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# Specific rotation

The specific rotation of a chemical compound [α] is defined as the observed angle of optical rotation α when plane-polarized light is passed through a sample with a path length of 1 decimeter and a sample concentration of 1 gram per 1 deciliter. The specific rotation of a pure material is an intrinsic property of that material at a given wavelength and temperature. Values should always be accompanied by the temperature at which the measurement was performed and the solvent in which the material was dissolved. Often the temperature is not specified; in these cases it is assumed to be room temperature. The formal unit for specific rotation values is deg cm² g-1 but scientific literature uses just degrees. A negative value means levorotatory rotation and a positive value means dextrorotatory rotation. Some examples:

Optical rotation is measured with an instrument called a polarimeter. There is a linear relationship between the observed rotation and the concentration of optically active compound in the sample. There is a non-linear relationship between the observed rotation and the wavelength of light used. Specific rotation is calculated using either of two equations, depending on the sample you are measuring:

For pure liquids:

$[\alpha]_\lambda^T = \frac{\alpha}{l \times d}$

In this equation, l is the path length in decimeters, and d is the density of the liquid in g/mL, for a sample at a temperature T (given in degrees Celsius) and wavelength λ (in nanometers). If the wavelength of the light used is 589 nanometer (the sodium D line), the symbol “D” is used. The sign of the rotation (+ or -) is always given.

$[\alpha]_D^{20} = +6.2$°

For solutions, a different equation is used:

$[\alpha]_\lambda^T = \frac{100 \alpha}{l \times c}$

In this equation, l is the path length in decimeters and c is the concentration in g/100mL, for a sample at a temperature T (given in degrees Celsius) and wavelength λ (in nanometers). If the wavelength of the light used is 589 nanometer (the sodium D line), the symbol “D” is used. The sign of the rotation (+ or -) is always given. When using this equation, the concentration and the solvent are always provided in parentheses after the rotation. The rotation is reported using degrees, and no units of concentration are given (it is assumed to be g/100mL).

For example:

$[\alpha]_D^{20} = +6.2$° (c 1.0, EtOH)

This solution equation is incorrectly represented in many textbooks and on many websites as:

$[\alpha]_\lambda^T = \frac{\alpha}{l \times c}$

(concentration in g/mL)

Mathematically, the two forms are the same, but chemically they are very different. Using the incorrect form of the equation will produce problems because the concentration will have the incorrect units. Because the units are not reported, this can produce difficulties for those trying to use the data later.

If a compound has a very large specific rotation or a sample is very concentrated, the actual rotation of the sample may be larger than 180°, and so a single polarimeter measurement cannot detect when this has happened (for example, the values +270° and –90° are not distinguishable, nor are the values 361° and 1°). In these cases, measuring the rotation at several different concentrations allows one to determine the true value.

In cases of very small or very large angles, one can also use the variation of specific rotation with wavelength to facilitate measurement. Switching wavelength is particularly useful when the angle is small. Many polarimeters are equipped with a mercury lamp (in addition to the sodium lamp) for this purpose.

The variation of specific rotation with wavelength is the basis of optical rotary dispersion (ORD) that can be used to elucidate the absolute configuration of certain compounds.

Measuring optical rotation provides, in theory, a way to assess optical purity of a sample containing a mixture of enantiomers. For example, if a sample of bromobutane measured under standard conditions has an observed rotation of −9.2°, this indicates that the net effect is due to (100%)(9.2°/23.1°)=40% of the R enantiomer. The remainder of the sample is a racemic mixture of the enantiomers (30% R and 30% S), which has no net contribution to the observed rotation. The enantiomeric excess is 40%; the total concentration of R is 70%. However, in practice the utility of this method is limited, as the presence of small amounts of highly rotating impurities can greatly affect the rotation of a given sample. For this reason other methods of determining the enantiomeric ratio such as gas chromatography or HPLC with a chiral column is generally preferred.