# Molar refractivity

Molar refractivity, $A$, 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

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

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

Substituting the molar refractivity into the Lorentz-Lorenz formula gives

$A = \frac{R T}{p} \frac{n^2 - 1}{n^2 + 2}$

For a gas, $n^2 \approx 1$, so the molar refractivity can be approximated by

$A = \frac{R T}{p} \frac{n^2 - 1}{3}.$

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

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

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

## 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