A tall cylindrical vessel with gaseous nitrogen is located in a uniform gravitational field in which the free-fall acceleration is equal to `g`. The temperature of the nitrogen varies along the height `h` so that its density is the same throught the volume. find the temperature gradient `dT//dh`.

Explanation

In the container shown in figure-2.15, consider a layer of width
at a height x above its bottom. If P is the pressure just below this layer then just above this layer, we can consider pressure to be P-dP. Here dP is the pressure due to the gas layer of width dx which can be given as
`dP=-dx rho g`
[If `rho` is the density. of gas] ...(2.28)
From open atmospheric form of gas law, we have
`P=(rhoRT)/(M)`
As it is given that density of gas is constant, we have
`dP=(rhoR)/(M)dT" "...(2.29)`
From (2..28) and (2.29) we have
`(rhoR)/(M)dT=-dx rhog`
or `" "(dT)/(dx)=-(Mg)/(R)`