Question

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
$n_2=n_{1}exp[-mg(h_2-h_1)/k_{B}T]$
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:
$n_2=n_{1}exp[-mg{N}_A(\rho -{\rho }^{\prime }\left)\right(h_2-h_1\left)/\right(\rho RT\left)\right]$
where $\rho$ is the density of the suspended particle and ${\rho }^{\prime }$ that of surrounding medium. [N${}_A$ is Avogadro's number, and R the universal gas constant.][Hint : Use Archimedes principle to find the apparent weight of the suspended particle.]

Answer 1

According to the law of atmospheres, we have:

$n_2=n_{1}exp[-mg(h_2-h_1\left)/k_{B}T\right]$ ...(i)
where,
$n_1$ is the number density at height $h_1$, and $n_2$ is the number density at height $h_2$
mg is the weight of the particle suspended in the gas column
Density of the medium $={\rho }^{\prime }$
Density of the suspended particle $=\rho$

Mass of one suspended particle $=m^{\prime }$
Mass of the medium displaced $=m$
Volume of a suspended particle $=V$
According to Archimedes’ principle for a particle suspended in a liquid column, the effective weight of the suspended particle is given as:
Weight of the medium displaced – Weight of the suspended particle
$=mg-m^{\prime }g$

$=mg-V{\rho }^{\prime }g=mg-\left(m/\rho \right){\rho }^{\prime }g$
$=mg(1-({\rho }^{\prime }/\rho \left)\right)$ ... (ii)

Gas constant, $R=k_{B}N$
$k_B=R/N$ ...(iii)
Substituting equation (ii) in place of mg in equation (i) and then using equation (iii), we get:

$n_2=n_{1}exp[-mg(h_2-h_1\left)/k_{B}T\right]$

$=n_{1}exp[-mg(1-{\rho }^{\prime }/\rho \left)\right)(h_2-h_1)\left(N/RT\right)]$
$=n_{1}exp[-mg(\rho -{\rho }^{\prime }\left)\right(h_2-h_1\left)\right(N/RT\rho \left)\right]$