Problem 110 - A Complete Resource Book in Chemistry for JEE Main

Question

Which is true regarding the \(\mathrm{HOCH}_{2}-(\mathrm{CHOH})_{4}\) \(-\mathrm{CH}_{2} \mathrm{OH} ?\) (1) the number of racemic mixtures will be four for the compound. (2) the total number of optical isomers is ten and number of meso isomer is two. (3) the total number of optical isomer is sixteen and number of meso isomer is zero. (4) the optically active isomers are eight while meso isomers are two. (a) 1,2 and 3 (b) 2,3 and 4 (c) 1,2 and 4 (d) 1 and 2 only

Step-by-Step Solution

Answer

Optically active isomers are 8; meso isomers are 2, thus option (4) is true.

Step 1: Understanding the Compound Structure

The compound given is a sugar alcohol commonly known as sorbitol. Structurally, it can be written as \( HOCH_2 - (CHOH)_4 - CH_2OH \). This indicates that we have six carbon atoms with the first and last as primary alcohols and four secondary alcohols in between.

Step 2: Determine the Stereocenters

In the molecular structure, the four \( CHOH \) groups each represent a stereocenter, as the presence of both the \( OH \) and hydrogen groups means these carbons are chiral.

Step 3: Calculating Possible Stereoisomers

For a compound with \( n \) stereocenters, the maximum number of stereoisomers is \( 2^n \). Here, \( n = 4 \), so the maximum number of stereoisomers is \( 2^4 = 16 \).

Step 4: Identify Meso Isomers

Meso compounds are those that have multiple stereocenters but also an internal plane of symmetry causing them to be non-chiral. Sorbitol has two meso isomers among these 16 isomers.

Step 5: Calculate Optical Isomers

Optical isomers are those which are chiral, meaning they exist in forms that are non-superimposable mirror images. From the 16 total isomers, after excluding the 2 meso compounds, we have \( 16 - 2 = 14 \) optical isomers.

Step 6: Determine Racemic Mixtures

A racemic mixture consists of equal amounts of left- and right-handed enantiomers. Since we have 14 optical isomers, these can form \( \frac{14}{2} = 7 \) racemic mixtures.

Step 7: Analyze Options Against Findings

Comparing with the options: (1) is not true because there are 7 racemic mixtures, not 4. (2) is incorrect because there are 14 optical, not 10. (3) is incorrect because there are 16 stereoisomers, not 16 optical isomers. (4) is true as there are 8 optically active isomers when considering two sets of enantiomers. Thus, option (c) 1,2 and 4 is incorrect, option (b) 2,3 and 4 is incorrect, option (d) 1 and 2 only is incorrect.

Key Concepts

Stereocenters

Stereocenters are specific atoms within a molecule where the interchange of two groups produces a stereoisomer. In simpler terms, they are carbon atoms bonded to four different groups, making the molecule chiral around that point. A chiral center, or stereocenter, results in molecules having mirror images that cannot be superimposed on each other, like left and right hands.
In the context of sorbitol, a sugar alcohol compound, there are four carbon atoms designated as stereocenters. These are the internal carbon atoms within the structure denoted by the formula \( HOCH_2 - (CHOH)_4 - CH_2OH \). Each of these carbon atoms has an \( OH \) group and an \( H \) group bound to it, qualifying as stereocenters.
The presence of these stereocenters is what allows sorbitol to have multiple optical isomers. For any molecule, the number of possible stereoisomers is given by the formula \( 2^n \), where \( n \) is the number of stereocenters. For sorbitol, with four stereocenters, this leads to a maximum of \( 2^4 = 16 \) possible stereoisomers.

Meso Isomers

Meso isomers are a special type of stereoisomers. They are unique because, even though they contain multiple stereocenters, these compounds are achiral due to an internal plane of symmetry. This symmetry essentially makes one half of the molecule a mirror image of the other half. Thus, they do not exhibit optical activity in polarized light.
In the case of sorbitol, among the 16 possible stereoisomers, two of these are meso isomers. The presence of an internal plane of symmetry within these two particular isomers means that they do not rotate plane-polarized light, distinguishing them from other chiral isomers.
Understanding meso isomers is crucial because they reduce the actual number of optically active isomers a compound can have. Without the consideration of meso isomers, chemists might overestimate the number of active forms of a compound.

Racemic Mixtures

Racemic mixtures consist of equal parts of two enantiomers - that is, a pair of optical isomers that are mirror images of each other. When you mix these isomers in equal amounts, their optical activities cancel each other out, making the mixture optically inactive despite containing chiral molecules.
For sorbitol, after identifying the 16 possible stereoisomers and recognizing the two as meso isomers, 14 optical isomers remain. These 14 are collected into pairs of enantiomers, giving us the potential for 7 racemic mixtures (since each racemic mixture is made from one pair of enantiomers).
Racemic mixtures are significant in chemistry because they often have different properties from the individual enantiomers. Understanding how to separate or utilize these mixtures is important in areas like pharmaceuticals, where the two enantiomers may have different biological activities.