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 V_vdw

$V_{\rm {vdw}}={\frac {4}{3}}\pi r^{3}={\frac {\pi d^{3}}{6}}$

Loschmidt's genius was to realize that he could estimate V_vdw from the macroscopic properties of liquids and gases, notably their densities. Let us take V_gas as the volume occupied by a molecule in the gas phase, that is 1/n₀, and V_liq as the volume occupied by a molecule in the liquid phase. Obviously, V_liq ≪ V_gas, 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 φ_APF: 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 φ_APF for a liquid must be only slightly lower than φ_APF for a solid, because liquids are only slightly less dense than solids. By estimating φ_APF,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 V_vdw (= πd³/6) and V_gas (= 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.