A very tall cylindrical vessel with an ideal gas of molar mass M, filled in it is placed in a region of uniform gravity ‘g’. In this container gas temperature varies in a manner such that density of gas remain constant throughout. Find the temperature gradient dT/dh, in container.  

Let say pressure at height h is p and at height (h+dh) pressure is (p+dp).

Considering the equilibrium of differential layer of a thickness dh at height h from bottom, we have

$(\mathrm{p}+\mathrm{dp}) \mathrm{A}+\mathrm{dmg}=\mathrm{pA}$

here, $\mathrm{dm}=\mathrm{\rho }\left(\mathrm{Adh}\right)$

$\Rightarrow \mathrm{dp}=-\mathrm{\rho gdh}$

Negative sign indicates that as height increases, pressure decreases.

$\begin{array}{l}\text{Using ideal gas equation,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{p}=\frac{\mathrm{\rho RT}}{\mathrm{M}}\\ \Rightarrow \mathrm{dp}=\frac{\mathrm{\rho R}}{\mathrm{M}}\mathrm{dT}\\ \hspace{0.17em}\mathrm{Thus}, \frac{\mathrm{dT}}{\mathrm{dh}}=\frac{-\mathrm{Mg}}{\mathrm{R}}\end{array}$