The amount of rotation depends on the number of molecules that a light wave encounters. This
depends on two factors: the concentration of the optically active compound and the length of the
tube through which the light passes. Chemists have set standard conditions of for
concentration and 1 dm (10 cm) for length to compare the optical activities of different compounds.
Rotations measured at different concentrations and tube lengths can be converted to a
standardized specific rotation using the following equation:
Equation 2.1where [α] is specific rotation in degrees, αobs is the observed rotation in degrees, c is the
concentration in , and l is the path length in dm.
When both (+) and (–) enantiomers are present in equal concentrations, they form a racemic
mixture. In these solutions, the rotations cancel each other out, and no optical activity is observed.
If enantiomerism is analogous to handedness, racemic mixtures are the equivalent of
ambidexterity. These solutions possess no handedness overall and will not rotate plane-polarized
light.
KEY CONCEPT
A racemic mixture displays no optical activity.The fact that enantiomers have identical physical and chemical properties prompts a question
about racemic mixtures: how can one separate the mixture into its two constituent isomers? The
answer lies in the relationship between enantiomers and diastereomers. Reacting two enantiomers
with a single enantiomer of another compound will, by definition, lead to two diastereomers.
Imagine, for example, two enantiomers that contain only one chiral carbon; these compounds could
be labeled (+) and (–). If each is reacted with only the (+) enantiomer of another compound, two
products would result: (+,+) and (–,+). Because these two products differ at some—but not all—chiral
centers, they are necessarily diastereomers. Diastereomers have different physical properties, as we