How far apart is the dust in the Sombrero Galaxy's dust lane?

The sizes of dust particles varies enormously, and thus the density of dust particles depend upon which sizes we are talking about. The dust size distribution can be described by a power law with a slope of roughly $-3.5$; that is, the density of 0.1 µm particles of is $10^{-3.5}$ times that of 0.01 µm particles, or roughly 3000 times smaller.

Moreover, the discussion is complicated by the fact that we don't really have a formal definition of what is dust. But one "definition" takes the minumum size as a conglomeration of a few molecules, and the maximum size as "what has time to grow in a formation process" (these dust particles can later "stick together" and grow to form pebbles, rocks, asteroids, and planets. That's why any definition will be arbitrary).

In a dust-dense region of the interstellar medium, this results in a typical mean density of roughly one dust particle per cubic centimeter.