something that is trigonal planar, it is 'flat' and all the atoms are in the same plane or as explained above all parallel.
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