The correct statements about the following compound are $$ \mathrm{CH}_{3}(\mathrm{OH}) \mathrm{CH}-\mathrm{CH}=\mathrm{CH}-\mathrm{CH}(\mathrm{OH}) \mathrm{CH}_{3} $$ (a) The total number of stereoisomers possible is 6 . (b) The total number of diastereomers possible is \(3 .\) (c) The number of enantiomers possible for the cis isomer is \(2 .\) (d) The number of enantiomers possible for the trans isomer is 4 .

Chiral centers, also known as stereocenters, are atoms in a molecule that have four different groups attached to them. This arrangement means that the molecule can exist in two non-superimposable mirror images, similar to the relationship between left and right hands.

For example, consider a chiral center with four different groups labeled A, B, C, and D. Due to the different spatial arrangements of these groups, two distinct configurations, or enantiomers, can be formed. These are designated as R (from the Latin rectus, meaning right) or S (from the Latin sinister, meaning left), depending on the arrangement of the groups in relation to each other.

When calculating the number of stereoisomers for a molecule, one must consider the presence of these chiral centers. The general formula is \(2^n\), where \(n\) is the number of chiral centers. If a molecule has two chiral centers, theoretically, there can be up to four stereoisomers, comprising a mix of enantiomers and possibly diastereomers as well.

Geometric isomers, also known as cis-trans isomers, are a type of stereoisomerism that arises from the inflexibility of double bonds. Since atoms cannot rotate around double bonds, this leads to two distinct configurations:

1. Cis-isomer: Identical or similar groups are on the same side of the double bond.
2. Trans-isomer: Identical or similar groups are on opposite sides of the double bond.

In the context of the exercise, the molecule contains a double bond which allows for these two configurations. This kind of isomerism adds another layer of complexity to stereochemistry by creating distinct molecules that have different physical and chemical properties despite having the same molecular formula.

Diastereomers are stereoisomers that are not mirror images of each other, unlike enantiomers. This difference is crucial as diastereomers can have completely different physical properties, such as melting points, boiling points, and solubilities.

In the exercise, diastereomers arise from the combination of chiral centers and the presence of cis-trans isomerism. For instance, a trans-isomer of a molecule with chiral centers could be a diastereomer to the cis-isomer of that same molecule. Moreover, each cis or trans form can have additional diastereomers due to the different configurations at the chiral centers.

The practical importance of recognizing diastereomers lies in their separability. Unlike enantiomers, which often require specialized methods for separation, diastereomers can usually be separated by conventional means due to their differing physical properties.

Enantiomers are pairs of molecules that are non-superimposable mirror images of each other. Just like our hands, enantiomers are related through reflection, not rotation. The existence of enantiomers is a direct consequence of chirality in molecules.

An important characteristic of enantiomers is that they have identical physical properties, such as melting points and boiling points, but they can have different chemical behaviors, especially in chiral environments. For instance, enantiomers may have different reactions with other chiral molecules, including biological structures like enzymes, which can lead to significantly different biological activities.

In the given exercise, for each geometric isomer (cis and trans), there are two pairs of enantiomers. Since they only differ by their spatial configuration around their chiral centers, they are chemically identical in achiral environments but can have drastic differences in chiral ones. When solutions state the number of enantiomers for a given isomer, as seen in the exercise, it's crucial to remember that enantiomers always come in pairs.