Specific rotation
= Specific rotation
{title2=$[\alpha]_{\lambda}^{T}$}
For a <chemical solution> in a validated linear regime,
$$
\alpha=[\alpha]_{\lambda}^{T}\,l\,c,
$$
where $l$ is path length in $\mathrm{dm}$ and $c$ is <solute> <mass> per final solution <volume (thermodynamics)> in $\mathrm{g\,mL^{-1}}$. These conventions give <specific rotation> units $\mathrm{deg\,mL\,g^{-1}\,dm^{-1}}$. Specify <solvent>, <temperature> $T$ and <wavelength> $\lambda$: <specific rotation> is conditional on the sample environment.<Organic Chemistry by John McMurry/5.3 Optical Activity>[McMurry, Organic Chemistry, §5.3: observed rotation divided by path length and mass concentration]{ref}
For an original model, stipulate an unnamed <enantiomer> pair in the same <achiral> <solvent> at $T=20^\circ\mathrm C$ and <wavelength> $\lambda=589\,\mathrm{nm}$. Assign the positive member <specific rotation> $+40\,\mathrm{deg\,mL\,g^{-1}\,dm^{-1}}$, assumed constant over the modeled <concentrations>. With $l=1\,\mathrm{dm}$ and $c=0.200\,\mathrm{g\,mL^{-1}}$,
$$
\alpha=(+40)(1)(0.200)^\circ=+8^\circ.
$$
These are chosen model parameters, not measured properties of a named material. Doubling only $l$ or only $c$ doubles $\alpha$ within this stipulated linear model; doubling both multiplies it by four. A real <chemical solution> must be checked for <concentration> dependence rather than assuming that $0.200\,\mathrm{g\,mL^{-1}}$ is dilute enough.
For an additive mixture containing only this <enantiomer> pair as solutes, put $k=[\alpha]_+$ and write $c_+,c_-$ for their mass <concentrations>. Then
$$
\alpha=kl(c_+-c_-),\qquad c=c_++c_-.
$$
Because <enantiomers> have the same <molar mass>, their relative <mole fractions> within the pair also equal their relative mass fractions: $c_+=x_+c$, $c_-=x_-c$. Thus $\alpha=klc(x_+-x_-)$, which connects <optical rotation> to <enantiomeric excess>.