Question

The correct statement \((\mathrm{s})\) about the compound \(\mathrm{H}_{3} \mathrm{C}(\mathrm{HO}) \mathrm{HC}-\mathrm{CH}=\mathrm{CH}-\mathrm{CH}(\mathrm{OH}) \mathrm{CH}_{3}(\mathbf{X})\) is(are) (A) The total number of stereoisomers possible for \(\mathbf{X}\) is 6 (B) The total number of diastereomers possible for \(\mathbf{X}\) is 3 (C) If the stereochemistry about the double bond in \(\mathbf{X}\) is trans, the number of enantiomers possible for \(\mathbf{X}\) is 4 (D) If the stereochemistry about the double bond in \(\mathbf{X}\) is cis, the number of enantiomers possible for \(\mathbf{X}\) is 2

Short answer

The total number of stereoisomers is 4 (A is incorrect), the number of diastereomers is 2 (B is incorrect), there are 2 enantiomers for both cis and trans configurations (C and D are incorrect).

Step-by-step solution

- Identify Chirality Centers and Double Bond Configuration

Examine the molecular structure given to determine the number of chiral centers and whether the double bond is cis or trans. Chiral centers are carbon atoms with four different groups attached. The compound has two chiral centers represented by the hydroxyl-bearing carbons. The double bond also can have trans or cis configuration, but it does not produce enantiomers.

- Calculate Number of Stereoisomers

The number of stereoisomers for a compound can be calculated using the formula 2^n, where n is the number of chiral centers. Here, n is 2, so the total number of stereoisomers is 2^2 = 4.

- Determine Number of Diastereomers

Diastereomers are stereoisomers that are not mirror images of each other. There are 4 stereoisomers in total. Out of these, any two that are enantiomers (mirror images) are not considered diastereomers of each other. Therefore, there are (4 - 2) = 2 diastereomers, one pair for each enantiomeric pair.

- Analyze the Impact of Double Bond Configuration

The configuration of the double bond (cis or trans) does not affect the number of chiral centers. Since enantiomers are mirror images, for each cis or trans configuration, there will be 2 enantiomers, making a total of 2 for cis and 2 for trans.

Key concepts

Stereoisomers

Stereoisomers are molecules that have the same molecular formula and sequence of bonded atoms (constitution), but differ in the three-dimensional orientations of their atoms in space. This phenomenon arises due to the presence of chiral centers, geometric isomerism, or other types of stereo-specific arrangements.

There are several types of stereoisomers, including enantiomers, diastereomers, and cis-trans isomers, each with unique properties and biological activities. Understanding these distinctions is important in fields like chemistry, pharmacology, and molecular biology because the spatial arrangement of atoms can significantly influence the behavior of a molecule.

Chiral Centers

A chiral center, often a carbon atom, is a central atom that has four different substituents attached to it, leading to non-superimposable mirror images called enantiomers. The presence of a chiral center in a molecule means that there are at least two configurations (stereoisomers) that are mirror images of each other. These configurations are referred to as 'R' or 'S' based on the Cahn-Ingold-Prelog priority rules.

Chiral molecules are not identical to their mirror images, much like how left and right hands are mirror images but not identical. This property has significant effects on the way these molecules interact with other chiral entities, such as enzymes or receptors in biological systems.

Diastereomers

Diastereomers are a type of stereoisomer that are not mirror images of each other. Unlike enantiomers, diastereomers can have different physical properties and chemical reactivities. They occur in molecules with two or more chiral centers that have the same connectivity but different arrangements of their atoms in space. Since they are not mirror images, they are not optically active in pairs and have different melting points, boiling points, and solubilities.

In the given compound, the diastereomers arise from the combination of the configurations at the chiral centers. With two chiral centers, you can have a maximum of 2^2 (four) stereoisomers, which includes both enantiomers and diastereomers.

Enantiomers

Enantiomers are stereoisomers that are non-superimposable mirror images of each other, much like one's left and right hands. These isomers have identical physical properties in a symmetric environment, but they can exhibit different behaviors in a chiral environment, such as interacting with other chiral molecules. One notable difference between enantiomers is the way they rotate plane-polarized light; one isomer will rotate the light to the left (levorotatory) while the other will rotate it to the right (dextrorotatory).

When related to the compound under discussion, because there are two chiral centers, there are two pairs of enantiomers. The presence of a double bond does not create additional enantiomers since the cis or trans configuration of a double bond does not yield mirror images.

Cis-Trans Isomerism

Cis-trans isomerism is a form of stereoisomerism where atoms or groups of atoms are positioned either on the same side (cis) or opposite sides (trans) of a double bond or a cycloalkane ring. The terms 'cis' and 'trans' are Latin for 'on this side' and 'acroad', respectively. This type of isomerism is significant because it affects the physical and chemical properties of molecules.

In the compound in question, the double bond could adopt either a cis or trans configuration. The configuration impacts the spatial arrangement of the remaining atoms and how the molecule might interact with other compounds. However, a double bond itself, without considering the chirality of surrounding groups, does not produce enantiomers but may affect the overall chiral nature of the molecule when considered in conjunction with chiral centers.