Question

Calculate the equilibrium ratio of \(\alpha\) - and \(\beta\)-D-glucopyranose from the specific rotations of the pure anomers \(\left(\alpha=+112^{\circ}, \beta=+18.7^{\circ}\right.\); equilibrium value \(=\) \(\left.+52.7^{\circ}\right)\). You might compare your calculation with the values given in Table 20.2.

Short answer

The equilibrium ratio is approximately 0.364 : 0.636.

Step-by-step solution

Write down the given values

First, note down the given specific rotations for the pure anomers and the equilibrium value: \[ \alpha = +112^{\circ} \]\[ \beta = +18.7^{\circ} \]\[ \text{equilibrium value} = +52.7^{\circ} \]

Set up the equation for the equilibrium specific rotation

The equilibrium rotation is a weighted average of the rotations of the two forms. Let \( x \) be the fraction of \( \alpha \)-D-glucopyranose and \( y \) be the fraction of \( \beta \)-D-glucopyranose. Since only these two forms are present, we have:\[ x + y = 1 \]The equilibrium specific rotation can be expressed as: \[ x \cdot 112^{\circ} + y \cdot 18.7^{\circ} = 52.7^{\circ} \]

Express y in terms of x

Since \( y = 1 - x \), substitute \( y \) into the equilibrium rotation equation:\[ x \cdot 112^{\circ} + (1 - x) \cdot 18.7^{\circ} = 52.7^{\circ} \]

Solve for x

Expand and simplify the equation:\[ 112x + 18.7 - 18.7x = 52.7 \]Combine like terms:\[ 112x - 18.7x = 52.7 - 18.7 \]\[ 93.3x = 34 \]Divide both sides by 93.3 to solve for \( x \):\[ x = \frac{34}{93.3} \approx 0.364 \]

Solve for y

Since \( y = 1 - x \):\[ y = 1 - 0.364 \approx 0.636 \]

Conclusion

Therefore, the equilibrium ratio of \( \alpha \)-D-glucopyranose to \( \beta \)-D-glucopyranose is approximately 0.364 : 0.636.

Key concepts

specific rotation

Specific rotation is a fundamental property in chemistry, especially when dealing with chiral compounds like sugars. It is the angle by which the plane of polarized light is rotated when it passes through a solution of a chiral molecule. The specific rotation is given by the formula: \[ [\text{α}] = \frac{α_{obs}}{l \times c} \] where:

• \(α_{obs}\) is the observed rotation.


• \(l\) is the path length of the cell containing the sample (in decimeters).


• \(c\) is the concentration of the solution (in g/mL).

For instance, D-glucopyranose, a common sugar, has different specific rotations for its anomers (α-D-glucopyranose and β-D-glucopyranose). The specific rotations for the pure α and β forms help in determining their equilibrium ratio in a mixture.

D-glucopyranose

D-glucopyranose is one of the most common sugars and is a form of glucose. It usually exists in two cyclic forms, called anomers, which differ at the carbon 1 position. These anomers are known as α-D-glucopyranose and β-D-glucopyranose. The structure of D-glucopyranose can be described as a six-membered ring with five carbon atoms and one oxygen atom.

• **α-D-glucopyranose**: In this form, the OH group on carbon 1 is trans (opposite) to the CH2OH group at carbon 5.
• **β-D-glucopyranose**: In this form, the OH group on carbon 1 is cis (same side) to the CH2OH group at carbon 5.

The specific structural differences lead to different physical properties, including their specific rotations. Understanding these forms is crucial for calculations involving their equilibrium ratio and specific rotation.

anomers

Anomers are a type of stereoisomer found in cyclic sugars. They differ only at the anomeric carbon (carbon 1 in glucose), which is the carbon derived from the carbonyl carbon (the carbon part of the aldehyde or ketone group) during the cyclization process.

• **Equilibrium Ratio**: When anomeric forms are dissolved in water, they can interconvert. Over time, an equilibrium mixture of these forms is established. The ratio of α to β anomers is determined by factors such as specific rotation.


• **Calculation**: To calculate the equilibrium ratio, use the specific rotation values of the pure anomers and the mixture. Assume equilibrium specific rotation is a weighted average of the specific rotations of the two anomers.


• Set up the equation \( x \times [\text{α}]_α + y \times [\text{α}]_β = [\text{α}]_{eq} \)


• Where \( x \) is the fraction of α-D-glucopyranose and \( y \) is the fraction of β-D-glucopyranose.


• Use \( x + y = 1 \) to solve for \( x \) and \( y \).

Mastering the concept of anomers helps in understanding how sugars behave in solutions and how their properties can be precisely measured and manipulated.