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

Question

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.

However, when reading about optical activity, I came across the definition:

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 ?

Answer 1

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 $[\alpha]_\mathrm D^\pu{20^\circ C}=0$.