A very tall vertical cylinder is filled with a gas of molar mass M under isothermal conditions temperature T. the density and pressure of the gas at the base of the container is `rho_(0)` and `P_(0)`, respectively
Select the incorrect statement if gravity is assumed to be constant throughout the container

Options

A. Both pressure and density decreases exponetially with height
B. The variation of pressure is `P=P_(0)e^(-(Mgh)/(RT))`
C. The variation of density `rho=rho_(0)e^((Mgh)/(RT))`
D. The molecular density decreases as one moves upwards.

Answer

C

Explanation

Considering the equilibrium of differential layer of a thickness dh and height h, we have
`(P+dP)A+dmg=PA`
or `A dP=-g dm=0grhoA dh`
From ideal gas equation, we have
`P=(rhoRT)/(M)` or `rho=(PM)/(RT)`
From eqns. (1) and (2), we get
`dp=-(PM)/(RT)g dh`
or `In (P)/(P_(0))=-(Mg)/(RT)h`
or `P=P_(0)e^(-Mgh//RT)`
In a similar manner if we eliminate P with the of ideal gas equation, we get
`dP=(drhoRT)/(M)`
The eqn. becomes
`drho(RT)/(M)=-rhog dh`
or `int_(P_(0))^(rho) (drho)/(rho)=-(Mg)/(RT)int_(0)^(h)dh`
or `rho=rho_(0)e^(-Mgh//RT)`