__Question 13.13__:
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

Where ρ is the density of the suspended particle, and ρ’ that of surrounding medium. [

*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}(*ρ**- P*′) (*h*_{2}–*h*_{1})/ (*ρ**RT*)]Where ρ is the density of the suspended particle, and ρ’ 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.]

__Solution__:*n*

_{2}=

*n*

_{1}exp [-

*mg*(

*h*

_{2 }–

*h*

_{1}) /

*kBT*] … (

*i*)

Where,

*n*

_{1 }is thenumber density at height

*h*

_{1}, and

*n*

_{2}is the number density at height

*h*

_{2}

*m*g is the weight of the particle suspended in the gas column

Density of the medium =

*ρ*'

Density of the suspended particle =

*ρ*

Mass of one suspended particle =

*m*'

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*'

*g*

*=*mg - V ρ' g = mg - (m/ρ)ρ'g

= mg(1 - (ρ'/ρ) ) ....(ii)

Gas constant, R =

*k*

_{B}

*N*

k

_{B}= R / N ....(iii)

Substituting equation (

*ii*) in place of

*m*g in equation (

*i*) and then using equation (

*iii*), we get:

*n*

_{2}=

*n*

_{1}exp [-

*mg*(

*h*

_{2 }–

*h*

_{1}) /

*k*

_{B}

*T*]

= n

_{1}exp [-mg (1 - (ρ'/ρ) )(

*h*

_{2 }–

*h*

_{1})(N/RT) ]

= n

_{1}exp [-mg (ρ - ρ')(

*h*

_{2 }–

*h*

_{1})(N/RTρ) ]

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