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 4 optically active isomers.

Step 1: Identify the Chiral Centers

A chiral center is a carbon atom that is bonded to four different groups. In the compound \(\mathrm{CH}_{2}\mathrm{OH}-\mathrm{CHBr}-\mathrm{CHOH}-\mathrm{CHBr}-\mathrm{CH}_{2}\mathrm{OH}\), we identify two chiral centers: the second and fourth carbon atoms, which are \(\mathrm{CHBr}\) atoms.

Step 2: Calculate the Number of Stereoisomers

The maximum number of stereoisomers for a molecule with \(n\) chiral centers is given by the formula \(2^n\). In this compound, there are 2 chiral centers, so the maximum number of stereoisomers is \(2^2 = 4\).

Step 3: Identify Optically Active Isomers

Optically active isomers are those that can rotate plane-polarized light. Since none of the chiral centers are identical and there is no internal mirror plane making the compound a meso compound, all the 4 stereoisomers are optically active.

Key Concepts

Chiral Centers

Chiral centers are essential in understanding optically active compounds. They are specific carbon atoms in a molecule where four different groups are attached. This unique configuration leads to the possibility of different spatial arrangements of these groups, resulting in unique isomers. The presence of chiral centers is what imparts chirality to a compound, allowing it to have non-superimposable mirror images, much like left and right hands.

In the given compound, \(\mathrm{CH}_{2}\mathrm{OH}-\mathrm{CHBr}-\mathrm{CHOH}-\mathrm{CHBr}-\mathrm{CH}_{2}\mathrm{OH}\), we identify the chiral centers as the second and fourth carbon atoms where \(\mathrm{CHBr}\) occurs. Here’s why this identification is crucial:

• Each chiral center acts like a switch, where the spatial arrangement can lead to different configurations - all potentially optically active.
• The chirality of these centers means that the molecule cannot be superimposed on its mirror image, leading to distinct isomers.

Identifying chiral centers is the first step in understanding a compound's stereochemistry and predicting its number of stereoisomers.

Stereoisomers

Stereoisomers are isomers that differ in the spatial arrangement of atoms, rather than the sequence of bonded atoms. They share the same structural formula but differ from each other in the three-dimensional orientation of their atoms. This is where chiral centers come into play, as they provide the possible variations in space.

For our compound, with two chiral centers, we apply the formula \(2^n\) to calculate the maximum number of stereoisomers. Here, \(n\), the number of chiral centers, is 2, resulting in \(2^2 = 4\) stereoisomers for this compound.

Consider this about stereoisomers:

• They include both enantiomers, which are mirror images that do not superimpose.
• Also includes diastereomers, which are not mirror images.

Understanding stereoisomers is crucial for predicting the different three-dimensional forms a molecule can take and their potential unique properties, including optical activity.

Plane-Polarized Light

The concept of plane-polarized light is essential in the study of optical activity. Plane-polarized light is light that oscillates in a single plane, as opposed to normal light, which vibrates in multiple planes. Chiral molecules interact with this light in a way that causes it to rotate, a phenomenon quantified as optical rotation. This unique behavior is a characteristic feature of optically active compounds.

In context, for the compound \(\mathrm{CH}_{2}\mathrm{OH}-\mathrm{CHBr}-\mathrm{CHOH}-\mathrm{CHBr}-\mathrm{CH}_{2}\mathrm{OH}\), determining its optical activity involves exploring how its stereoisomers interact with plane-polarized light.

• Compounds with no internal plane of symmetry (not meso) have optical activity.
• If a compound’s stereoisomers can rotate plane-polarized light, they are referred to as optically active isomers.

By identifying the compound’s chiral centers, we know that all the stereoisomers in this scenario are optically active because no two chiral centers are historically identical and no meso form is present, guaranteeing interaction with plane-polarized light.