Animated Solution for Chemistry - Organic Chemistry: The number of stereoisomers possible for 1,2-dimethylcyclopropane is

Imagine you are looking at a simple three-membered carbon ring. This is the core of our molecule, cyclopropane. Now, we attach two methyl groups ($-C{H}_3$) to adjacent carbons, specifically at positions $1$ and $2$.

This gives us $1,2$-dimethylcyclopropane. Immediately, your stereochemistry radar should go off! Carbons $1$ and $2$ are attached to four different groups: a hydrogen atom, a methyl group, and two different paths around the cyclopropane ring. Therefore, both of these carbons are chiral centers.

A common mistake is to blindly apply the $2^n$ formula, where $n$ is the number of chiral centers. If we did that, we would guess $2^2=4$ stereoisomers. But wait! We must always check for symmetry.

Let's draw the cis-isomer, where both methyl groups are pointing in the same direction (let's say, downwards). If you look closely, you can draw a line right down the middle of the molecule, bisecting the $C_3$ carbon and the $C_1-C_2$ bond. The left half is a perfect mirror reflection of the right half.

Because of this internal plane of symmetry, the cis-isomer is superimposable on its mirror image. It is optically inactive. We call this a meso compound. So, the cis configuration only gives us $1$ stereoisomer.

Now, let's flip one of the methyl groups to create the trans-isomer. One methyl group points up, and the other points down.

Try to find a plane of symmetry now. You can't! The "up" methyl group does not reflect into the "down" methyl group. Because it lacks any plane of symmetry, the trans-isomer is optically active.

This means its mirror image is non-superimposable. Therefore, the trans-isomer exists as a pair of enantiomers (often referred to as the $d$ and $l$ forms). This configuration gives us $2$ stereoisomers.

Let's tally them up. We have $1$ meso form from the cis configuration, and $2$ enantiomers from the trans configuration.

Pro-Tip: What if the two groups were different? For example, $1$-chloro-$2$-methylcyclopropane. In that case, the cis-isomer would not have a plane of symmetry (since a chlorine atom doesn't reflect into a methyl group). You would then have $2$ enantiomers for the cis form and $2$ enantiomers for the trans form, giving a total of $4$ stereoisomers. Always check for symmetry!