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
`n_2=n_1 exp [-mg(h_2-h_1)//(k_BT)]`
where `n_2`, n1 refer to number density at heights` h_2` and `h_1` respectively. Use this relation to derive the equation for sedimentation equilibrium of a suspension in a liquid column:
`n_2=n_1 " exp " [-mg N_A(rho-rho')(h_2-h_1)//(rho " RT")]` where `rho` is the density of the suspended particle, and `rho` 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.]

Explanation

According to the law of atmospheres, we have:
`n_2 = n_1 exp [-mg (h_2 – h_1)// kBT]` … (i)
Where,
`n_1` is thenumber density at height `h_1`, and `n_2` is the number density at height `h_2` mg is the weight of the particle suspended in the gas column
Density of the medium `= rho`‘
Density of the suspended particle `= rho`

Mass of one suspended particle = m‘
Mass of the medium displaced = m
Volume of a suspended particle = V
According to Archimedes’ principle for a particle suspended in a liquid column, the effective weight of the suspended particle is given as:
Weight of the medium displaced – Weight of the suspended particle
= mg – m‘g
`=mg-V rho'g=mg-((m)/rho)rho'g`
`=mg(1-(rho')/(rho))`
Gas constant ,`R=K_BN`
` k_B=R/N`
Substituting equation (ii) in place of mg in equation (i) and then using equation (iii), we get:
`n_1exp[-mg(1-(rho')/(rho))(h_2-H_1)(N)/(RT)]`
`n_1exp[-mg(rho-rho'))(h_2-H_1)(N)/(RT)]`