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In it's most general form, the Lorentz-Lorenz equation is given by $$\frac{n^2 - 1}{n^2 + 2}= \frac{4 \pi}{3} N \alpha_\mathrm{m},$$ where $n$ is the refractive index, $N$ is the number of molecules per unit volume, and $\alpha_\mathrm{m}$ is the mean polarizability (Wikipedia).

How can we rewrite this equation for molar mass?

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Here $N$ is the number density, so $$ N=\frac{n\cdot N_\text{A}}{V} \\ n=\frac{\text{wt. of compoud}}{\text{Molar mass}}=\frac{w}{M} \\ N=\left(\frac{w}{V}\right)\cdot\left(\frac{N_\text{A}}{M}\right) \\ N=\frac{\rho N_\text{A}}{M} $$

Therefore, $$ \frac{n^2-1}{n^2 + 2}=\frac{4\pi\alpha_\text{m}}{3}\cdot \left(\frac{\rho N_\text{A}}{M}\right) $$

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