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,   