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)/ kBT]


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 NA(ρ - P) (h2 h1)/ (ρ RT)]


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


[NA 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 = kBN


kB = ………………..(ii)


Given,


The law of atmosphere states that:


n2 = n1 exp [ -mg (h2 h1)/ kBT] ……………(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.


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