Problem 89 - Advanced Problems in Organic Chemistry for Competitive Examinations

Question

The specific rotation of \((2 \mathrm{R}, 3 \mathrm{R})-(+)-\) tartaric acid is \(+12.4^{\circ}\left(\mathrm{c}=2, \mathrm{H}_{2} \mathrm{O}\right)\). The optical rotation of a solution made up from \(10 \mathrm{~g}\) of a mixture \(\mathrm{B}\) and \(\mathrm{C}\) in \(20 \mathrm{ml}\) of water was measured in a \(10 \mathrm{~cm}\) cell. The observed rotation was \(-3.1^{\circ} .\) What is the specific rotation of the mixture? (a) \(-3.1^{\circ}\) (b) \(-6.2^{\circ}\) (c) \(-0.62^{\circ}\) (d) \(+12.4^{\circ}\)

Step-by-Step Solution

Answer

The specific rotation of the mixture is (b) -6.2\,^{\circ}\,\text.

Step 1: Understand the Formula for Specific Rotation

The specific rotation (D_{[20]} 25) is calculated using the formula:\[[\alpha] = \frac{\alpha_{obs}}{c \times l}\]where \(\alpha_{obs}\) is the observed optical rotation, \(c\) is the concentration of the solution in g/mL, and \(l\) is the path length in dm (1 dm = 10 cm).

Step 2: Define Given Values

From the problem, \(\alpha_{obs} = -3.1^{\circ}\), the solution contains \(10\,\text{g}\) of mixture in \(20\,\text{mL}\) of water, making \(c = \frac{10}{20} = 0.5\,\text{g/mL}\), and \(l = 10\,\text{cm} = 1\,\text{dm}\).

Step 3: Substitute Values into the Formula

Substitute the known values into the formula:\[[\alpha] = \frac{-3.1}{0.5 \times 1} = \frac{-3.1}{0.5} = -6.2\,^{\circ}\]This calculation determines the specific rotation of the mixture.

Key Concepts

Specific Rotation

Understanding specific rotation is key to exploring the behavior of mixtures and solutions in optical activity. Specific rotation is the standardized measure of a compound's ability to rotate plane-polarized light. It provides a unique value for each compound, enabling chemists to identify and differentiate substances.
The formula to calculate specific rotation is:

• \([\alpha] = \frac{\alpha_{obs}}{c \times l}\)

Here:

• \(\alpha_{obs}\) is the observed optical rotation, indicating how much rotation occurs in the solution being studied.
• \(c\) is the concentration of the solution in grams per milliliter.
• \(l\) is the path length the light travels through the solution, measured in decimeters (1 dm = 10 cm).

In the context of the original exercise, specific rotation was calculated using an observed rotation of \(-3.1^{\circ}\), a concentration of \(0.5 \, \text{g/mL}\), and a path length of 1 dm. This led to a specific rotation of \(-6.2^{\circ}\). This calculated value gives insight into the optical properties and identity of the mixture studied.

Chiral Compounds

Chiral compounds are fascinating because they can exist in two mirror-image forms that are not superimposable, much like your left and right hands. This characteristic is due to the presence of what’s known as a chiral center, typically a carbon atom with four different groups attached to it.
Chirality is a key concept in chemistry because:

• It affects how a compound interacts with other chiral substances, which is crucial in biological systems.
• Chiral compounds often have different properties and activities despite having the same molecular formula. For example, one enantiomer of a chiral drug might be effective, while its counterpart could be harmful.

Chiral compounds often show optical activity, meaning they can rotate the plane of polarized light. The direction and degree of this rotation can distinguish between different chiral compounds, playing a key role in the field of stereochemistry and chemical identification.

Enantiomers

Enantiomers are special forms of chiral compounds. They are a pair of molecules that are non-superimposable mirror images of each other. This means that, even though the molecules have the same structure and properties, they behave differently in chiral environments.
Key characteristics of enantiomers include:

• They rotate plane-polarized light in equal but opposite directions. One enantiomer will rotate light to the right (\(+\) direction), and the other will rotate light to the left (\(-\) direction). This property is crucial in distinguishing between enantiomers.
• Despite having identical chemical properties, enantiomers differ in how they interact with other chiral molecules, which can have vast implications, especially in biological systems and pharmaceuticals.

In the context of the original exercise, understanding enantiomers helps in interpreting optical rotation data. The specific arrangement of atoms in each enantiomer affects how they interact with light, thus impacting the measured optical rotation. Recognizing enantiomers and their properties is essential to evaluating the chiral nature and composition of chemical mixtures.