A chemist finds that the addition of (+)-epinephrine to the catalytic reduction of butan-2-one (Figure 5-17 ) gives a product that is slightly optically active, with a specific rotation of +0.45°. Calculate the percentages of (+)-butan-2-ol and (−)-butan-2-ol formed in this reaction.

Verified step by step guidance

1. Understand the problem: The reaction involves the catalytic reduction of butan-2-one to butan-2-ol in the presence of (+)-epinephrine, which results in a product that is optically active. This indicates that the reaction produces an unequal mixture of the enantiomers (+)-butan-2-ol and (−)-butan-2-ol. The specific rotation of the product is given as +0.45°.
2. Recall the formula for specific rotation: The specific rotation of a mixture of enantiomers is given by the equation: $\mathrm{[\alpha ]}$_mixture = (f₊ × [α]₊) + (f_- × [α]_-) , where $f$₊ and $f$_- are the fractions of the (+) and (−) enantiomers, respectively, and $\mathrm{[\alpha ]}$₊ and $\mathrm{[\alpha ]}$_- are their specific rotations. Note that $\mathrm{[\alpha ]}$_- = -[α]₊ for enantiomers.
3. Set up the equation: The specific rotation of pure (+)-butan-2-ol is a known value (look up this value in a reference, typically around +13.5°). Let $f$₊ represent the fraction of (+)-butan-2-ol and $f$_- represent the fraction of (−)-butan-2-ol. Since the mixture is optically active with a specific rotation of +0.45°, substitute the known values into the equation: $0.45=\; (f$₊ × 13.5) + (f_- × -13.5).
4. Use the relationship between fractions: The total fraction of the two enantiomers must equal 1, so $f$₊ + f_- = 1. Solve for $f$_- in terms of $f$₊: $f$_- = 1 - f₊. Substitute this into the specific rotation equation.
5. Solve for the fractions: Substitute $f$_- = 1 - f₊ into the equation $0.45=\; (f$₊ × 13.5) + ((1 - f₊) × -13.5). Simplify and solve for $f$₊. Once $f$₊ is determined, calculate $f$_- using $f$_- = 1 - f₊. Convert these fractions to percentages by multiplying by 100.

Key Concepts

Optical Activity

Optical activity refers to the ability of chiral compounds to rotate plane-polarized light. This property is crucial in organic chemistry for distinguishing between enantiomers, which are non-superimposable mirror images of each other. The specific rotation, measured in degrees, quantifies this rotation and helps determine the composition of a mixture of enantiomers.

Chirality and Enantiomers

Chirality is a property of a molecule that makes it non-superimposable on its mirror image, often due to the presence of a chiral center, typically a carbon atom bonded to four different groups. Enantiomers are pairs of chiral molecules that are mirror images of each other, and they can exhibit different biological activities and properties, making their separation and quantification important in synthesis and analysis.

Catalytic Reduction

Catalytic reduction is a chemical reaction that involves the addition of hydrogen to a compound, often facilitated by a catalyst, to reduce functional groups such as ketones to alcohols. In the context of butan-2-one, the reaction produces butan-2-ol, and the presence of a chiral catalyst like (+)-epinephrine can influence the stereochemical outcome, leading to the formation of enantiomers in varying proportions.