Problem 16 - Advanced Problems in Organic Chemistry for Competitive Examinations

Question

Identify specific rotation of a mixture have \(5 \mathrm{~g}\) of \((+) 2\) -butanol and \(7 \mathrm{~g}\) of \((-) 2\) -butanol (If specific rotation of \(100 \%\) pure compound is \(13.5^{\circ}\) ) (a) \(-2.25^{\circ}\) (b) \(-1.6^{\circ}\) (c) \(12.2^{\circ}\) (d) \(4.45^{\circ}\)

Step-by-Step Solution

Answer

The specific rotation of the mixture is \(-2.25^{\circ}\), which corresponds to option (a).

Step 1: Understanding Specific Rotation

Specific rotation is given as the angle through which plane-polarized light is rotated when it passes through a solution of a chiral compound. It is defined by the formula: \[ [\alpha] = \frac{\alpha}{l \cdot c} \]where \( \alpha \) is the observed rotation, \( l \) is the path length in decimeters, and \( c \) is the concentration in grams per milliliter.

Step 2: Calculate Total Mass

First, calculate the total mass of the mixture: \[ \text{Total mass} = 5 \text{ g} + 7 \text{ g} = 12 \text{ g} \]

Step 3: Calculate Contribution of Each Enantiomer to Rotation

Since we need the effect of individual enantiomers:- For \((+) 2\)-butanol, use its portion of the total mass: \( \frac{5}{12} \times 13.5^{\circ} = 5.625^{\circ} \)- For \((- ) 2\)-butanol, use its portion: \( \frac{7}{12} \times (-13.5^{\circ}) = -7.875^{\circ} \).

Step 4: Combine Contributions for Net Specific Rotation

The net specific rotation is the algebraic sum of the contributions from both enantiomers:\[ 5.625^{\circ} + (-7.875^{\circ}) = -2.25^{\circ} \]

Step 5: Single Absorption Path Length/Zeroth Concentration Assumption

Assume standard conditions where path length \( l \) is 1 dm and total concentration \( c \) equals the total mass. Then specific rotation directly aligns with the net rotation value from the mixture contributions.

Step 6: Identify the Correct Option

Review the calculated specific rotation of \( -2.25^{\circ} \). Match it with the provided options. The closest match is (a) \(-2.25^{\circ}\).

Key Concepts

Chiral Compounds

Chiral compounds are fascinating molecules that are not superimposable on their mirror images. Imagine your left and right hands. They look similar, but no matter how you turn them, they won't perfectly overlay each other. That's chirality! In chemistry, chiral compounds feature a central carbon atom bonded to four different groups or atoms.
This central carbon is known as a stereocenter or chiral center. It's this asymmetry that enables chiral compounds to interact differently with polarized light, leading to unique properties.
You'll frequently encounter chiral compounds in the world of organic chemistry and biology. They are crucial because certain biological receptors are selective to the chirality of the compound they bind with. Thus, understanding chiral compounds helps us grasp concepts in many applications including pharmacology, where the chirality can affect a drug's therapeutic efficacy.

Optical Activity

Optical activity is a unique phenomenon associated with chiral compounds. When plane-polarized light passes through a solution containing a chiral compound, the light's plane of polarization is rotated. This property is precisely what optical activity refers to.
The degree of rotation is measured using a polarimeter and is influenced by several factors, including the concentration of the solution, the length of the path the light travels through, and the intrinsic properties of the chiral compound itself.

• Positive rotations (2+2) are known as dextrorotatory.
• Negative rotations (2-2) are referred to as levorotatory.

For the specific rotation equation \( [\alpha] = \frac{\alpha}{l \cdot c} \) we'll define each part:

• \( \alpha \): the observed rotation
• \( l \): path length (in dm)
• \( c \): concentration (in g/mL)

When optically active compounds are mixed, they can counteract each other's rotation, leading to various observable effects.

Enantiomers

Enantiomers are like the left and right hands of molecular structures. They are chiral compounds that are mirror images of each other but cannot be superimposed. Essentially, they are pairs of molecules with exactly opposite configurations at each chiral center.
Each enantiomer will rotate plane-polarized light, but in opposite directions with equal magnitudes. This characteristic is crucial when dealing with mixtures of enantiomers, as seen in racemic mixtures, where the rotations cancel out resulting in no net optical activity.
Understanding enantiomers is vital for industries such as pharmaceuticals. Often, only one enantiomer is therapeutically active while the other may be inactive or even harmful. Correct identification and usage can thus dictate the efficacy and safety of a drug. Additionally, they are useful in food flavors and fragrances, impacting how compounds smell and taste. Recognizing and distinguishing enantiomers leads to advancements and precision in various scientific fields.