### Q.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

*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.]

**Answer:-**

According to the law of atmospheres, we have:

*n*_{2} = *n*_{1} exp [-*mg *(*h*_{2 }– *h*_{1}) / *kBT*] … **(i)**

where,

*n*_{1 }is the number 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 [-

*m*

*g*(1 – (ρ’/ρ) )(

*h*

_{2 }–

*h*

_{1})(

*N*/

*RT*) ]

=

*n*

_{1}exp [-

*m*

*g*(ρ – ρ’)(

*h*

_{2 }–

*h*

_{1})(

*N*/

*R*T

*ρ*) ]