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If you have some sample in solution (of a pure R substance), wouldn't the individual molecules all have a random orientation in space? Or do the photons emitted into the solution cause all of the molecules to align? Or is it just that, statistically, a given percentage of the particles will have an orientation which corresponds to the polarized light source.

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Short answer: The light beam interacts with all of the randomly aligned molecules in its path. There are roughly $\ce{6 x 10^{23}}$ molecules in a mole of material. If we were to dissolve a micro mole of this compound in 10 ml (1 deciliter is a typical polarimetry sample size) of solvent we would have $\ce{6 x 10^{17}}$ molecules in our sample cell. Since the molecules are randomly oriented, there are enough molecules so that the polarized light beam will sample all possible orientations of the molecule with respect to the light beam many, many times over. This allows for measurements to be made in an accurate and statistically reproducible manner.

Detailed answer: A beam of light has associated with it an electric and magnetic field. In simple polarimetry experiments it is the electric field (or vector) associated with the light beam that is of interest. Let's say the beam is propagating in the Z direction, then the propagating electric vector lies in an infinite number of planes that contain the Z axis. When we plane-polarize the electric vector associated with the light beam, we remove all of the planes that the electric vector resides in except for one.

The next important point is to realize that a plane polarized electric vector associated with a beam of light can also be represented by (is equivalent to) two circularly polarized beams (or vectors), rotating in opposite directions (clockwise and counter-clockwise), each one-half the intensity of the plane polarized electric vector. What we wind up with is two helices, one right-handed and one left-handed, traveling through the sample in our polarimeter.

Let's say our polarimeter sample contains a significant concentration of chiral molecules in solution. Although the chiral molecules are randomly aligned with respect to the light beam, each chiral molecule will interact with the right-handed or clockwise rotating helical electric vector differently than it will interact with the left-handed or counter-clockwise rotating helical electric vector. Our sample must contain a large number of molecules so that, despite all of the different molecular alignments between our molecules and the beam, when averaged over the entire sample, the difference in interaction of the right- and left-handed helices with our chiral sample will be repeatable. It is not hard to have a significant concentration of molecules because a reasonably long sample path length is used.

The unequal (dissymmetric) interaction between our chiral sample and the right- and left-handed helices results in a rotation of the plane of polarization of the emerging light beam from its original position - and this angle of rotation is what the polarimeter measures.

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