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Are there any approximate relationships between the (static) dielectric constant of a liquid and properties of the individual molecules?

For example, suppose I know the polarizability and dipole moment of a molecule, as well as it's density in the liquid phase. Can I estimate the dielectric constant of the liquid from these quantities (or other similar ones)?

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Yes, such relations exist.

Notably there are two general theories for polar and non-polar fluids that connect the polarizability $P$ with the dielectric constant $\epsilon$ as reviewed in this article by Harvey and Lennon, International Journal of Thermophysics, Vol. 26, No. 1, January 2005.

The Clausius-Mosotti equation for non-polar liquids is: $$P = \frac{\epsilon-1}{\epsilon +2}$$

Whereas an equation developed by Kirkwood for polar liquids is: $$ P = \frac{(\epsilon-1)(2\epsilon+1)}{9 \epsilon}$$

The article then continues to explain that you can estimate the polarizability from theoretical relations involving the molar density $\rho$ and the dipole moment $\mu$, but in your case the two equations above are sufficient since you assume that you know $P$.

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