Question

Specific rotation of equilibrium mixture of the three forms of glucose is (A) \(52.2^{\circ}\) (B) \(112^{\circ}\) (C) \(19.5^{\circ}\) (D) \(-90^{\circ}\)

Short answer

The specific rotation of the equilibrium mixture of the three forms of glucose is approximately \(13.83^\circ\), which is closest to option C (\(19.5^\circ\)). This is based on the assumption that the mixture contains equal proportions of the three forms and considering the provided specific rotation values for each form.

Step-by-step solution

Understand Specific Rotation

Specific rotation is a measure of a substance's ability to rotate the plane of polarized light. It is defined as the rotation in degrees of the plane of polarized light per millimeter path length and per gram of substance per milliliter of solution. In this exercise, we have the specific rotation values for three forms of glucose, so the specific rotation of the equilibrium mixture is a combination of these three values.

Determine the Relative Concentrations of the Three Forms at Equilibrium

To find the specific rotation of the equilibrium mixture, we need to know the relative concentrations of the individual forms of glucose at equilibrium. Assuming that the equilibrium mixture is equal parts of each form of glucose (since we are not given information about their actual concentrations), we can proceed with calculating the specific rotation of the mixture.

Calculate the Specific Rotation of the Mixture

Use the following formula to calculate the specific rotation of the mixture: Specific rotation of the mixture = (fraction of form 1 * specific rotation of form 1) + (fraction of form 2 * specific rotation of form 2) + (fraction of form 3 * specific rotation of form 3) Given the specific rotation values for the three forms of glucose: Form 1: \(112^\circ\) (Option B) Form 2: \(19.5^\circ\) (Option C) Form 3: \(-90^\circ\) (Option D) Since the equilibrium mixture contains equal proportions of the three forms of glucose, the specific rotation of the mixture will be: Specific rotation of the mixture = \(\frac{1}{3}(112 + 19.5 + (-90))\)

Solve for the Specific Rotation

Now, calculating the specific rotation of the equilibrium mixture: Specific rotation of the mixture = \(\frac{1}{3}(112 + 19.5 - 90)\) = \(\frac{1}{3}(41.5)\) Specific rotation of the mixture = \(13.83^\circ\) The answer is closest to option C, \(19.5^\circ\). Since none of the given options is exactly \(13.83^\circ\), we can assume that there might be minor inaccuracies or differences in the relative concentrations of the three forms at equilibrium. However, based on the given information, option C is the best approximation for the specific rotation of the equilibrium mixture of the three forms of glucose.

Key concepts

Glucose

Glucose is a simple sugar that plays a crucial role in biology, serving as a primary source of energy for cells. It's a type of carbohydrate and is known scientifically as a monosaccharide. In the context of chemistry and this specific exercise, glucose can exist in multiple forms known as isomers. These isomers differ in the arrangement of atoms but have the same molecular formula.
Typically, glucose can be found in two primary forms in solution: alpha ( \(\alpha\)-D-glucose) and beta ( \(\beta\)-D-glucose). Each form has a unique specific rotation value, which means they rotate polarized light differently.
However, a third form, known as the open chain form of glucose, can also exist. All these forms reach an equilibrium in an aqueous solution. This balance between isomers is an essential concept because it affects the glucose's overall specific rotation when measuring mixtures.

Equilibrium Mixture

An equilibrium mixture refers to a solution where different chemical entities exist in proportions that no longer change with time. For glucose, the equilibrium mixture consists of its various isomers. These isomers interconvert between each other until they reach a dynamic balance known as chemical equilibrium.
In the equilibrium mixture of glucose forms, each isomer rotates polarized light to a different extent. By accurately measuring this rotation, we can calculate the overall specific rotation of the mixture.
The concept of equilibrium is important because it determines the behavior and properties of the solution. Understanding equilibrium helps in predicting the composition of the mixture, which in this exercise is simplified by assuming equal parts of each glucose form.

Polarized Light

Polarized light is light waves that oscillate in a single plane, as opposed to ordinary light which oscillates in multiple planes. When light is polarized, it becomes more concentrated along a particular orientation.
Specific rotation is the measure of how much a substance, like the glucose mixture, can rotate a plane of polarized light. This property is significant in sugar chemistry because it provides insights into the structure and dynamics of sugar molecules.
The use of polarized light allows scientists to deduce various aspects of molecular structures and their interactions. In experiments involving specific rotation, a polarimeter is often used. This device measures the angle of rotation, thereby offering crucial information about the concentration and chiral properties of the substances in the mixture.