Using a 1.0-cm polarimeter cell and a solution prepared by dissolving 2.54 g of glucose in 1 L of H₂O, the specific rotation of d-glucose was calculated to be +52.7° at 20 °C using the sodium D line as the source of light. What was the observed rotation of the solution?

Verified step by step guidance

1. Understand the formula for specific rotation: \( [\alpha] = \frac{\alpha}{l \cdot c} \), where \([\alpha]\) is the specific rotation, \(\alpha\) is the observed rotation, \(l\) is the path length in decimeters, and \(c\) is the concentration in g/mL.
2. Convert the path length from cm to dm. Since 1 cm = 0.1 dm, the path length \(l\) is \(1.0 \times 0.1 = 0.1\, \text{dm}\).
3. Calculate the concentration \(c\) of the solution. The concentration is given by \(c = \frac{\text{mass of solute}}{\text{volume of solution}}\). Here, the mass of glucose is 2.54 g, and the volume of the solution is 1 L (or 1000 mL). Thus, \(c = \frac{2.54}{1000}\, \text{g/mL}\).
4. Rearrange the specific rotation formula to solve for the observed rotation \(\alpha\): \(\alpha = [\alpha] \cdot l \cdot c\).
5. Substitute the known values into the formula: \([\alpha] = +52.7\, ^\circ\), \(l = 0.1\, \text{dm}\), and \(c = \frac{2.54}{1000}\, \text{g/mL}\). Perform the multiplication to find the observed rotation \(\alpha\).

Key Concepts

Specific Rotation

Specific rotation is a property of chiral compounds that quantifies their ability to rotate plane-polarized light. It is defined as the observed rotation of light (in degrees) divided by the path length of the sample (in decimeters) and the concentration of the solution (in grams per milliliter). This value is temperature and wavelength dependent, and for glucose, it is typically reported at 20 °C using the sodium D line.

Observed Rotation

Observed rotation refers to the actual angle by which plane-polarized light is rotated when it passes through a sample in a polarimeter. It is influenced by the concentration of the optically active substance, the length of the polarimeter cell, and the specific rotation of the substance. The observed rotation can be calculated using the formula: [α] = α / (l * c), where [α] is specific rotation, α is observed rotation, l is the path length, and c is concentration.

Concentration Calculation

Concentration in this context refers to the amount of solute (glucose) present in a given volume of solution. It is typically expressed in grams per liter (g/L) or grams per milliliter (g/mL). For the given problem, the concentration of glucose can be calculated by dividing the mass of glucose (2.54 g) by the volume of the solution (1 L), which is essential for determining the observed rotation using the specific rotation formula.