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

Consider a thin layer at a height `h` and thickness `dh`. Let `p` and `dp + p` be the pressure on the two sides of the layer. The mass of the layer us `Sdh rho`. Equating vertical downward force to te upward force acting on the layer.
`S dh rho g + (p + dp) S = p S`
So, `(dp)/(dh) = - rho g` ...(1)
But, `p = (rho)/(M) RT`. we have `dp = (rho R)/(M) dT`,
or, `- (rho R)/(M) dT = rho g dh`
So, `(dT)/(dh) = - (g M)/( R) = - 34 K//km`
That means, temperature of air drops by `34^@ C` at a height of `1 km` above bottom.
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