Question

Does 1,2 dimethylcyclopropane exhibit optical isomerism?if no...why?if yes...why?

Answer 1

Yes.

Each of the carbon atoms with a methyl group on it is, in fact, a chiral center. The four different groups attached to it are:

No 4. A hydrogen atom

No 3. The methyl group that's attached outside the ring

And...

No 2, No 1. The ring going clockwise and going counterclockwise, as viewed from "above" the C-H bond, are counted as two different attached groups, as if the ring were "cut off" in either direction before rejoining the carbon atom you're looking at.

If you look at the ring attachment that way, you see that the two different directions rank 1 and 2, in some order depending on where the second carbon with the methyl group is, hence my numbering above. So you can figure out an R or S configuration for each chiral center. In the cis isomer you end up with one R and one S (thus mirror-image configurations), and a mirror plane does go between them making a meso rather than a chiral molecule. But the trans isomer gives both R or both S, which never allows a mirror plane and so it's chiral.