Question

Optical rotation of \(\alpha\)-form of a pyranose is \(+150.7^{\circ}\), that of the \(\beta\)-form is \(+52.8^{\circ}\). In solution an equilibrium mixture of these anomers have an optical rotation of \(+80.2^{\circ}\). The \(\%\) of the \(\alpha\)-form in equilibrium mixture is (A) \(28 \%\) (B) \(32 \%\) (C) \(68 \%\) (D) \(72 \%\)

Short answer

The percentage of the α-form in the equilibrium mixture is approximately \(53.1\%\). The closest answer among the given choices is: (C) \(68 \%\).

Step-by-step solution

Set up the weighted average formula for optical rotation.

Let x be the percentage of the α-form in the equilibrium mixture. Therefore, the percentage of the β-form is (100 - x). The formula for the optical rotation of the mixture: Optical rotation of mixture = (Optical rotation of α-form * percentage of α-form) + (Optical rotation of β-form * percentage of β-form)

Plug in the given values.

Optical rotation mixture = +80.2° Optical rotation α-form = +150.7° Optical rotation β-form = +52.8° We can substitute the values in the formula to get: 80.2 = (150.7 * x) + (52.8 * (100 - x))

Solve the equation for x.

Now, we will solve the equation for x, which represents the percentage of the α-form in the mixture: 80.2 = 150.7x - 52.8x + 5280 Combine like terms: 80.2 = 97.9x + 5280 Subtract 5280 from both sides: -5199.8 = 97.9x Divide by 97.9: x = -5199.8 / 97.9 x ≈ 53.1

Find the percentage of the α-form in the equilibrium mixture.

Now that we have found the value of x, we can find the percentage of the α-form in the equilibrium mixture: Percentage of α-form = x ≈ 53.1% This means that the percentage of the α-form in the equilibrium mixture is approximately 53.1%. The closest answer among the given choices is: (C) 68 %

Key concepts

Optical Isomerism

Optical isomerism is a form of stereoisomerism that occurs when molecules have the same molecular formula and sequence of bonded atoms (constitution), but differ in the three-dimensional orientations of the atoms in space. This kind of isomerism is due to the presence of a chiral center, typically a carbon atom with four different groups attached to it, which gives rise to non-superimposable mirror images. These non-superimposable mirror images are called enantiomers.

Each enantiomer of a chiral compound can rotate plane-polarized light, but in opposite directions; one is known as the 'right-handed' (dextrorotatory or 'D') enantiomer and rotates light to the right, whereas the other is the 'left-handed' (levorotatory or 'L') enantiomer and rotates light to the left. The extent of this rotation is measured as the optical rotation of the substance, which is a unique physical property. This concept becomes particularly relevant in the analysis of compounds such as those found in sugars, which can exist as various enantiomers, called anomers, that affect their optical rotation. The α- and β-forms of pyranose sugars are examples of such anomers.

Pyranose Anomers

Pyranose anomers refer to the different structural forms of a pyranose, the six-membered sugar ring resembling pyran, which can arise due to the different spatial arrangements of the hydroxyl group attached to the anomeric carbon (carbon number one in the ring). In the context of glucose, for instance, the α-anomer has the OH group below the plane of the ring, while the β-anomer has the OH group above the plane of the ring.

These small changes in structure have a large impact on the properties of the sugar, including their optical rotation, solubility, and reactivity. Anomers can interconvert in solution through a process called mutarotation, where the α- and β-forms of a sugar interconvert in the presence of water, and the resulting mixture will exhibit an optical rotation that is a weighted average of the two anomeric forms. This concept is fundamental to understanding the exercise provided, where the equilibrium mixture displays an optical rotation stemming from a combination of both the α- and β-pyranose forms.

Chemical Equilibrium

Chemical equilibrium pertains to the state of a chemical reaction in which the rate of the forward reaction equals the rate of the reverse reaction. At this point, the concentrations of reactants and products remain constant over time, but they are not necessarily equal. The concept of equilibrium is intrinsic to many areas of chemistry and can explain the behavior of reactions under different conditions.

Equilibrium can be shifted by changing concentrations, pressure, volume, or temperature, according to Le Chatelier's principle. In the case of optical isomers in pyranose sugars, equilibrium refers to the dynamic balance between the α and β anomers in solution. When these sugars are dissolved, they reach a state where the rate at which α-anomers convert to β-anomers is equal to the rate at which β-anomers convert back to α-anomers. The observed optical rotation at equilibrium is the result of this balance and can be calculated using the concept of a weighted average, as demonstrated by the problem and solution discussed.