# Is specific rotation defined for mixtures of enantiomers or meso compounds?

I recently had to do an experiment where we had to make a racemic mixture of D and L complexes and then resolve the enantiomers, and measure the optical rotation of the racemic mixture, and the individual salts. After that we had to calculate the specific rotation in case.

For an optically active substance, defined by $$\ce{[α]^θ_λ = α/γl}$$, where α is the angle through which plane polarized light is rotated by a solution of mass concentration γ and path length l. Here θ is the Celsius temperature and λ the wavelength of the light at which the measurement is carried out.

Source: CRC handbook of Chemistry and Physics

Also, specific rotation is treated as an intensive property of a compound (i.e. one enantiomer). Given that, does it make any sense to calculate the specific optical rotation for a mixture of enantiomers (whether racemic or not)?

And applying the same logic, can we say that specific rotation has no meaning for a meso-compound ?

• It would help if you could please put references when you answer, because my confusion is about the definition itself. – Shoubhik R Maiti Dec 17 '19 at 21:18
• If you knew for certain that you only had a mixture of two enantiomers in a solution, but didn't know the relative concentrations, you could use OR to determine them. The specific rotation for a meso compound would be zero, so it does have meaning, at least in the sense that it tells you the compound is not optically active. – Tyberius Dec 17 '19 at 21:28
• An observed rotation of zero degrees does not necessarily mean a racemate, meso or achiral compound. If zero is obtained, the sample should be diluted and the rotation repeated to assure that the original reading was not a multiple of 360 (0) degrees. – user55119 Dec 17 '19 at 22:52
• @user55119, Is there any real example of 360 degree rotation due to a mixture? Never heard of it. – M. Farooq Dec 18 '19 at 1:27
• @M. Farooq: In part it is a philosophical issue. Observed rotation depends on concentration. I was not referring to specific rotation. – user55119 Dec 18 '19 at 3:06

Nothing stops you from taking the equation and calculating a specific rotation of a mixture. Indeed, as far as I am aware when establishing a new asymmetric reaction, the e.e. is measured by something like chiral HPLC (which will unambiguously give two peaks that can be integrated and compared) while the specific rotation is measured for the resulting mixture of enantiomers as is. So the paper will say something like $$99~\%$$ yield ($$98~\%\ \text{e.e.}$$) but below there will be a standard calculation of $$[\alpha]_\mathrm D^\pu{20^\circ C}$$.
So obviously you can measure a specific rotation of the racemate or meso compound but whatever value you measure should fall into the error of the polarimeter so that the final value of $$[\alpha]_\mathrm D^\pu{20^\circ C}=0$$.