# Molar refractivity

Molar refractivity, ${\displaystyle A}$ , is a measure of the total polarizability of a mole of a substance and is dependent on the temperature, the index of refraction, and the pressure.

The molar refractivity is defined as

${\displaystyle A={\frac {4\pi }{3}}N_{A}\alpha ,}$

where ${\displaystyle N_{A}\approx 6.022\times 10^{23}}$ is the Avogadro constant and ${\displaystyle \alpha }$ is the mean polarizability of a molecule.

Substituting the molar refractivity into the Lorentz-Lorenz formula gives, for gasses

${\displaystyle A={\frac {RT}{p}}{\frac {n^{2}-1}{n^{2}+2}}}$

where ${\displaystyle n}$ is the refractive index, ${\displaystyle p}$ is the pressure of the gas, ${\displaystyle R}$ is the universal gas constant, and ${\displaystyle T}$ is the (absolute) temperature. For a gas, ${\displaystyle n^{2}\approx 1}$ , so the molar refractivity can be approximated by

${\displaystyle A={\frac {RT}{p}}{\frac {n^{2}-1}{3}}.}$

In SI units, ${\displaystyle R}$ has units of J mol−1 K−1, ${\displaystyle T}$ has units K, ${\displaystyle n}$ has no units, and ${\displaystyle p}$ has units of Pa, so the units of ${\displaystyle A}$ are m3 mol−1.

In terms of density ρ, molecular weight M, it can be shown that:

${\displaystyle A={\frac {M}{\rho }}{\frac {n^{2}-1}{n^{2}+2}}\approx {\frac {M}{\rho }}{\frac {n^{2}-1}{3}}.}$

## References

• 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