Question

Enantiomers of a compound is taken in a container and mixture is prepared contan?ng \(\mathrm{X}_{d}\) mole fraction of dextroisomer with its specific angle of rotation \(\theta_{d}\). The observed angle of rotation of mixture when plane polarised light passed in container is (A) \(\mathrm{X}_{d} \theta_{d}\) (B) \(\theta_{d}\left(2 \mathrm{X}_{d}-1\right)\) (C) \(\theta_{d}\left(3 \mathrm{X}_{d}-2\right)\) (D) \(\theta_{d}\left(1-X_{d}\right)\)

Short answer

The observed angle of rotation of the mixture when plane-polarized light is passed through the container is \(\theta_{obs} = \theta_{d}(2\mathrm{X}_d - 1)\). Thus, the correct answer is (B) \(\theta_{d}\left(2 \mathrm{X}_{d}-1\right)\).

Step-by-step solution

Recall the formula for angle of rotation of a mixture

The observed angle of rotation (\(\theta_{obs}\)) of a mixture of enantiomers is given by the product of the mole fraction and specific rotation of each enantiomer in the mixture, which can be written as: \(\theta_{obs} = \mathrm{X}_d \theta_{d} + \mathrm{X}_l \theta_{l}\) where \(\mathrm{X}_d\) and \(\mathrm{X}_l\) are the mole fractions of the dextroisomer and laevoisomer respectively, and \(\theta_{d}\) and \(\theta_{l}\) are their specific angles of rotation.

Relate \(\mathrm{X}_d\) and \(\mathrm{X}_l\)

Since \(\mathrm{X}_d\) and \(\mathrm{X}_l\) are mole fractions, they must sum up to 1, so we can write: \(\mathrm{X}_l = 1 - \mathrm{X}_d\)

Use the relationship between specific rotations of enantiomers

For enantiomers, their specific angles of rotation are opposite in sign and equal in magnitude, so we can write: \(\theta_{l} = -\theta_{d}\)

Substitute \(\mathrm{X}_l\) and \(\theta_{l}\) into the observed angle of rotation formula

Now, let's substitute the values of \(\mathrm{X}_l\) and \(\theta_{l}\) into the formula for \(\theta_{obs}\): \(\theta_{obs} = \mathrm{X}_d \theta_{d} + (1 - \mathrm{X}_d) (-\theta_{d})\)

Simplify the equation to find the final expression for \(\theta_{obs}\)

Finally, let's simplify the equation to find the expression for \(\theta_{obs}\): \(\theta_{obs} = \theta_{d}( \mathrm{X}_d - 1 + \mathrm{X}_d)\) \(\theta_{obs} = \theta_{d}(2\mathrm{X}_d - 1)\) So, the observed angle of rotation of the mixture when plane-polarized light is passed through the container is: \(\theta_{obs} = \theta_{d}(2\mathrm{X}_d - 1)\) Thus, the correct answer is (B) \(\theta_{d}\left(2 \mathrm{X}_{d}-1\right)\).

Key concepts

Optical Isomerism

Optical isomerism is a fascinating type of stereoisomerism where molecules exist as non-superimposable mirror images. These mirror image molecules are called enantiomers. Enantiomers have identical physical properties, like boiling points and solubility, but they differ in the way they interact with polarized light.
When a compound is optically active, it can rotate plane-polarized light. Enantiomers will rotate light in opposite directions by equal amounts. This means one enantiomer will rotate light to the right (dextrorotatory), while the other will rotate it to the left (levorotatory).
Optical isomerism is crucial because even though enantiomers are chemically identical, biological systems may interact differently with each form, leading to differing effects. This property is especially important in pharmaceuticals, where the efficacy or safety of a drug can depend on which enantiomer is present.

Mole Fraction

Mole fraction is a simple yet vital concept in chemistry that helps us understand the composition of mixtures. It is a way to express the concentration of a component in a mixture. In mathematical terms, mole fraction ( ext{X}) of a component is defined as:

• The ratio of the number of moles of that component to the total number of moles in the mixture.

For a mixture containing two components, like dextroisomer and levoisomer, we can write:

• \(\text{X}_d + \text{X}_l = 1\)

where \(\text{X}_d\) is the mole fraction of the dextroisomer and \(\text{X}_l\) is the mole fraction of the levoisomer.
Understanding and using mole fractions are crucial because it normalizes the composition of mixtures irrespective of the amount. This makes mole fraction a very useful tool in calculating various properties of mixtures, including optical activity.

Specific Rotation

Specific rotation is a property that quantifies the extent to which a chiral compound can rotate plane-polarized light. The specific rotation (\(\theta_d^{[\alpha]}\) ) of a pure enantiomer is defined as the observed angle of rotation when light passes through a sample of known path length and concentration.
It is calculated using the formula:\[\alpha = \frac{\theta}{l \cdot c}\]where:

• \(\theta\) is the observed rotation in degrees,
• \(l\) is the path length in decimeters,
• \(c\) is the concentration in grams per milliliter.

In mixtures of enantiomers, specific rotation helps determine both the composition and optical purity. When the mole fractions of the enantiomers are known, we can calculate the observed rotation using the specific rotations of the individual enantiomers. Importantly, the rotations of enantiomers are equal in magnitude but opposite in direction, so specific rotation provides valuable insights into the proportion of each enantiomer in the mixture.