# Optical Activity

Molecules with a chirality center can exist as a pair of stereoisomers called enantiomers that have identical physical properties. However, in a chiral environment (such as polarized light), a pair of enantiomers will exhibit different properties, such as equal but opposite rotation of polarized light.

Unlike diastereomers, enantiomers cannot be separated based on their physical properties (boiling points, solubilities, infrared spectra, and so on). The only way that enantiomers differ from each other is their behavior toward plane-polarized light. When a beam of plane-polarized light is passed through a solution of a single enantiomer, the plane of polarization rotates. Additionally, separate enantiomers will rotate the plane of light in equal but opposite directions. From the point of view of the observer, these directions are clockwise and counterclockwise. Enantiomers that rotate the light clockwise are designated dextrorotatory and labeled D or (+). Enantiomers that rotate the light counterclockwise are designated levorotatory and labeled L or (-). While these designations are related to the enantiomers' chirality, there is no correlation between these designations and the R,S naming system. An (R)-enantiomer can be the dextrorotatory enantiomer in one case and the levorotatory enantiomer in another.

Enantiomers that rotate light are optically active compounds. An optically active compound is a substance that can rotate the plane of polarized light that is passed through it. An optically inactive compound is a substance that is not able to rotate the plane of polarized light that is passed through it. The rotation of plane-polarized light (light waves that vibrate only in one plane) by solutions of optically active compounds is measured in degrees by a polarimeter. A polarimeter is an instrument used to measure the effect of optically active compounds on plane-polarized light. This instrument is made up of a light source, a polarizer, a tube for holding the optically active substance, and a detector for measuring the angle (in degrees) that the plane-polarized light is rotated by the optically active substance.

#### Example of a Polarimeter

Optically active molecules cause the rotation of plane-polarized light by an amount specific to each molecule. The measurement generated by a polarimeter is known as the observed rotation or $\alpha$ (observed) because it is dependent on the concentration of the compound, the length of the tube holding the solution, and the temperature. By applying the formula for specific rotation to these values (observed rotation, concentration of compound, and cell length), the specific rotation can be obtained. The specific rotation is a physical property, like melting point and boiling point, and can, therefore, be located in reference texts. Dextrorotatory compounds have positive specific rotations, and levorotatory compounds have negative specific rotations. Using the known specific rotation for one enantiomer, the enantiomeric excess, or %ee, can be calculated from the observed rotation, or $\alpha$ (observed), of a mixture of enantiomers, where $c$ is the concentration in grams per milliliter (mL) and $l$ is the length of the sample cell, or path length, in decimeters (dm):
$\lbrack\alpha\rbrack = \frac{\alpha(\text{observed})}{c \times l}$
$\%{\rm{ee}}=\frac{\vert\alpha({\text{observed}})\vert}{\vert\alpha({\text{pure\ enantiomer}})\vert}\times100\%$
However, if the solution contains equimolar amounts of the two enantiomers, it is a racemic mixture. A racemic mixture is a 50/50 mixture of enantiomers and will cause no net rotation to plane-polarized light.
Step-By-Step Example
Calculating Enantiomeric Excess
The specific rotation of (R)-2-butanol is –13.5º. If an enantiomeric mixture of 2-butanol has an observed specific rotation of +8.7º, what is the %ee? Is the predominant form R or S?
Step 1

Since the observed rotation is positive and the specific rotation of the R isomer is negative, the mixture is mostly S.

Use the Enantiomeric Excess formula to determine the %ee. Substitute the specific rotation observed in an experiment, $\alpha$ (observed), with the 2-butanol observed specific rotation, +8.7º. Then substitute the accepted value of the specific rotation of a pure sample, $\alpha$ (pure enantiomer), with (R)-2-butanol’s specific rotation, –13.5º.
\begin{aligned}\%{\rm{ee}}&=\frac{\vert\alpha({\text{observed}})\vert}{\vert\alpha({\text{pure\ enantiomer}})\vert}\times100\%\\&=\frac{\vert+8.7^{\degree}\vert}{\vert-13.5^{\degree}\vert}\times 100\%\\&=64\%\;S\end{aligned}
Step 2

Examine the results of the calculation. Determine the percentage of pure S.

The calculation shows that the mixture is 64% pure S.
$\text{Pure}\;S=64\%$
Step 3

Determine the percentage of R and S.

The rest of the mixture is a 50/50 mixture of R and S.
$50/50 \; {\text{mixture}}=100\%-{\text{Pure}\;}S=100\%-64=36\%$
The rest of the mixture is 18% S and 18% R.
Solution
Add the percentages for S to determine the total %ee of the mixture:
$64\%+18\%=82\%$
If a mixture is 64% S, then 64% of the mixture is pure S, and the other 36% is a 50/50 mixture of R and S. Therefore, the mixture is 82% S and 18% R ($82-18=64$).