Derivation of Specific Rotation formula?

Question

This is sort of a weird question, so bear with me:

I was reading about how chiral molecules rotate plane polarized light and decided to think of a "logical" formula for expressing the amount that light rotates when passing through a solution:

      A_true(angle of rotation) = A_obs(observed angle of rotation) * C(concentration)

Turns out there IS a formula and I was kinda far off:

                             [alpha] = [alpha]_obs/(C*l)

where alpha is the angle of rotation, C is concentration, and l is path length.

I am curious as to how exactly this equation was derived and who proposed it? Why did they decide to divide angle_observed by the parameters as opposed to multiply (because the important linear relationship between alpha and alpha_obs is preserved either way)? Was it arbitrary? For practical reasons?

Two quick supplemental questions:

1.) How do we know if the light was rotated clockwise 45 degrees or counterclockwise -135 degrees? (cause the pictures depict the polarized light as a line)

2.) How do we know if the light was rotated 15 degrees or 375 degrees (or does it not matter for practical reasons cause its the difference that is useful)?

Answer 1

For your first question you want to find a way to express the observered angle in polarization in a form that is independent of the concentration of the molecule and pathlength of the light. In other words you are looking for units of angle per concentraion pathlength and you need to divide by the concentration and pathlength. Of course when you want to calculate the angle of a certain solution you multiply the concentration and pathlength with a (tabulated) standard rotation angle.

Typically, the rotation of the polarization of the light by chiral molecules is very small (only a few percent), so you can probably take the smallest angle. Of course you can vary the concentration or pathlength to see how the angle changes if you increase or decrease those parameters. This also applies to your second additional question.

Answer 2

What is the difference between "observed" and "true" angles of rotation?

What I think you mean is that the "true" angle is "specific" to the compound. It is defined like absorption coefficient $\beta$ as a constant of proportionality :

$absorption \space A = coefficient \space \beta \times concentration \space C \times path \space length \space L$
$rotation \space angle \space A = coefficient \space \alpha \times concentration \space C \times path \space length \space L$

Like Ohm's Law and Hooke's Law, these formulae are based on experimental observation and the reasonable expectation that equal small increases in C or L have the same additive effect on absorption (or rotation angle) regardless of the initial value of C or L. These assumptions might break down for large concentration (ie the Law is only valid for weak concentrations) or small path lengths.

I think your two supplementary questions are in fact the same. You can distinguish a +45 degree rotation from a -135 degree rotation (or a +405 degree rotation, which is also equivalent) by examining what happens as the path length or concentration is increased from small values. You are correct that polarisation is linear so the two rotations have the same end result. Whether there is any practical difference depends on whether you are only interested in the end result or some effect which depends on intermediate rotation angles.