Editing Molar refractivity

'''Molar refractivity''',<ref>{{cite journal | last1=Van Rysselberghe | first1=Pierre | title=Remarks concerning the Clausius-Mossotti Law | journal=The Journal of Physical Chemistry | publisher=American Chemical Society | volume=36 | issue=4 | year=1932 | issn=0092-7325 | doi=10.1021/j150334a007 | pages=1152-1155 | url=https://doi.org/10.1021/j150334a007 | url-access=subscription }}</ref><ref>{{cite journal |last1=Achtermann  |first1=H. J. |last2=Hong |first2=J. G. |last3=Magnus |first3=G. |last4=Aziz |first4=R. A. |last5=Slaman |first5=M. J. |date=1993 |title=Experimental determination of the refractivity virial coefficients of atomic gases |url=https://doi.org/10.1063/1.464212 |journal=The Journal of Chemical Physics |volume=98 |issue=3 |pages=2308-2318 |doi=10.1063/1.464212 |access-date=2024-11-24 |issn=0021-9606|url-access=subscription }}</ref><!-- IT SEEMS LIKE THE FOLLOWING REFERENCE IS A BROKEN LINK. HIDING IT: <ref>W. Foerst et.al. ''Chemie für Labor und Betrieb'', '''1967''', ''3'', 32-34.
https://organic-btc-ilmenau.jimdo.com/app/download/9062135220/molrefraktion.pdf?t=1616948905</ref>--> <math>R_m</math>, is a measure of the total [[polarizability]] of a [[mole (unit)|mole]] of a substance.

For a [[perfect gas|perfect dielectric]] which is made of one type of molecule, the molar refractivity is proportional to the polarizability of a single molecule of the substance. For real materials, intermolecular interactions (the effect of the induced dipole moment of one molecule on the field felt by nearby molecules) give rise to a density dependence.

The molar refractivity is commonly expressed as a sum of components, where the leading order is the value for a perfect dielectric, followed by the density-dependent corrections:

:<math> R_m = A + B\cdot\rho + C\cdot\rho^2 + ... </math>

The coefficients <math> A, B, C, ... </math> are called the refractivity virial coefficients. Some research papers are dedicated to finding the values of the subleading coefficients of different substances. In other contexts, the material can be assumed to be approximately perfect, so that the only coefficient of interest is <math> A </math>.

The coefficients depend on the wavelength of the applied field (and on the type and composition of the material), but not on [[thermodynamics|thermodynamic]] [[state variable]]s such as [[temperature]] or [[pressure]].

The leading order (perfect dielectric) molar refractivity is defined as
:<math> A = \frac{4 \pi}{3} N_A \alpha_\mathrm{m}, </math>
where <math>N_A \approx 6.022 \times 10^{23}</math> is the [[Avogadro constant]] and <math>\alpha_\mathrm{m}</math> is the mean [[polarizability]] of a molecule.

Substituting the molar refractivity into the [[Lorentz-Lorenz]] formula gives, for gasses
:<math> \frac{n^2 - 1}{n^2 + 2} = A \frac{p}{RT}</math>

where <math>n</math> is the [[refractive index]], <math>p</math> is the pressure of the gas, <math>R</math> is the [[universal gas constant]], and <math>T</math> is the (absolute) temperature; the [[ideal gas law]] was used here to convert the particle density (appearing in the Lorentz-Lorenz formula) to pressure and temperature.

For a gas, <math>n^2 \approx 1</math>, so the molar refractivity can be approximated by
:<math>A = \frac{R T}{p} \frac{n^2 - 1}{3}.</math>
As mentioned above, despite the relation imposed by the last expression on <math> A, T, p </math> and <math> n </math>, the molar refractivity <math> A </math> is a function of the substance itself and not of its conditions, and therefore does not depend on the three state variables appearing in the right hand side of the expression.{{efn|<math> n </math> depends on <math> T </math> and <math> p </math>, and their variations cancel out in this expression; the advantage in extracting <math> A </math> and presenting it as a combination of the other quantities is that it gives an experimental way to measure <math> A </math> (one simply needs to measure <math> n </math> for a gas with known temperature and pressure).}}

In terms of [[density]] ρ and [[molecular weight]] M, it can be shown that:
:<math>A = \frac{M}{\rho} \frac{n^2 - 1}{n^2 + 2} \approx \frac{M}{\rho} \frac{n^2 - 1}{3}.</math>

==Notes==
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==References==
{{Reflist}}

==Bibliography==
* [[Max Born|Born, Max]], and Wolf, Emil, ''[[Principles of Optics]]: Electromagnetic Theory of Propagation, Interference and Diffraction of Light'' (7th ed.), section 2.3.3, Cambridge University Press (1999) {{ISBN|0-521-64222-1}}

[[Category:Physical chemistry]]
[[Category:Optical quantities]]
[[Category:Molar quantities]]