The Total solution for NCERT class 6-12
A gas inequilibrium has uniform density and pressure throughout its volume. This isstrictly true only if there are no external influences. A gas column undergravity, for example, does not have a uniform density (and pressure). Asyou might expect, its density decreases with height. The precisedependence is given by the so-called law of atmospheresn2= n1 exp[ -mg (h2 – h1)/ kBT]where n2, n1 referto number density at heights h2 andh1respectively. Use this relationto derive the equation for sedimentation equilibrium of a suspension in aliquid column:n2 = n1 exp[ -mg NA (ρ – ρ′) (h2–h1)/ (ρ RT)]where ρ is the density of thesuspended particle, and ρ′ ,that of surrounding medium. [NA is Avogadro’snumber, and R the universal gas constant.] [Hint : Use Archimedes principleto find the apparent weight of the suspended particle.]
Law of atmosphere
n2= n1 exp [ -mg (h2 – h1)/ kBT] ———(1)
The suspended particle experiences an apparent weight because ofthe liquid displaced
According to Archimedes principle
Apparent weight = Weight of the water displaced – weight of thesuspended particle
= mg – m’g
= mg – Vρ’g = mg – (m/ρ) ρ’g
= mg (1 – (ρ’/ρ)) ——-(2)
ρ′= Density of the waterρ = Density of the suspended particlem′ = Mass of the suspended particlem = Mass of the water displacedV = Volume of a suspended particle
Boltzmann’s constant (K) = R/NA ——(3)
Substituting equation (2) and equation (3) in equation (1)
n2= n1 exp [ -mg (h2 – h1)/ kBT]
n2 =n1 exp [ -mg (1– ρ′/ ρ ) (h2–h1)NA / (RT)]
n2 =n1 exp [ -mg NA (ρ – ρ′ ) (h2–h1)/ (ρ RT)]