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Ch. 20 - The Organic Chemistry of Carbohydrates
Bruice - Organic Chemistry 8th Edition
Bruice8th EditionOrganic ChemistryISBN: 9780135213711Not the one you use?Change textbook
All textbooksBruice 8th EditionCh. 20 - The Organic Chemistry of CarbohydratesProblem 65
Chapter 21, Problem 65

Calculate the percentages of α\(\alpha\)-D-glucose and β\(\beta\)-D-glucose present at equilibrium from the specific rotations of α\(\alpha\)-D-glucose, β\(\beta\)-D-glucose, and the equilibrium mixture. Compare your values with those given in Section 20.10. (Hint: The specific rotation of the mixture equals the specific rotation of α\(\alpha\)-D-glucose times the fraction of glucose present in the a-form plus the specific rotation of β\(\beta\)-D-glucose times the fraction of glucose present in the β\(\beta\) -form.)

Verified step by step guidance
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Step 1: Understand the problem. The goal is to calculate the percentages of α-D-glucose and β-D-glucose at equilibrium using their specific rotations and the specific rotation of the equilibrium mixture. The hint provided suggests using a weighted average formula based on the fractions of each form.
Step 2: Write the formula for the specific rotation of the equilibrium mixture. Let the specific rotation of α-D-glucose be denoted as \( [\alpha]_\text{α} \), the specific rotation of β-D-glucose as \( [\alpha]_\text{β} \), and the specific rotation of the equilibrium mixture as \( [\alpha]_\text{eq} \). The formula is: \( [\alpha]_\text{eq} = [\alpha]_\text{α} \cdot f_\text{α} + [\alpha]_\text{β} \cdot f_\text{β} \), where \( f_\text{α} \) and \( f_\text{β} \) are the fractions of α-D-glucose and β-D-glucose, respectively.
Step 3: Recognize that the fractions \( f_\text{α} \) and \( f_\text{β} \) must add up to 1, since they represent the total glucose present. Therefore, \( f_\text{β} = 1 - f_\text{α} \). Substitute \( f_\text{β} \) into the formula: \( [\alpha]_\text{eq} = [\alpha]_\text{α} \cdot f_\text{α} + [\alpha]_\text{β} \cdot (1 - f_\text{α}) \).
Step 4: Rearrange the equation to solve for \( f_\text{α} \), the fraction of α-D-glucose. Expand the equation: \( [\alpha]_\text{eq} = [\alpha]_\text{α} \cdot f_\text{α} + [\alpha]_\text{β} - [\alpha]_\text{β} \cdot f_\text{α} \). Combine terms: \( [\alpha]_\text{eq} = f_\text{α} \cdot ([\alpha]_\text{α} - [\alpha]_\text{β}) + [\alpha]_\text{β} \). Solve for \( f_\text{α} \): \( f_\text{α} = \frac{[\alpha]_\text{eq} - [\alpha]_\text{β}}{[\alpha]_\text{α} - [\alpha]_\text{β}} \).
Step 5: Calculate \( f_\text{β} \) using \( f_\text{β} = 1 - f_\text{α} \). Convert the fractions \( f_\text{α} \) and \( f_\text{β} \) into percentages by multiplying by 100. Compare the calculated percentages with the values provided in Section 20.10 to verify accuracy.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Specific Rotation

Specific rotation is a property of chiral compounds that quantifies the degree to which they rotate plane-polarized light. It is defined as the observed rotation of light divided by the path length and the concentration of the solution. This property is crucial for distinguishing between different anomers of sugars, such as a-D-glucose and b-D-glucose, as each form has a unique specific rotation.
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Equilibrium Mixture

An equilibrium mixture in the context of sugar anomers refers to the state where the rates of conversion between a-D-glucose and b-D-glucose are equal, resulting in constant concentrations of both forms. At equilibrium, the proportions of each anomer can be determined using their specific rotations, allowing for the calculation of their respective percentages in the mixture.
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Fraction Calculation

Fraction calculation involves determining the proportion of each component in a mixture based on their contributions to a measurable property, such as specific rotation. In this case, the specific rotation of the equilibrium mixture is a weighted average of the specific rotations of a-D-glucose and b-D-glucose, multiplied by their respective fractions. This relationship is essential for calculating the percentages of each anomer present at equilibrium.
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Related Practice
Textbook Question

How many aldaric acids are obtained from the 16 aldohexoses?

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Textbook Question

An unknown disaccharide gives a positive Tollens' test. A glycosidase hydrolyzes it to D-galactose and D-mannose. When the disaccharide is treated with methyl iodide and Ag2O and then hydrolyzed with dilute HCl, the products are 2,3,4,6-tetra-O-methylgalactose and 2,3,4-tri-O-methylmannose. Propose a structure for the disaccharide.

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Textbook Question

The specific rotation of α-D-galactose is 150.7 and that of β-D-galactose is 52.8. When an aqueous mixture that was initially 70% α-D-galactose and 30% β-D-galactose reaches equilibrium, the specific rotation is 80.2. What is the percentage of α-D-galactose and β-D galactose at equilibrium?

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Textbook Question

Draw each of the following:

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b. α-D-idopyranose  

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Draw the mechanism for the elimination step in the Wohl degredation.

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A hexose is obtained when the residue of a shrub Sterculia setigeria undergoes acid-catalyzed hydrolysis. Identify the hexose from the following experimental information: it undergoes mutarotation; it does not react with Br2; and D-galactonic acid and D-talonic acid are formed when it reacts with Tollens’ reagent.

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