Why does atmospheric pressure decrease more rapidly than linearly with altitude?

Fluid statics, often known as hydrostatics, is a branch of fluid mechanics that investigates the state of balance of floating and submerged body, as well as the pressure in a fluid, or imposed by a fluid, on an immersed body.

Atmospheric pressure is the pressure exerted by the air molecules. This can be mathematically expressed as;

$\mathit{P}\mathbf{=}\mathit{h}\mathit{\rho }\mathit{g}$

Where, h is the height of the air column, ρ is the density of air and g is the acceleration due to gravity.

So, atmospheric pressure is inversely related to height, density and acceleration due to gravity.

At higher altitudes, the density of air is very low. The gravity of the earth attracts air molecules towards its surface. So, as we ascend to higher altitudes, the number of air molecules will be less. Since density is the ratio of mass to volume, it decreases with the deficiency of air molecules. As the pressure depends on the density, it also decreases with an increase in altitude.

Therefore, atmospheric pressure rapidly decreases with altitude due to a decrease in density.