Polarimetry

Intro to the problem
Polarimetry, optical rotation or alpha-D is the fastest and simplest method for having a peek into the chiral world. The interaction of plane polarised light and stereocentres is one of the first concepts covered in any course or text on chirality. Despite this, the lab procedure is still a source of great confusion.

This owes to the bizarre units that we use in our equations.

Theory

When the sample goes in the machine, the machine reports back a number (the unstandardised alpha) which is the angle the light was rotated (unit = degrees).

Alpha varies with all sorts of factors. Temperature, solvent, concentration and length of the optical cell all add to or subtract from alpha.

To account for this, we standardise alpha.

Alpha has a linear relationship with both the concentration of the sample and the length of the cell - thus, doubling one of these will double alpha. We can therefore obtain a standardised value by dividing alpha by both the concentration of the sample and the length of the cell. This new value is the alpha-D. For any given sample, alpha-D should remain the same whether the sample's concentration is changed, or cells of different length are used.

It is not possible to standardise with respect to temperature and solvent, so these are both reported alongside the alpha-D.

In truth, the sample concentration can affect alpha-D in a non-linear way. Part of the chirality is often a chiral cage of solvent molecules around the chiral molecule, too high a concentration will disrupt this cage and change alpha-D in a non-linear fashion. For this reason, the concentration is also often reported.

Equation and units

alpha-D =                   alpha                   
                    concentration * cell length

alpha -                  in degrees, straight off the polarimeter
concentration -    in units of density (grams per millilitre)
cell length -          in decimetres (so 10cm = 1dm)

Perhaps the most confusing unit is the concentration. We are more used to dealing with concentration in terms of moles, here it is grams. Another unusual factor is that it is in grams rather than milligrams. Some equations that you will find revert to milligrams, which is why you find multiplications by 1000 turning up in some equations.

Reporting


Here's one of mine. Note subscript D and superscript temperature. There are no units for the alpha-D. Concentration (g per ml) in the solvent is given.


Further Information

The optical purity is defined as    op = α(sample)/α(max)

The optical purity does not have a direct relationship to the enantiomeric ratio of the sample, but should have a direct relationship to the enantiomeric excess. This is why enantiomeric excess was defined.

ee = (R-S)/(R+S)

The relationship should, ideally, be as simple as    op = ee

There are complications. For example, for a 2-ethyl-2-methyl succinic acid in an R:S ratio of 92.5:7.5 the ee is 5%. As concentration of the sample changes, ee stays the same, and so should the measured op. In reality, op varied significantly with concentration - and even changed in direction of rotation at higher concentration! [Krow, G. and Hill, R. K. Chem. Commun., (1968), 430-431.]
Gawley, R. E., J. Org. Chem., 71, (2006), 2411-2416. 

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