Physical Properties of Stereoisomers
How can it be determined whether or not a given optically active sample contains only a pure enantiomer or a mixture of enantiomers of a chiral compound? The plane of polarized light is rotated by an amount which is specially designated for each enantiomer, as it passes through a chiral compound solution. The other enantiomer rotates the plane of polarized light by an equal amount in the opposite direction. An enantiomer mixture rotates the plane by a different amount. Since the amount of the optical rotation α depends on the concentration of the chiral compound, the temperature, the solvent, the pathlength of the cuvet, and the wavelength of the polarized light, the specific amount of a given enantiomer's optical rotation is recorded for certain values of these parameters. This specific amount of optical rotation is called specific rotation [α]λT (λ = wavelength, T = temperature). Usually, the specific rotation is noted for a solution of 1.0 g of a pure enantiomer per 1 mL solvent, a temperature of 25 °C, a pathlength of 1.0 dm, and a wavelength of 589.3 nm (sodium D line). As far as certain measurement conditions are concerned, the specific rotation is a material constant.
The specific rotations of the enantiomers of 2-butanol are, for instance, +13.52° and -13.52°. The specific rotations of two enantiomers have the same amount, though opposite signs. That is, the enantiomers show opposite rotational senses. The sense of rotation is called dextrorotatory, which is indicated by (+), or levorotatory, which is indicated by (-). The absolute configuration of an enantiomer cannot be deduced from the sense of rotation, as they are not interdependent.
- The specific rotation of a solution of a chiral compound or a mixture of chiral compounds is calculated by
- [α]λT = α / (c · l)
- where α is the observed rotation, [α] is the specific rotation of the enantiomer, T is the temperature in °C, λ is the wavelength of the polarized light, c is the concentration of the solution in grams per milliliter, and l is the pathlength of the cuvet in decimeters. ([α]λT is a material constant for the given temperature, wavelength, solvent, and enantiomer or mixture of enantiomers).
When two enantiomers are dissolved together, the solution's specific rotation is smaller than that of a pure enantiomer. If the specific rotation of the mixture is negative, the solution contains an excess of the levorotatory (-) enantiomer. In a positive specific rotation, the dextrorotatory (+) enantiomer exceeds the (-) enantiomer. The optical rotation of a racemate, a 1:1 mixture of two enantiomers, is zero degrees, since the optical rotations of the enantiomers cancel each other out. A sample that contains a solution of only one of the enantiomers is called optically pure or enantiomerically pure.
- The optical purity of a mixture of two enantiomers is calculated from the measured specific rotations by
- optical purity (%) = (observed specific rotation of the sample) / (specific rotation of the pure enantiomer).
- The enantiomeric excess (ee) is the percent excess of a pure enantiomer over the racemate in a mixture of two enantiomers. It is calculated by
- ee = [(mole fraction of the excess enantiomer - mole fraction of the other enantiomer) / (mole fraction of the excess enantiomer + mole fraction of the other enantiomer)] · 100 %
- or, as the denominator is equal 1 (in case of a solution of a chemically (not enantiomerically) pure compound)
- ee = (mole fraction of the excess enantiomer - mole fraction of the other enantiomer) · 100 %
Example: if a sample contains eight moles of the excess enantiomer and two moles of the other enantiomer, the enantiomeric excess is ee = (0.8 - 0.2) · 100 % = 60 %. If a sample contains nine moles of the excess enantiomer and six moles of the other enantiomer the enantiomeric excess is ee = (9 / 15 - 6 / 15) · 100 % = (0.6 - 0.4) · 100 % = 20 %. The enantiomeric excess of a racemate, a 1:1 mixture of two enantiomers, is (0.5 - 0.5) · 100 % = 0.
In a solution of a chemically (not enantiomerically) pure compound, the values of optical purity and enantiomeric excess are identical. If the solution does not contain any other compound (except for the solvent) and the specific rotation of the pure enantiomer is known, the enantiomeric excess may therefore be determined by measuring the optical rotation of a solution of a two-enantiomer mixture. In order for the optical purity and the enantiomeric excess to be calculated, the specific rotation of the pure enantiomer must first be noted.