Physics:Molar refractivity

Molar refractivity, [math]\displaystyle{ A }[/math], 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

[math]\displaystyle{ A = \frac{4 \pi}{3} N_A \alpha, }[/math]

where [math]\displaystyle{ N_A \approx 6.022 \times 10^{23} }[/math] is the Avogadro constant and [math]\displaystyle{ \alpha }[/math] is the mean polarizability of a molecule.

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

[math]\displaystyle{ A = \frac{R T}{p} \frac{n^2 - 1}{n^2 + 2} }[/math]

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

[math]\displaystyle{ A = \frac{R T}{p} \frac{n^2 - 1}{3}. }[/math]

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

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

[math]\displaystyle{ A = \frac{M}{\rho} \frac{n^2 - 1}{n^2 + 2} \approx \frac{M}{\rho} \frac{n^2 - 1}{3}. }[/math]

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