# 19.2: Specific Rotation

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

Optical rotation is the usual and most useful means of monitoring enantiomeric purity of chiral molecules. Therefore we need to know what variables influence the magnitude of optical rotation.

The measured rotation, $$\alpha$$, of a chiral substance varies with the concentration of the solution (or the density of a pure liquid) and on the distance through which the light travels. This is to be expected because the magnitude of $$\alpha$$ will depend on the number as well as the kind of molecules the light encounters. Another important variable is the wavelength of the incident light, which always must be specified even though the sodium D line $$\left( 589.3 \: \text{nm} \right)$$ commonly is used. To a lesser extent, $$\alpha$$ varies with the temperature and with the solvent (if used), which also should be specified. The optical rotation of a chiral substance usually is reported as a specific rotation $$\left[ \alpha \right]$$, which is expressed by the Equations 19-1 or 19-2

For solutions:

$\left[ \alpha \right]^t_{\lambda} =\dfrac{100 \alpha}{l \times c} \tag{19-1}$

For neat liquids:

$\left[ \alpha \right]^t_{\lambda} =\dfrac{\alpha}{l \times d} \tag{19-2}$

with

• $$\alpha$$ is the measured rotation in degrees
• $$t$$ is the temperature
• $$\lambda$$ is the wavelength of light
• $$l$$ is the length in decimeters of the light path through the solution
• $$c$$ is the concentration in grams of sample per 100 ml of solution
• $$d$$ is the density of liquid in grams ml-I

For example, quinine (Section 19-3A) is reported as having $$\left[ \alpha \right]_{D} = -117^\text{o} \left( c = 1.5, \: \ce{CHCl_3} \right) \left( t = 17^\text{o} \right)$$, which means that it has a levorotation of 117 degrees for sodium D light $$\left( 589.3 \: \text{nm} \right)$$ at a concentration of $$1.5 \: \text{g}$$ per $$100 \: \text{mL}$$ of chloroform solution at $$17^\text{o} \text{C}$$ when contained in a tube 1 decimeter long.

Frequently, molecular rotation, $$\left[ M \right]$$, is used in preference to specific rotation and is related to specific rotation by Equation 19-3:

$\left[ M \right]^t_{\lambda} =\dfrac{\left[ \alpha \right]^t_{\lambda} \times M}{100} \tag{19-3}$

in which $$M$$ is the molecular weight of the compound. Expressed in this form, optical rotations of different compounds are directly comparable on a molecular rather than a weight basis. The effects of wavelength of the light in the polarized beam on the magnitude and sign of the observed optical rotation are considered in Section 19-9.