Specific rotation and observed rotation
If S glyceraldehyde has a specific rotation of \([\\alpha]^{20}_{D} = -8.7^{\\circ}\), the sign indicates a counterclockwise (left) rotation of plane-polarized light under the stated conditions. Enantiomers share the same magnitude of specific rotation but have opposite signs when measured at the same temperature, wavelength, solvent, and concentration definition.
\[ [\alpha]^{T}_{\lambda}=\frac{\alpha_{\text{obs}}}{l\cdot c} \qquad\Longleftrightarrow\qquad \alpha_{\text{obs}}=[\alpha]^{T}_{\lambda}\cdot l\cdot c \]
The common convention is \(l\) in decimeters (dm) and \(c\) in \(\text{g/mL}\), giving \(\alpha_{\text{obs}}\) in degrees. Some references use \(c\) in \(\text{g}/100\ \text{mL}\); mixing conventions changes the numerical result by a factor of 100.
Assumed measurement conditions
- Temperature notation: \(20^{\circ}\text{C}\) in \([\\alpha]^{20}_{D}\).
- Wavelength notation: sodium D line (\(D\), about \(589\ \text{nm}\)).
- Path length: \(l=1.00\ \text{dm}\).
- Concentration definition: \(c=0.200\ \text{g/mL}\) of glyceraldehyde in solution.
Expected observed rotation for a pure enantiomer sample
A pure (S)-glyceraldehyde solution under these conditions has an observed rotation proportional to \(l\cdot c\).
\[ \alpha_{\text{obs, pure}} = (-8.7^{\circ})\cdot(1.00\ \text{dm})\cdot(0.200\ \text{g/mL}) = -1.74^{\circ} \]
Enantiomeric excess from an observed rotation
For mixtures of only the two enantiomers (R and S), the observed rotation scales with enantiomeric excess because the two rotations partially cancel. The sign of \(\alpha_{\text{obs}}\) identifies which enantiomer is in excess when the sign of the pure enantiomer is known.
\[ \text{ee}=\frac{\alpha_{\text{obs}}}{\alpha_{\text{obs, pure}}}\times 100\% \]
With \(\alpha_{\text{obs}}=-1.22^{\circ}\) measured at the same \(l\) and \(c\):
\[ \text{ee}=\frac{-1.22^{\circ}}{-1.74^{\circ}}\times 100\% \approx 70.1\% \]
The mixture composition follows from \(\text{ee}=\%S-\%R\) with \(\%S+\%R=100\%\):
\[ \%S=\frac{100\%+\text{ee}}{2}\approx \frac{100\%+70.1\%}{2}=85.05\% \qquad \%R=\frac{100\%-\text{ee}}{2}\approx 14.95\% \]
| Quantity | Expression | Value |
|---|---|---|
| Pure-sample rotation | \(\alpha_{\text{obs, pure}}=[\alpha]^{20}_{D}\cdot l\cdot c\) | \(-1.74^{\circ}\) |
| Enantiomeric excess | \(\text{ee}=(\alpha_{\text{obs}}/\alpha_{\text{obs, pure}})\times 100\%\) | \(70.1\%\) (S in excess) |
| Composition (two-enantiomer mixture) | \(\%S=(100\%+\text{ee})/2,\ \%R=(100\%-\text{ee})/2\) | \(\%S\approx 85.05\%,\ \%R\approx 14.95\%\) |
Visualization: polarimetry sign and mixture cancellation
Common pitfalls
- Concentration convention mismatch: values tabulated with \(c\) in \(\text{g}/100\ \text{mL}\) produce rotations 100× larger than the \(\text{g/mL}\) convention for the same \(l\).
- Wavelength and temperature dependence: \([\\alpha]^{20}_{D}\) is not interchangeable with measurements made at different \(T\) or \(\lambda\).
- Mixture assumption: \(\text{ee}=\alpha_{\text{obs}}/\alpha_{\text{obs, pure}}\) holds for two-enantiomer mixtures; additional chiral impurities break the proportionality.
- Sign interpretation: dextrorotatory/levorotatory labels relate to the sign of \(\alpha_{\text{obs}}\), not directly to R/S configuration in general; glyceraldehyde is a special historical reference point, not a universal mapping.
Result summary
With \([\\alpha]^{20}_{D}(S)=-8.7^{\circ}\), \(l=1.00\ \text{dm}\), and \(c=0.200\ \text{g/mL}\), the expected pure-sample rotation is \(\alpha_{\text{obs, pure}}=-1.74^{\circ}\). An observed rotation of \(-1.22^{\circ}\) under the same conditions corresponds to \(\text{ee}\approx 70.1\%\) in favor of (S)-glyceraldehyde, giving approximately \(85.05\%\) S and \(14.95\%\) R.