Question

What is the relationship between the specific rotations of \((2 R, 3 R)-\) dichloropentane and \((2 S, 3 S)\) -dichloropentane? Between \((2 R, 3 S)-\) dichloropentane and ( \(2 R, 3 R\) )-dichloropentane?

Short answer

Enantiomers have equal and opposite specific rotations, while diastereomers do not have a predictable relationship.

Step-by-step solution

Understanding Optical Isomers

Specific rotation is the measure of a compound's ability to rotate plane-polarized light. In the case of chiral molecules, the orientation of groups around chiral centers determines the direction and magnitude of rotation.

Checking Enantiomer Relationship

(2 R, 3 R)-dichloropentane and (2 S, 3 S)-dichloropentane are enantiomers because they are non-superimposable mirror images of each other. Enantiomers have equal but opposite specific rotations.

Determining Specific Rotations of Enantiomers

Since (2 R, 3 R)-dichloropentane and (2 S, 3 S)-dichloropentane are enantiomers, if the specific rotation of (2 R, 3 R) is \[ + \alpha \], then the specific rotation of (2 S, 3 S) will be \[ - \alpha \].

Checking Diastereomer Relationship

(2 R, 3 S)-dichloropentane and (2 R, 3 R)-dichloropentane are diastereomers as they have different configurations at one of multiple chiral centers. Diastereomers do not necessarily have equal and opposite specific rotations.

Comparing Specific Rotations of Diastereomers

The relationship between the specific rotations of diastereomers, like (2 R, 3 S) and (2 R, 3 R)-dichloropentane, depends on the spatial arrangement of atoms, but they are independent of each other and not equal and opposite.

Key concepts

Specific Rotation

Specific rotation is a fundamental concept in the study of optical activity in chiral compounds. It provides insight into how a particular compound interacts with light. When plane-polarized light passes through an optically active compound, the light is rotated either clockwise or counterclockwise. This ability to rotate light is quantified as the specific rotation, usually denoted by \( [\alpha] \).

• Measurement: Specific rotation is measured using a polarimeter, a device designed to measure the angle through which the light is rotated.
• Formula: The specific rotation is calculated using the formula \([\alpha] = \frac{\alpha}{l \times c}\), where \(\alpha\) is the observed rotation in degrees, \(l\) is the path length of the light through the sample in decimeters, and \(c\) is the concentration of the sample in grams per milliliter.
• Sign: If the specific rotation is positive, the compound rotates light to the right (dextrorotatory). If negative, it rotates light to the left (levorotatory).

Understanding specific rotation is crucial as it helps to differentiate between enantiomers, which have specific rotations that are equal in magnitude but opposite in direction.

Enantiomers

Enantiomers are a fascinating type of stereoisomers. They are molecules that are mirror images of each other but cannot be superimposed. This is similar to how your left and right hands are mirror images yet cannot perfectly overlay each other. Enantiomers are closely related to the concept of chirality in chemistry.

• Chirality: A molecule is chiral if it has an asymmetrical carbon atom with four different groups attached. This lack of symmetry at one or more centers leads to the existence of two non-superimposable mirror images, namely enantiomers.
• Optical Activity: Enantiomers exhibit unique optical activities. Each enantiomer will rotate plane-polarized light to an equal degree, but in opposite directions.
• Example: The specific patterns of optical rotation in enantiomers explain how (2 R, 3 R)-dichloropentane and (2 S, 3 S)-dichloropentane have specific rotations of \(+\alpha\) and \(-\alpha\) respectively.

Understanding enantiomers is essential in various fields, including pharmacology, as the biological effects of enantiomers can differ significantly.

Diastereomers

Diastereomers are another intriguing class of stereoisomers. Unlike enantiomers, they are not mirror images of each other. Diastereomers occur when two or more stereocenters in a molecule are altered in configuration.

• Characteristics: Diastereomers have different physical properties, such as melting points and boiling points, unlike enantiomers which share these properties.
• Optical Activity: The specific rotations of diastereomers are not necessarily related in a symmetrical manner. For example, the specific rotations of (2 R, 3 S)-dichloropentane and (2 R, 3 R)-dichloropentane do not correlate as directly as enantiomers do.
• Comparison: In contrast with enantiomers, the spatial arrangement and not mere reflection contributes to the differences in diastereomers. This results in varied but unrelated optical activity.

Diastereomers play a significant role in chemical synthesis and the study of stereochemistry, providing valuable insights into the spatial configuration of molecules.