Talk:Loschmidt constant/Derivation

The following is a slightly modified version of Loschmidt's derivation of the relationship between molecular diameter and macroscopic properties.

We start from Maxwell's equation for the mean free path ℓ of gas molecules of diameter d when the number density of molecules (the number of molecules per unit volume) is n0:
 * $$\ell = \frac{3}{4n_0\pi d^2}$$

Next we rearrange this to find the quantity 1/n0. As n0 is the number of molecules per unit volume, 1/n0 is the volume per "unit molecule" in the gas phase.
 * $$\frac{1}{n_0} = \frac{4\pi\ell d^2}{3}$$

At this point, Loschmidt makes a slight digression, but we need simply to say that "true volume" of a spherical molecule of diameter d (and hence radius r) is given by simple geometry. Nowadays, this quantity is known as the van der Waals volume, so we shall denote it Vvdw
 * $$V_{\rm vdw} = \frac{4}{3}\pi r^3 = \frac{\pi d^3}{6}$$

Loschmidt's genius was to realize that he could estimate Vvdw from the macroscopic properties of liquids and gases, notably their densities. Let us take Vgas as the volume occupied by a molecule in the gas phase, that is 1/n0, and Vliq as the volume occupied by a molecule in the liquid phase. Obviously, Vliq ≪ Vgas, as liquids are much denser than gases. For N molecules, each of mass m, the densities are defined as:
 * $$\rho_{\rm liq} = \frac{Nm}{NV_{\rm liq}}$$, $$\rho_{\rm gas} = \frac{Nm}{NV_{\rm gas}}$$

Hence
 * $$\frac{V_{\rm liq}}{V_{\rm gas}} = \frac{\rho_{\rm gas}}{\rho_{\rm liq}}$$

However, there is still a small amount of free space between the molecules in a liquid. The closest that spherical molecules can approach one another is by close-packing of spheres, as might be expected in a solid (and is found in many solids, although Loschmidt could only surmise this). In a close-packed arrangement the spheres only occupy 74% of the volume: This figure is now known as the atomic packing factor, so we shall denote it as φAPF: for a regular arrangement of spheres, it can be calculated by geometry (although Loschmidt's geometry seems to have failed him here, as he used an incorrect value for close-packing). Loschmidt realized that the atomic packing factor φAPF for a liquid must be only slightly lower than φAPF for a solid, because liquids are only slightly less dense than solids. By estimating φAPF,liq, he could state
 * $$V_{\rm vdw} = \phi_{\rm APF,liq}V_{\rm liq}\,$$

and
 * $$\frac{V_{\rm vdw}}{V_{\rm gas}} = \phi_{\rm APF,liq}\frac{\rho_{\rm gas}}{\rho_{\rm liq}}$$

By substituting the definitions of Vvdw (= πd3/6) and Vgas (= 1/n0), we obtain
 * $$\frac{\pi d^3 n_0}{6} = \phi_{\rm APF,liq}\frac{\rho_{\rm gas}}{\rho_{\rm liq}}$$

Substituting the unknown number density n0 by its definition in terms of the (measurable) mean free path:
 * $$n_0 = \frac{3}{4\pi\ell d^2}$$

gives
 * $$\frac{d}{8\ell} = \phi_{\rm APF,liq}\frac{\rho_{\rm gas}}{\rho_{\rm liq}}$$

which trivially rearranges to Loschmidt's result.