13.13 A gas in equilibrium has uniform density and pressure throughout its volume. This is strictly true only if there are no external influences. A gas column under gravity, for example, does not have a uniform density (and pressure). As you might expect, its density decreases with height. The precise dependence is given by the so-called law of atmospheres

${\mathrm{n}}_2={\mathrm{n}}_1{\mathrm{e}}^{\left[-\mathrm{mg}\raisebox{1ex}{$\left({\mathrm{h}}_2 – {\mathrm{h}}_1\right)$}\!\left/ \!\raisebox{-1ex}{${\mathrm{k}}_{\mathrm{B}}\mathrm{T}$}\right.\right]}$

where ${\mathrm{n}}_2$, ${\mathrm{n}}_1$ refer to number density at heights ${\mathrm{h}}_2$ and ${\mathrm{h}}_1$ respectively.

Use this relation to derive the equation for sedimentation equilibrium of a suspension in a liquid column:

${\mathrm{n}}_2={\mathrm{n}}_1{\mathrm{e}}^{\left[-{\mathrm{mgN}}_{\mathrm{A}}\raisebox{1ex}{$\left(\mathrm{\rho }-\mathrm{\rho }\text{'}\right)\left({\mathrm{h}}_2 – {\mathrm{h}}_1\right)$}\!\left/ \!\raisebox{-1ex}{$\mathrm{\rho RT}$}\right.\right]}$

where ρ is the density of the suspended particle, and $\mathrm{\rho }\text{'}$, that of the surrounding medium. [${\mathrm{N}}_{\mathrm{A}}$ is Avogadro’s number and R the universal gas constant.]

[Hint: Use Archimedes principle to find the apparent weight of the suspended particle.]