Question

Let's say you are cleaning out your low-temperature freezer and you discover a bottle with a label that has been partially eaten by low-temperature, label- eating bugs. Fortunately, enough of the label remains for you to see that the bottle contains either \((R, R)-1,2\)-dimethylcyclohexane or \((R, S)-1,2\)-dimethylcyclohexane. To resolve (pun intended) the identity of the substance, you remove the bottle from the freezer, and the next day you take an optical rotation measurement, thinking you can match it to the known value for one of the two possible structures. Much to your surprise you find that the optical rotation value is 0 . There is no optical rotation. Draw \((R, R)-1,2\)-dimethylcyclohexane and \((R, S)-1,2-\) dimethylcyclohexane. Which stereoisomer is in the bottle?

Short answer

The stereoisomer in the bottle is (R, S)-1,2-dimethylcyclohexane.

Step-by-step solution

Understanding Optical Rotation

Optical rotation is a property of chiral molecules that causes them to rotate plane-polarized light. For a substance to be optically active, it must be chiral and not a racemic mixture.

Draw (R, R)-1,2-dimethylcyclohexane

Draw the structure of (R, R)-1,2-dimethylcyclohexane, ensuring both methyl groups are on chiral centers and following Cahn-Ingold-Prelog priority rules. The molecule will not be symmetrical and will exhibit optical activity.

Draw (R, S)-1,2-dimethylcyclohexane

Draw the structure of (R, S)-1,2-dimethylcyclohexane. This molecule is a meso compound because it has a plane of symmetry, making it optically inactive despite having chiral centers.

Identify Optical Activity

Compare the optical activity of the two molecules. (R, R)-1,2-dimethylcyclohexane will be optically active, while (R, S)-1,2-dimethylcyclohexane will be optically inactive as it is a meso compound.

Conclusion

Since the optical rotation measurement is 0, the compound in the bottle must be the meso compound. Therefore, the stereoisomer in the bottle is (R, S)-1,2-dimethylcyclohexane.

Key concepts

chiral molecules

Chiral molecules are a fascinating aspect of chemistry. A molecule is considered chiral if it cannot be superimposed on its mirror image. An easy way to visualize this is by thinking about your hands.
Your left hand and right hand are mirror images of each other but cannot be perfectly overlapped, making them chiral.
Chiral molecules usually have at least one carbon atom bonded to four different groups. These special carbons are called chiral centers or stereocenters.
Take, for example, (R,R)-1,2-dimethylcyclohexane; it has chiral centers because two carbons in its ring structure are each bonded to different groups.
It's important to remember that not all molecules with chiral centers are optically active. For a molecule to be optically active, it must lack internal symmetry. Otherwise, it might cancel its own optical activity.

meso compounds

Meso compounds present a unique twist in the world of chirality. Although they contain chiral centers, they are optically inactive due to an internal plane of symmetry.
This means that the left side of the molecule is a mirror image of the right side, effectively cancelling out any optical activity.
To illustrate, let's consider (R,S)-1,2-dimethylcyclohexane. Despite having two chiral centers, this molecule is a meso compound because of its symmetry.
When you draw it, you'll notice that a plane divides the molecule into two symmetrical halves. These halves reflect each other perfectly.
This symmetry makes the overall compound optically inactive, even though it contains chiral centers. It's a surprising and important concept in stereochemistry.

optical rotation

Optical rotation refers to the ability of chiral molecules to rotate plane-polarized light. If a chiral molecule is placed in a beam of this kind of light, the plane of polarization will rotate to either the left or the right.
The degree and direction of this rotation depend on the specific molecular structure and concentration.
To measure optical rotation, a polarimeter is used. It quantifies how much the plane of light is rotated by the sample.
For instance, (R,R)-1,2-dimethylcyclohexane would show a measurable optical rotation because it is chiral and lacks symmetry. However, if you found that the rotation was 0, this indicates the substance is optically inactive.
In our exercise, the optical rotation measurement was 0, pinpointing that the bottle contained a meso compound, specifically (R,S)-1,2-dimethylcyclohexane.
This illustrates how understanding optical rotation helps in identifying unknown molecules, emphasizing its practical importance in chemistry.