$A$ 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:
$n_{2}=n_{1} \exp \left[-m g\left(h_{2}-h_{1}\right) / k_{B} T\right]$
where $n_{2}, n_{1}$ refer to number density at heights $h_{2}$ and $h_{1}$ respectively. Use this relation to derive the equation for sedimentation equilibrium of a suspension in a liquid column:
$n_{2}=n_{1} \exp \left[-m g N_{A}\left(\rho-\rho^{\prime}\right)\left(h_{2}-h_{1}\right) /(\rho R T)\right]$
where $\rho$ is the density of the suspended particle,and $\rho^{\prime}$ that of the surrounding medium. [$N_{A}$ is Avogadro's number,and $R$ the universal gas constant.]

Vedclass pdf generator app on play store
Vedclass iOS app on app store
(N/A) According to the law of atmospheres,the number density is given by:
$n_{2}=n_{1} \exp \left[-m g\left(h_{2}-h_{1}\right) / k_{B} T\right]$
For a particle of mass $m$ and density $\rho$ suspended in a medium of density $\rho^{\prime}$,the effective weight $W_{eff}$ is the actual weight minus the buoyant force (Archimedes' principle).
Let $V$ be the volume of the particle. Then $m = V\rho$.
The buoyant force is equal to the weight of the displaced medium: $F_{B} = V\rho^{\prime}g = (m/\rho)\rho^{\prime}g$.
The effective weight is:
$W_{eff} = mg - F_{B} = mg - (m/\rho)\rho^{\prime}g = mg(1 - \rho^{\prime}/\rho) = mg(\rho - \rho^{\prime})/\rho$.
Substituting this effective weight into the law of atmospheres and using $k_{B} = R/N_{A}$:
$n_{2} = n_{1} \exp \left[ -\frac{mg(\rho - \rho^{\prime})}{\rho} \frac{(h_{2} - h_{1})}{k_{B}T} \right]$
Substituting $k_{B} = R/N_{A}$:
$n_{2} = n_{1} \exp \left[ -\frac{mg(\rho - \rho^{\prime})}{\rho} \frac{(h_{2} - h_{1}) N_{A}}{RT} \right]$
Rearranging the terms,we get:
$n_{2} = n_{1} \exp \left[ -\frac{mg N_{A} (\rho - \rho^{\prime}) (h_{2} - h_{1})}{\rho RT} \right]$

Explore More

Similar Questions

$A$ body floats in a liquid contained in a beaker. If the whole system as shown in the figure falls freely under gravity, then the upthrust on the body due to the liquid is

Equal masses of three liquids are kept in three identical cylindrical vessels $A$, $B$ and $C$. The densities are $\rho_A$, $\rho_B$ and $\rho_C$ with $\rho_A$ < $\rho_B$ < $\rho_C$. The force on the base will be maximum in vessel:

$A$ cubical block of wood of edge $10$ $cm$ and mass $0.92$ $kg$ floats in a tank of water with a layer of oil of relative density $0.6$ to a depth of $4$ $cm$ above the water. When the block attains equilibrium with four of its side edges vertical,which of the following is true?

Difficult
View Solution

Two solids $A$ and $B$ float in water. It is observed that $A$ floats with $\frac{1}{2}$ of its body immersed in water and $B$ floats with $\frac{1}{4}$ of its volume above the water level. The ratio of the density of $A$ to that of $B$ is

$A$ $0.5\ kg$ mass of lead is submerged in a container filled to the brim with water and a block of wood floats on top. The lead mass is slowly lifted from the container by a thin wire and as it emerges into air the level of the water in the container drops a bit. The lead mass is now placed on the block of wood. As the lead is placed on the wood,what happens to the water level in the container?

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial
For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free
For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo