Problem 26 - A Complete Resource Book in Chemistry for JEE Main

Question

(+)-mandelic acid has a specific rotation of \(+158^{\circ}\). What would be the observed specific rotation of a mixture containing \(25 \%\) (-)-mandelic acid and \(75 \%\) (+)-mandelic acid? (a) \(+79^{\circ}\) (b) \(-118.5^{\circ}\) (c) \(-79^{\circ}\) (d) \(+118.5^{\circ}\)

Step-by-Step Solution

Answer

The observed specific rotation of the mixture is \(+79^{\circ}\).

Step 1: Understanding Specific Rotation

Specific rotation is a measure of how much a compound rotates plane-polarized light. The specific rotation of a pure enantiomer is given, and for a racemic mixture (50:50 of enantiomers), the specific rotation is zero.

Step 2: Recognize the Contributions of Each Enantiomer

In this problem, we have 25% of (-)-mandelic acid and 75% of (+)-mandelic acid. The (-)-mandelic acid will have a specific rotation of \(-158^{\circ}\) as it is the opposite enantiomer to (+)-mandelic acid, which has a specific rotation of \(+158^{\circ}\).

Step 3: Calculate the Net Specific Rotation

Calculate the specific rotation of the mixture by considering the contributions from both enantiomers: \[(75\%)(+158^{\circ}) + (25\%)(-158^{\circ}) = (+118.5^{\circ}) + (-39.5^{\circ}) = +79^{\circ}\]

Step 4: Choose the Correct Answer

After calculating, the observed specific rotation of the mixture is +79°. Therefore, the correct option is (a) \(+79^{\circ}\).

Key Concepts

Specific Rotation Calculation

Specific rotation is an important concept in stereochemistry. It measures how much a chiral substance can rotate plane-polarized light.
The value of specific rotation is unique for every chiral compound and is represented by the Greek letter \( \alpha \).
The formula used to calculate specific rotation is:

• \[ \alpha = \frac{\text{observed rotation (degrees)}}{\text{concentration (g/mL)} \times \text{path length (dm)}} \]

Pure enantiomers, which are non-superimposable mirror images of each other, have specific rotations that are equal in magnitude but opposite in direction.
For example, if (+)-mandelic acid has a specific rotation of \(+158^{\circ}\), then its enantiomer, (-)-mandelic acid, will have a specific rotation of \(-158^{\circ}\).
A racemic mixture, which contains equal amounts of each enantiomer, will have a specific rotation of zero because the effects of the two enantiomers cancel each other out.

Enantiomers

Enantiomers are a type of stereoisomer.
They are molecules that are mirror images of each other, like left and right hands, but cannot be superimposed onto one another.
Due to this property, they are also called chiral molecules, and are crucial in the world of optical isomerism. Enantiomers typically have identical physical properties such as melting point and boiling point.
However, they differ in how they interact with plane-polarized light, rotating it in opposite directions.

• An enantiomer that rotates light clockwise is called dextrorotatory and is designated as (+).
• An enantiomer that rotates light counterclockwise is called levorotatory and is designated as (-).

This optical rotation is the distinctive characteristic that helps in identifying enantiomers in a mixture.

Racemic Mixture

A racemic mixture is an equimolar mixture of two enantiomers of a chiral molecule.
In such a mixture, the properties like optical rotation are nullified because the rotations induced by each enantiomer cancel out.
This means the overall specific rotation of a racemic mixture is zero. Racemic mixtures are often denoted with the prefix "rac-" or as "(±)" to signify the presence of both enantiomers in equal proportions.

• An important aspect of racemic mixtures in a laboratory setting is that they can sometimes be resolved back into their individual enantiomers, allowing for the separation of chiral compounds.
• This separation process is crucial in pharmaceuticals as only one enantiomer may be biologically active or required for medication.

Understanding racemic mixtures and how to manipulate them is essential in the field of chemistry, especially in synthesis and drug development.