Specific rotation and measurement convention
“Tartaric acid has a specific rotation of 12.0” is interpreted as a pure enantiomer of tartaric acid having \([ \alpha ]^{20}_{D} = +12.0^{\circ}\) under the stated or implied conditions (temperature, wavelength, solvent, and concentration convention). The observed rotation \(\alpha_{\text{obs}}\) depends linearly on path length \(l\) and concentration \(c\) in solution.
\[ [\alpha]^{T}_{\lambda}=\frac{\alpha_{\text{obs}}}{l\cdot c} \qquad\Longleftrightarrow\qquad \alpha_{\text{obs}}=[\alpha]^{T}_{\lambda}\cdot l\cdot c \]
The convention \(l\) in decimeters (dm) and \(c\) in \(\text{g/mL}\) is used, so \(\alpha_{\text{obs}}\) is obtained in degrees. A two-enantiomer mixture (dextrorotatory and levorotatory tartaric acid only) is assumed for enantiomeric-excess calculations.
Expected observed rotation for the stated concentration and tube length
Given \([ \alpha ]^{20}_{D} = +12.0^{\circ}\), \(c = 0.150\ \text{g/mL}\), and \(l = 2.00\ \text{dm}\), the proportionality \(\alpha_{\text{obs}}=[\alpha]\cdot l\cdot c\) gives the expected observed rotation for a pure enantiomer sample.
\[ \alpha_{\text{obs, pure}} = ( +12.0^{\circ} )\cdot(2.00\ \text{dm})\cdot(0.150\ \text{g/mL}) = +3.60^{\circ} \]
Enantiomeric excess from an observed rotation
In a mixture containing only two enantiomers, the observed rotation is proportional to the excess of one enantiomer over the other. The ratio \(\alpha_{\text{obs}}/\alpha_{\text{obs, pure}}\) equals the fraction of optical purity (as a decimal), which is the enantiomeric excess.
\[ \text{ee}=\frac{\alpha_{\text{obs}}}{\alpha_{\text{obs, pure}}}\times 100\% \]
With \(\alpha_{\text{obs}}=+3.00^{\circ}\) under the same \(l\) and \(c\):
\[ \text{ee}=\frac{+3.00^{\circ}}{+3.60^{\circ}}\times 100\% \approx 83.3\% \]
The mixture composition follows from \(\text{ee}=\%(+)-\%(-)\) and \(\%(+)+\%(-)=100\%\):
\[ \%(+)=\frac{100\%+\text{ee}}{2}\approx \frac{100\%+83.3\%}{2}=91.65\% \qquad \%(-)=\frac{100\%-\text{ee}}{2}\approx 8.35\% \]
| Quantity | Expression | Value |
|---|---|---|
| Specific rotation (assumed) | \([ \alpha ]^{20}_{D}\) | \(+12.0^{\circ}\) |
| Expected pure-sample rotation | \(\alpha_{\text{obs, pure}}=[\alpha]\cdot l\cdot c\) | \(+3.60^{\circ}\) |
| Enantiomeric excess from \(+3.00^{\circ}\) | \(\text{ee}=(\alpha_{\text{obs}}/\alpha_{\text{obs, pure}})\times 100\%\) | \(83.3\%\) toward the \(+\) enantiomer |
| Approximate composition | \(\%(+)=\frac{100\%+\text{ee}}{2},\ \%(-)=\frac{100\%-\text{ee}}{2}\) | \(\%(+)\approx 91.65\%,\ \%(-)\approx 8.35\%\) |
Visualization: polarimeter sign and linear mixing law
Common pitfalls
- Concentration definition mismatch: some tables use \(c\) in \(\text{g}/100\ \text{mL}\) rather than \(\text{g/mL}\), shifting \(\alpha_{\text{obs}}\) by a factor of 100 for the same numeric \(c\).
- Condition dependence: \([ \alpha ]^{20}_{D}\) changes with temperature, wavelength, solvent, and sometimes concentration; numerical substitution requires matching conditions.
- Non-enantiomer mixtures: meso-tartaric acid is optically inactive and a mixture containing meso form does not follow the two-enantiomer cancellation model.
- Sign interpretation: the \(+\) or \(−\) sign describes rotation direction, not a universal assignment of absolute configuration without additional context.
Result summary
With \([ \alpha ]^{20}_{D}=+12.0^{\circ}\), \(l=2.00\ \text{dm}\), and \(c=0.150\ \text{g/mL}\), the expected observed rotation for a pure enantiomer sample is \(\alpha_{\text{obs, pure}}=+3.60^{\circ}\). An observed rotation of \(+3.00^{\circ}\) under the same conditions corresponds to \(\text{ee}\approx 83.3\%\) toward the dextrorotatory enantiomer, giving approximately \(91.65\%\) (+) and \(8.35\%\) (−).