A substance which is optically active is one which is capable of rotating the plane of polarized light. In a pair of optically active enantiomers, each enantiomer will rotate the plane of polarized light in equal and opposite directions; the enantiomers can therefore be referred to as (+) and (-) enantiomers, depending upon the direction of the observed rotation. A racemic mixture, in which there are equal concentrations of both enantiomers, will display no net optical rotation.
The angle of rotation is measured in a device known as a polarimeter. The substance in question is typically dissolved in a solvent at a known concentration (c in grams per mL) and placed in an analysis tube of a known length (l given in decimeters; one dm = 10 cm). The rotation observed in the polarimeter is denoted with the Greek letter a, and the specific rotation for a molecule (denoted as [a]) is given by the equation shown below:
The specific rotation for a molecule is also dependent on the wavelength of the plane polarized light. A common light source for simple polarimeters is a lamp with enhanced output at the "sodium D-line"; in this instance, the specific rotation would be shown as [a]D.
Sample problem:
A pure enantiomer has an observed optical rotation of -0.82o when measured in a one dm tube at a concentration of 0.3 g/10 mL. Calculate the specific rotation for this molecule.
Solution:
The concentration of 0.3 g/10 mL is equivalent to 0.03 g/mL; c = 0.03 g/mL
The length of the analysis tube is one dm; l = 1.0 dm
The specific rotation is therefore:
[a] = -27.3o g-1 mL-1 dm-1
If the specific rotation of a pure enantiomer is known, the observed rotation can also be used to calculate optical purity, or the level of contamination of one compound with its enantiomer, using the simple convention:
The specific rotation for a pure enantiomer is known to be -39o g-1 mL-1 dm-1. A sample containing both enantiomers is found to have an observed rotation of -0.62o in a one dm tube at a concentration of 3.5 g/100 mL. What is the optical purity of the sample?
Solution:
For this sample, the apparent specific rotation is:
[a] = -17.7o g-1 mL-1 dm-1
If the fraction of the (-) enantiomer is x, then (1x) gives the fraction of the (+) enantiomer. For any mixture of the two, the apparent specific rotation will be given by:
x(-39o) + (1x)(+39o) = [a]apparent
For this mixture:
x(-39o) + (1x)(+39o) = -17.7o
(-39x) + 39 (-39x) = -17.7
-78x = -56.7
x = 0.73
Therefore, the mixture contains 73% of the (-) enantiomer and 27% of the (+) enantiomer.
