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(ρ - P′) (h₂ – h₁)/ (ρ RT)]

Where ρ is the density of the suspended particle, and ρ’ that of surrounding medium.

[N_A is Avogadro’s number, and R the universal gas constant.]

Let us suppose:

The density of the suspended particle is ρ

The density of the medium is ρ’

Mass of the medium which is displaced by the suspended particle is m

Mass of the suspended particle is m’

Volume of the suspended particle is V

Since, the equation for sedimentation equilibrium of a suspension in a liquid column is asked we will use the Archimedes’ principle.

The Archimedes’ Principle for a suspended particle in a liquid column, the effective weight W’ of the suspended particle in the liquid is given as:

W’ = Weight of the displaced medium – Weight of the suspended particle

⇒ W’ = mg – m’g

⇒ W’ = mg – Vρ’g =

⇒ ……..(i)

And we have,

Gas Constant, R = k_BN

⇒ k_B = ………………..(ii)

Given,

The law of atmosphere states that:

n₂ = n₁ exp [ -mg (h₂ – h₁)/ k_BT] ……………(iii)

where,

n₁ is the number density at the height ‘h₁’

n₂ is the number density at the height ‘h₂’

‘mg’ is the apparent weight of the particle suspended in the gas column.

Replacing the values from equation (i) and (ii) in equation (iii) we get:

⇒ (h₂ – h₁)

⇒(h₂ – h₁)

This is the equation for sedimentation equilibrium of a suspension in a liquid column.