Question

Mutarotation. The specific rotations of the \(\alpha\) and \(\beta\) anomers of \(\mathrm{D}\) -glucose are +112 degrees and +18.7 degrees, respectively. Specific rotation, \([\alpha]_{\mathrm{D}},\) is defined as the observed rotation of light of wavelength 589 nm (the \(D\) line of a sodium lamp) passing through \(10 \mathrm{cm}\) of a \(1 \mathrm{g} \mathrm{ml}^{-1}\) solution of a sample. When a crystalline sample of \(\alpha\) -D-glucopyranose is dissolved in water, the specific rotation decreases from 112 degrees to an equilibrium value of 52.7 degrees. On the basis of this result, what are the proportions of the \(\alpha\) and \(\beta\) anomers at equilibrium? Assume that the concentration of the open-chain form is negligible.

Short answer

At equilibrium, the fraction of the \( \alpha \) anomer is 0.364, and the \( \beta \) anomer is 0.636.

Step-by-step solution

Define the Variables and Understand the Problem

We need to find the proportions of the \( \alpha \) and \( \beta \) anomers of D-glucose at equilibrium given their specific rotations. \( [\alpha]_{\alpha} = +112^\circ \) and \( [\alpha]_{\beta} = +18.7^\circ \) are the specific rotations for \( \alpha \) and \( \beta \) anomers, respectively. At equilibrium, the mixture has a specific rotation of \( +52.7^\circ \). We will use these to find the proportions of the \( \alpha \) and \( \beta \) anomers.

Set Up the Equation

Using the equation for the specific rotation of a mixture: \[ [\alpha]_{\text{mixture}} = f_{\alpha} [\alpha]_{\alpha} + f_{\beta} [\alpha]_{\beta} \] where \( f_{\alpha} \) and \( f_{\beta} \) are the fractions of \( \alpha \) and \( \beta \) anomers, respectively. We know that \( f_{\alpha} + f_{\beta} = 1 \). Substitute the given values: \[ 52.7 = f_{\alpha} \times 112 + (1 - f_{\alpha}) \times 18.7 \]

Solve for the Fraction of the \( \alpha \) Anomer

First, expand the equation: \[ 52.7 = 112f_{\alpha} + 18.7 - 18.7f_{\alpha} \] Simplify it: \[ 52.7 = 112f_{\alpha} - 18.7f_{\alpha} + 18.7 \] Combine the \( f_{\alpha} \) terms:\[ 52.7 = (112 - 18.7)f_{\alpha} + 18.7 \] \[ 52.7 = 93.3f_{\alpha} + 18.7 \]

Rearrange and Solve for \( f_{\alpha} \)

Subtract 18.7 from both sides: \[ 52.7 - 18.7 = 93.3f_{\alpha} \] \[ 34 = 93.3f_{\alpha} \] Divide both sides by 93.3 to solve for \( f_{\alpha} \):\[ f_{\alpha} = \frac{34}{93.3} \approx 0.364 \] This is the fraction of the \( \alpha \) anomer at equilibrium.

Calculate the Fraction of the \( \beta \) Anomer

Knowing that \( f_{\alpha} + f_{\beta} = 1 \), we can find \( f_{\beta} \):\[ f_{\beta} = 1 - f_{\alpha} = 1 - 0.364 = 0.636 \] This is the fraction of the \( \beta \) anomer at equilibrium.

Key concepts

Specific Rotation

Specific rotation is a fundamental concept in the study of optical activity in solutions. It refers to the angle by which the plane of polarized light is rotated when it passes through a solution of chiral compounds. In this context, polarized light is light that oscillates in a single plane. When it enters a solution of optically active substances, the plane can be rotated either to the left or to the right.
For solutions, specific rotation is described by the equation:\[[\alpha] = \frac{\theta}{l \cdot c}\] where \( \theta \) is the observed angle of rotation in degrees, \( l \) is the length of the solution path in decimeters, and \( c \) is the concentration of the solution in grams per milliliter. This calculated value helps chemists understand the optical properties of various substances.
Different substances and their stereoisomers can produce unique specific rotations, making it a useful property for identifying compounds like sugars and amino acids.

D-glucose Anomers

D-glucose anomers are fascinating because they help illustrate the concept of mutarotation. By definition, anomers are types of stereoisomers differing at a new asymmetric carbon atom formed on ring closure. In D-glucose, the specific position and orientation of the hydroxyl group attached to the first carbon determines whether it's the \( \alpha \) or \( \beta \) anomer.
For D-glucose:

• The \( \alpha \)-anomer has the hydroxyl group on the opposite side of the ring compared to the 6th carbon's hydroxymethyl group. Its specific rotation is noted as +112 degrees.
• Meanwhile, the \( \beta \)-anomer has the hydroxyl group on the same side as the 6th carbon's hydroxymethyl group, with a specific rotation of +18.7 degrees.

When a solution of one anomer is prepared, it gradually changes to reach an equilibrium mixture of both anomers through a process called mutarotation. This transformation impacts their specific rotation.

Equilibrium Proportions

Equilibrium proportions involve determining the stable mixture of \( \alpha \) and \( \beta \) anomers in a solution over time. When D-glucose is dissolved, initially in the form of either pure \( \alpha \) or \( \beta \) anomer, the solution doesn't remain in that form over time. Instead, it undergoes mutarotation and reaches equilibrium, meaning the ratio of both anomers becomes constant.
The specific rotation observed at equilibrium, in this case, is 52.7 degrees. By using the formula for the specific rotation of a mixture:\[[\alpha]_{\text{mixture}} = f_{\alpha} [\alpha]_{\alpha} + f_{\beta} [\alpha]_{\beta}\]we can find the proportion of each anomer. From previous steps, it's calculated that at equilibrium, approximately 36.4% of the glucose is in the \( \alpha \) form, and 63.6% is in the \( \beta \) form, reflecting the balance reached in solution.

Optical Activity

Optical activity is a characteristic of chiral compounds that allows them to rotate the plane of polarized light. This property arises due to the asymmetry in the structure of molecules, making it a fundamental aspect of stereochemistry. Solutions of optically active substances like sugars adopt specific interactions with light, leading to observable specific rotations.
For D-glucose, both \( \alpha \) and \( \beta \) anomers show different levels of optical activity due to their structural arrangements. Interestingly, the overall optical rotation of their equilibrium mixture represents the net average of these effects. This interplay of orientation due to optical activity is not only crucial for understanding glucose but also important in other biological and chemical systems.
Thus, optical activity provides insights into molecular arrangements and their interactions with light, proving vital in fields like biochemistry and pharmacology, where understanding molecular orientation can affect reactivity and function.