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

where refer to number density at heights and respectively. Use this relation to derive the equation for sedimentation equilibrium of a suspension in a liquid column:
where is the density of the suspended particle, and ‘, that of surrounding medium. [ is Avogadro’s number, and the universal gas constant.] [Hint: Use Archimedes principle to find the apparent weight of the suspended particle.]
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

where refer to number density at heights and respectively. Use this relation to derive the equation for sedimentation equilibrium of a suspension in a liquid column:
where is the density of the suspended particle, and ‘, that of surrounding medium. [ is Avogadro’s number, and the universal gas constant.] [Hint: Use Archimedes principle to find the apparent weight of the suspended particle.]

Law of atmosphere states that,

According to Archimedes principle, we have,

Apparent weight Weight of the water displaced – weight of the suspended particle

is the Density of the water
is the Density of the suspended particle
is the Mass of the suspended particle
is the Mass of the water displaced
is the Volume of a suspended particle

Boltzmann’s constant

Substituting equation (2) and equation (3) in equation (1), we get,