Sugar Solution Optical Rotation Explained

Optical activity is a fascinating property exhibited by certain substances, such as sugar, where they have the ability to rotate the plane of plane-polarized light. This rotation is measured using an instrument called a polarimeter. The extent of this rotation depends on the specific properties of the substance, its concentration in a solution, the length of the path the light travels through the solution, and the temperature.

The specific rotation (\( \left[ \alpha \right] \)) is a fundamental physical constant for a given optically active compound under specific conditions (temperature and wavelength of light). It is defined by the following relationship:

\[ \left[ \alpha \right]_{D}^{T} = \frac{\alpha}{l \times c} \]

Where:

• \( \alpha \) (alpha) represents the observed angle of rotation of the plane of polarization, measured in degrees.
• \( \left[ \alpha \right]_{D}^{T} \) is the specific rotation. The subscript 'D' usually refers to the sodium D-line (589 nm), which is a common wavelength of light used, and 'T' refers to the temperature in degrees Celsius. The units are typically degrees decimeter\(^{-1}\) (gram/milliliter)\(^{-1}\).
• \( l \) is the path length of the polarimeter tube through which the light passes, measured in decimeters (dm).
• \( c \) is the concentration of the optically active substance in the solution, measured in grams per milliliter (g/mL).

Polarimeter Tube Data Analysis

Let's extract the given information from the question and convert units as necessary to match the formula for specific rotation.

• Specific Rotation (\( \left[ \alpha \right] \)): The specific rotation of the 10% sugar solution is given as 60°. This value is provided in a form that is directly applicable when length is in decimeters and concentration is in g/mL.
• Concentration (c): The sugar solution is described as a 10% sugar solution. This means that there are 10 grams of sugar dissolved in 100 mL of the solution. To use this in our formula, we need to convert it to grams per milliliter (g/mL): \[ c = \frac{10 \text{ g}}{100 \text{ mL}} = 0.1 \text{ g/mL} \]
• Length of Polarimeter Tube (l): The length of the polarimeter tube is given as 20 cm. For calculations involving specific rotation, the length should be in decimeters (dm). We convert centimeters to decimeters: \[ l = 20 \text{ cm} \times \frac{1 \text{ dm}}{10 \text{ cm}} = 2 \text{ dm} \]

Calculating Polarization Rotation Angle

Our goal is to find the angle of rotation (\( \alpha \)). We can rearrange the specific rotation formula to solve for \( \alpha \):

\[ \alpha = \left[ \alpha \right] \times l \times c \]

Now, we substitute the values we determined into this rearranged formula:

\[ \alpha = 60 \text{ deg dm}^{-1} (\text{g/mL})^{-1} \times 2 \text{ dm} \times 0.1 \text{ g/mL} \]

Multiplying the numerical values and observing the unit cancellation (dm cancels dm, g/mL cancels g/mL), we are left with degrees:

\[ \alpha = 60 \times 2 \times 0.1 \text{ deg} \]

\[ \alpha = 120 \times 0.1 \text{ deg} \]

\[ \alpha = 12 \text{ deg} \]

Therefore, the plane of polarization of light is rotated by 12° when it passes through the 10% sugar solution in the 20 cm polarimeter tube.

Summary of Optical Rotation Calculation

The key parameters and the calculated result are summarized in the table below:

This calculation demonstrates how the observed rotation of plane-polarized light is directly proportional to the specific rotation of the substance, its concentration, and the length of the path through which the light travels.