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

Option (4) is true; the correct answer is (d) 1 and 2 only.

Step 1: Identify compound structure

The given compound is glycerol with four chiral centers, represented as \(\text{HOCH}_2-(\text{CHOH})_4-\text{CH}_2\text{OH}\). Each carbon in \((\text{CHOH})_4\) is a chiral center, giving a total of four chiral centers.

Step 2: Determine possible stereoisomers

The number of possible stereoisomers given \( n \) chiral centers is \(2^n\). Here, \(n = 4\), so the compound can have \(2^4 = 16\) stereoisomers.

Step 3: Identify meso isomers

A portion of these stereoisomers will be meso isomers, which possess internal planes of symmetry. For a compound with four chiral centers, there are two meso forms, since not all configurations will result in asymmetry.

Step 4: Calculate optical isomers

Subtract the number of meso isomers from the total stereoisomers to get optical isomers: \(16 - 2 = 14\) optical isomers exist without symmetry.

Step 5: Analyze racemic mixtures

Racemic mixtures are a combination of optical isomers that result in non-optical activity. Typically, for each chiral center, half of the optical isomers will form racemic mixtures. For four chiral centers, there are \(\frac{16}{2} = 8\) racemic mixtures typically possible, but specific conditions depend on practical pairings.

Step 6: Verify options

(1) False, as the number of racemic mixtures isn't exactly four. (2) False, for given isomer counts. (3) False, incorrect count of meso isomers. (4) True, matching calculated values.

Key Concepts

Chiral Centers

Chiral centers are pivotal in understanding optical isomerism. A molecule with a chiral center has a carbon atom attached to four different groups. This configuration allows the molecule to exist in two non-superimposable mirror images, known as enantiomers. Think of your left and right hands; they are mirror images but cannot be perfectly aligned on top of each other.

In the compound \(\text{HOCH}_2-(\text{CHOH})_4-\text{CH}_2\text{OH}\), there are four chiral centers. Each carbon atom in \((\text{CHOH})_4\) has this characteristic, allowing the compound to have multiple stereoisomers. The principle is that with \(n\) chiral centers, a compound could have \(2^n\) stereoisomers due to varying possible configurations.

For this specific compound, \(n = 4\), so there are potentially \(2^4 = 16\) stereoisomers.

Meso Compounds

Meso compounds offer an intriguing twist to optical isomerism. Despite containing chiral centers, they remain non-optically active. This inactivity arises because they possess an internal plane of symmetry, allowing them to be superimposed on their mirror image.

In our compound example, not all configurations result in optical activity. Specifically, with four chiral centers, two configurations yield meso compounds. This symmetry effectively cancels out the optical activity usually seen in chiral isomers.

Identifying meso forms is essential, as it helps in accurately determining the number of optical isomers a compound can have.

Racemic Mixtures

Racemic mixtures are an important concept in optical isomerism. They consist of equal amounts of two enantiomers, making them optically inactive overall. This occurs because the optical activities of each enantiomer cancel each other out.

In the scenario of four chiral centers, the total number of stereoisomers is \(16\), but the practical count of racemic mixtures hinges on specific conditions, often seen as \(\frac{16}{2} = 8\).

Understanding racemic mixtures is critical in chemistry, especially when questions about a compound's optical activity arise. It's like a balance where equal amounts of left and right-handed configurations lead to no net optical activity.