Specific Rotation of Sugar Solution

The specific rotation of a substance is a fundamental property that quantifies its ability to rotate the plane of polarization of light. This property is crucial in understanding the optical activity of compounds like sugar solutions, which are known for their ability to rotate polarized light.

Understanding Optical Rotation in Sugar Solutions

When plane-polarized light passes through an optically active substance, such as a sugar solution, its plane of polarization is rotated by a certain angle. This phenomenon is known as optical rotation. The extent of this rotation angle depends on several factors:

• The intrinsic nature of the substance (e.g., the specific type of sugar).
• The concentration of the sugar solution.
• The tube length (path length) through which the light travels.
• The temperature of the solution.
• The wavelength of light used for the measurement.

For a given substance at a specific temperature and wavelength, the specific rotation, denoted as $[\alpha]$, is a constant value. It is typically defined by the formula used in polarimetry:

\[ [\alpha] = \frac{100 \times \alpha}{l \times c} \]

Where:

• $\alpha$ (alpha) is the observed rotation angle in degrees ($\text{deg}$).
• $l$ is the tube length of the sample container in decimeters ($\text{dm}$).
• $c$ is the concentration of the solution in grams per 100 milliliters ($\text{g/100 mL}$) or as a percentage ($\%$).

Given Information for Sugar Solution

From the question, we are provided with the following data for the sugar solution:

• Observed rotation angle ($\alpha$) = 21°
• Tube length ($l$) = 20 cm
• Concentration of sugar solution ($c$) = 15%

Converting Units for Specific Rotation Calculation

To use the formula for specific rotation, we need to ensure all units are consistent. The tube length ($l$) needs to be converted from centimeters ($\text{cm}$) to decimeters ($\text{dm}$), as per the standard unit requirement for the formula. The concentration ($c$) given as 15% is directly usable as the numerical value 15 for the calculation, representing 15 grams per 100 mL.

• Conversion of tube length ($l$):
• Since 1 decimeter ($\text{dm}$) = 10 centimeters ($\text{cm}$), we convert the given length:
• $l = 20 \text{ cm} = \frac{20}{10} \text{ dm} = 2 \text{ dm}$
• The concentration $c = 15\%$ is directly used as 15 in the formula.

Calculating Specific Rotation of the Sugar Solution

Now, we will substitute the given and converted values into the specific rotation formula:

\[ [\alpha] = \frac{100 \times \alpha}{l \times c} \]

Substitute the values:

\[ [\alpha] = \frac{100 \times 21^\circ}{2 \text{ dm} \times 15 \text{ (g/100 mL)}} \]

\[ [\alpha] = \frac{2100}{30} \]

\[ [\alpha] = 70 \text{ deg dm}^{-1} \text{ (g/100 mL)}^{-1} \]

Therefore, the specific rotation of the sugar solution is 70°. This value represents the intrinsic optical activity of the sugar under the specified conditions related to concentration and path length.