Qualitative Comparisons of Dielectric Constants of Organic Solvents

This question is inspired by the discussion around this earlier question.

It is relatively simple to do a qualitative comparison of the polarities of different molecules (for example by analyzing the presence of polar bonds alongside the geometry of the molecule).

Now, ethanol and acetic acid have almost identical dipole moments ($$\mu$$=1.68 for acetic acid, $$\mu$$ = 1.69 for ethanol) but dramatically different dielectric constants ($$\mathcal{E}$$ = 6.15 for acetic acid, $$\mathcal{E}$$ = 24.5).

Is there a similar qualitative analysis that can be done for dielectric constant of a solvent?

Yes, but it is much more complicated. The molecule does not have to have a dipole, in which case the (static) dielectric constant (relative permittivity) depends on the molecule's polarisability and if it does have a dipole it depends on both factors. The equation is $$\displaystyle \epsilon_r=\frac{1+2b}{1-b}$$ where $$b=\left(\alpha+\mu^2/(k_BT)\right)N/(3\epsilon_0)$$ and $$N=N_A\rho/M$$ where $$\alpha$$ is polarisability, $$\mu$$ dipole, $$\rho$$ density,$$M$$ molar mass, $$\epsilon_0$$ permittivity of free space. See Atkins, & Friedman, 'Molecular Quantum Mechanics'. The dielectric constant also depends on frequency of the applied electric field, but the value quoted is normally the static value measured by comparing the capacity of a capacitor with and without the solvent. The name relative permittivity arises from the potential a charge produces. In a vacuum at a distance $$r$$ this is $$\displaystyle \phi=\frac{q}{4\pi\epsilon_0 r}$$, in a dielectric medium of permittivity $$\epsilon$$ this becomes $$\displaystyle \phi'=\frac{q}{4\pi\epsilon r}$$ and $$\epsilon_r=\epsilon/\epsilon_0$$.
As an aside you can see why a polar molecule is generally insoluble in a non-polar solvent. If the permittivity is small (say hexane) the electric potential of a charge is higher at any given $$r$$ than for a solvent with a large $$\epsilon$$, such as acetonitrile. This means that polar molecules are less soluble in low dielectric solvents than they are in high dielectric ones.