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Mechanical Properties Of Fluids

Question
CBSEENPH11020190
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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
n2 = n1 exp [-mg (h2 – h1)/ kBT]
Where n2, n1 refer to number density at heights h2 and h1respectively. Use this relation to derive the equation for sedimentation equilibrium of a suspension in a liquid column:
n2 = n1 exp [-mg NA(ρ - P′) (h2 –h1)/ (ρRT)]
Where ρ is the density of the suspended particle, and ρ’ that of surrounding medium. [NA is Avogadro’s number, and R the universal gas constant.] [Hint: Use Archimedes principle to find the apparent weight of the suspended particle.]

Solution
According to the law of atmospheres, we have
n2 = n1 exp [-mg (h2 – h1) / kBT]                    … (i) 

where,
n1 is the number density at height h1, and 
n2 is the number density at height h2 

mg 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 = kBN 

kB = R / N                                                  ...(iii) 

Substituting equation (ii) in place of mg in equation (i) and then using equation (iii), we get: 
n2 = n1 exp [-mg (h2 – h1) / kBT] 
    = n1 exp [-mg (1 - (ρ'/ρ) )(h2 – h1)(N/RT) ] 
    = n1 exp [-mg (ρ - ρ')(h2 – h1)(N/RTρ) ]

Some More Questions From Mechanical Properties of Fluids Chapter

What is wave?

Is it logical to say that wave motion is a process of propagation of mass?

What is common property of all types of waves?

What are different types of wave motions?

Define transverse wave motion.

Define longitudinal wave motion.

Which of the two waves - elastic waves or electromagnetic waves does need any material media for its propagation?

What is sound?

How sound is produced?

Can sound travel in vacuum?

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