The observed rotation of 2.0 g of a compound in 50 mL of solution in a polarimeter tube 20 cm long is +138°. What is the specific rotation of the compound?

검증된 단계별 안내

Step 1: Recall the formula for specific rotation: \( [\alpha] = \frac{\alpha_{\text{obs}}}{l \cdot c} \), where \( [\alpha] \) is the specific rotation, \( \alpha_{\text{obs}} \) is the observed rotation, \( l \) is the path length in decimeters, and \( c \) is the concentration in g/mL.

Step 2: Convert the path length from centimeters to decimeters. Since 1 decimeter = 10 centimeters, divide the given path length (20 cm) by 10 to get \( l = 2 \, \text{dm} \).

Step 3: Calculate the concentration \( c \) of the solution. The concentration is given by \( c = \frac{\text{mass of compound}}{\text{volume of solution}} \). Use the given mass (2.0 g) and volume (50 mL) to find \( c = \frac{2.0}{50} \, \text{g/mL} \).

Step 4: Substitute the values into the formula for specific rotation. Use \( \alpha_{\text{obs}} = +138° \), \( l = 2 \, \text{dm} \), and \( c = \frac{2.0}{50} \, \text{g/mL} \).

Step 5: Simplify the expression to calculate \( [\alpha] \), ensuring the units are consistent throughout the calculation.

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주요 개념

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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 product of the concentration of the solution (in grams per milliliter) and the path length of the sample (in decimeters). This value allows for the comparison of optical activity among different substances under standardized conditions.

Optical Activity

Optical activity refers to the ability of chiral molecules to rotate the plane of polarized light. This phenomenon occurs because chiral molecules exist in two non-superimposable mirror images, known as enantiomers, which can rotate light in opposite directions. The degree of rotation depends on the nature of the compound, its concentration, and the wavelength of light used.

Polarimeter

A polarimeter is an instrument used to measure the angle of rotation of polarized light as it passes through a sample solution. It consists of a light source, a polarizer, a sample tube, and an analyzer. By measuring the angle of rotation, one can determine the specific rotation of a compound, which is essential for characterizing chiral substances in organic chemistry.