Problem 154 - A Complete Resource Book in Chemistry for JEE Main

Question

The total number of optically active isomers of the compound \(\mathrm{CH}_{2} \mathrm{OH}-\mathrm{CHBr}-\mathrm{CHOH}-\mathrm{CHBr}-\mathrm{CH}_{2} \mathrm{OH}\) is

Step-by-Step Solution

Answer

The compound has 2 optically active isomers.

Step 1: Identify Chiral Centers

The compound is given as \( ext{CH}_2 ext{OH-CHBr-CHOH-CHBr-CH}_2 ext{OH}\). To find the number of optically active isomers, start by identifying the chiral centers. A chiral center is a carbon atom that is bonded to four different groups. In this compound, the two carbon atoms \( ext{CHBr}\) and \( ext{CHOH}\) can be chiral centers if they are bonded to four different substituents.

Step 2: Assign Chiral Centers

Examine the structure: the second carbon \( ext{CHBr}\) is attached to bromine, hydrogen, and different parts of the chain (from the left and right sides). The third carbon \( ext{CHOH}\) is attached to a hydroxyl group, hydrogen, and different chain parts. These differences confirm that both \( ext{CHBr}\) and \( ext{CHOH}\) are chiral centers.

Step 3: Calculate Total Isomers

Each chiral center can exist in two different configurations (R and S, or left and right handedness). Therefore, if there are \(n\) chiral centers, the maximum number of stereoisomers is \(2^n\). Since there are two chiral centers here, \(n = 2\), we have \(2^2 = 4\) stereoisomers.

Step 4: Evaluate Optical Activity

Not all stereoisomers are optically active. Optical activity arises only when there are enantiomers (non-superimposable mirror images). The four stereoisomers include two pairs of enantiomers, which are optically active, while the other two are diastereomers, which may be racemic (optically inactive in their mixture). Therefore, the compound will have 2 optically active isomers corresponding to the possible enantiomers.

Key Concepts

Chiral Centers

A chiral center is a key concept in understanding optically active isomers. It refers to a carbon atom bonded to four distinct groups or atoms, making it asymmetric. This asymmetry prevents the carbon from being superimposed on its mirror image, much like how your left and right hands are mirror images but not identical. In the compound under discussion, chiral centers are identified at specific positions along the carbon chain:

• Each chiral center can be a potential point of isomerism.
• The configurations of these centers significantly impact the molecule's optical activity.
• In this example, the chiral centers are at the carbon atoms labeled CHBr and CHOH.

Identifying the number and positions of chiral centers is crucial because it helps predict the number of possible stereoisomers of a molecule.

Stereoisomers

Stereoisomers are molecules that have the same molecular formula and sequence of bonded atoms, but differ in the three-dimensional orientations of their atoms. This category of isomers includes both enantiomers and diastereomers, though only enantiomers affect optical activity. Here’s a quick breakdown:

• Stereoisomers emerge from different spatial arrangements, particularly around chiral centers.
• If a compound has \(n\) chiral centers, the maximum potential number of stereoisomers is given by the formula \(2^n\).
• For our given compound, with two chiral centers, there are \(2^2 = 4\) possible stereoisomers.

Understanding stereoisomers is essential to grasp how identical chemical compositions can lead to different physical or chemical properties.

Enantiomers

Enantiomers are a specific type of stereoisomer. They are non-superimposable mirror images of each other and play a critical role in determining whether a compound is optically active. Here's what you need to know:

• For a compound to be optically active, at least some of its stereoisomers must be enantiomers.
• Given that our compound has four stereoisomers, these include two pairs of enantiomers.
• Each pair of enantiomers affects polarized light, rotating it in opposite directions.

The presence of enantiomers confirms optically active forms, distinct from diastereomers, which might not influence light. Therefore, in the context of the problem, the compound has 2 optically active isomers due to these enantiomeric pairs.