Editing Circular dichroism (section)

====Molar ellipticity====

Although <math> \Delta A </math> is usually measured, for historical reasons most measurements are reported in degrees of ellipticity.
Molar ellipticity is circular dichroism corrected for concentration. Molar circular dichroism and molar ellipticity, <math> [\theta]</math>, are readily interconverted by the equation:

[[File:Electric Vectors 1.png|thumb|200px|right|Elliptical polarized light (violet) is composed of unequal contributions of right (blue) and left (red) circular polarized light.]]

:<math> [\theta] = 3298.2\,\Delta \varepsilon.\, </math>

This relationship is derived by defining the [[polarization (waves)|ellipticity of the polarization]] as:

:<math> \tan \theta = \frac{E_\mathrm R - E_\mathrm L}{E_\mathrm R + E_\mathrm L} \,</math>

where
:<math> E_\mathrm R</math> and <math> E_\mathrm L</math> are the magnitudes of the [[electric field]] [[vector (geometry)|vectors]] of the right-circularly and left-circularly polarized light, respectively.

When <math> E_\mathrm R</math> equals <math> E_\mathrm L</math> (when there is no difference in the absorbance of right- and left-circular polarized light), <math> \theta</math> is 0° and the light is [[linear polarization|linearly polarized]]. When either <math> E_\mathrm R</math> or <math> E_\mathrm L</math> is equal to zero (when there is complete absorbance of the circular polarized light in one direction), <math> \theta</math> is 45° and the light is [[circular polarization|circularly polarized]].

Generally, the circular dichroism effect is small, so <math> \tan\theta</math> is small and can be approximated as <math> \theta</math> in [[radian]]s. Since the [[Radiant intensity|intensity]] or [[irradiance]], <math> I</math>, of light is proportional to the square of the electric-field vector, the ellipticity becomes:

:<math> \theta (\text{radians}) = \frac{(I_\mathrm R^{1/2} - I_\mathrm L^{1/2})}{(I_\mathrm R^{1/2} + I_\mathrm L^{1/2})}\,</math>

Then by substituting for I using [[Beer–Lambert law|Beer's law]] in [[natural logarithm]] form:

:<math> I = I_0 \mathrm e^{-A\ln 10}\,</math>

The ellipticity can now be written as:

:<math> \theta (\text{radians}) = \frac{(\mathrm e^{\frac{-A_\mathrm R}{2}\ln 10} - \mathrm e^{\frac{-A_\mathrm L}{2}\ln 10})}{(\mathrm e^{\frac{-A_\mathrm R}{2}\ln 10} + \mathrm e^{\frac{-A_\mathrm L}{2}\ln 10})} = \frac{\mathrm e^{\Delta A \frac{\ln 10}{2}} - 1}{\mathrm e^{\Delta A \frac{\ln 10}{2}} + 1} \,</math>

Since <math> \Delta A \ll 1</math>, this expression can be approximated by expanding the exponentials in a [[Taylor series]] to first-order and then discarding terms of <math> \Delta A </math> in comparison with unity and converting from [[radian]]s to degrees:

:<math> \theta (\text{degrees}) = \Delta A \left( \frac {\ln 10}{4} \right) \left( \frac {180}{\pi} \right)\, </math>

The linear dependence of solute concentration and pathlength is removed by defining molar ellipticity as,

:<math> [\theta] = \frac {100\theta}{Cl}\, </math>

Then combining the last two expression with [[Beer–Lambert law|Beer's law]], molar ellipticity becomes:

:<math> [\theta]= 100 \,\Delta \varepsilon \left( \frac {\ln 10}{4} \right) \left( \frac {180}{\pi} \right) = 3298.2\,\Delta \varepsilon \,</math>

The units of molar ellipticity are historically (deg·cm<sup>2</sup>/dmol). To calculate molar ellipticity, the sample concentration (g/L), cell pathlength (cm), and the molecular weight (g/mol) must be known.

If the sample is a protein, the mean residue weight (average molecular weight of the amino acid residues it contains) is often used in place of the molecular weight, essentially treating the protein as a solution of amino acids. Using mean residue ellipticity facilitates comparing the CD of proteins of different molecular weight; use of this normalized CD is important in studies of protein structure.