# Molar refractivity

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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 α is the mean polarizability of a molecule.

Substituting the molar refractivity into the Lorentz-Lorenz formula gives

$A = \frac{N_A}{N} \frac{n^2 - 1}{n^2 + 2},$

where N is the number of molecules per unit volume and n is the refractive index. The ratio NA / N is simply the molar volume Vm. Evaluating the ideal gas law for 1 mole gives

$V_m = \frac{N_A}{N} = \frac{R T}{p},$

where R is the universal gas constant, T is the absolute temperature, and p is the pressure. Then the molar refractivity is

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

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

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