question_answer 13)

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': where, refer to number density at heights h-^ and h^ respectively. Use this relation to derive the       equation for sedimentation equilibrium of a suspension in a liquid column :               where, p is the density of the suspended particle, and p' that of surrounding medium. [is Avogadro's number, and R is the universal gas constant].  

Answer:

According to the law of atmospheres,                    ?..(1) where refer to number density of particles at heights and  respectively. If we consider the sedimentation equilibrium of suspended particles in a liquid, then in place of mg, we will have to take effective weight of the suspended particles. Let V = average volume of a suspended particle, = density of suspended particle,  = density of liquid, m = mass of one suspended particle, m' = mass of equal volume of liquid displaced. According to Archimede's principle, effective weight of one suspended particle   = actual weight - weight of liquid displaced = mg - m' g Also, Boltzmann constant, where, R is gas constant and is Avogadro's number. Putting, mg  in place of mg and value of  in (i), we get          Which is the required relation.