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 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

n₂ = n₁ exp [ -mg (h₂ – h₁)/ k_BT]

where n₂, n₁ refer to number density at heights h₂ and h1 respectively. Use this relation to derive the equation for sedimentation equilibrium of a suspension in a liquid column:

n₂ = n₁ exp [ -mg N_A ( $\mathit{\rho }$ - ${\mathit{\rho }}^{\mathit{\text{'}}}$ ) (h₂ –h₁₎/ ( $\mathit{\rho }$ RT)]

where $\mathit{\rho }$ is the density of the suspended particle, and  that of surrounding medium. [N_A is Avogadro's number, and R the universal gas constant.] [Hint : Use Archimedes principle to find the apparent weight of the suspended particle.]