13.4: Variation of Atmospheric Pressure
Change in atmospheric pressure with height is particularly interesting. The decrease in atmospheric pressure with increasing altitude is due to the decreasing gravitational force per unit area as we move away from the surface of the earth.
Assuming the air temperature is constant at a given altitude and that the ideal gas law of thermodynamics describes the atmosphere to a good approximation, one can find the variation of atmospheric pressure with height.
Let p(y) be the atmospheric pressure at height y. The density ρ at y, the temperature T in the Kelvin scale (K), and the mass m of a molecule of air are related to the absolute pressure by the ideal gas law, in the form:
Using density from the ideal gas law, the rate of variation of pressure with height is integrated from sea level, and the final expression is obtained as:
Atmospheric pressure drops exponentially with height, since the y-axis points up from the ground, and y has positive values in the atmosphere above sea level. The pressure drops by a factor of 1/e when the height is 1/α, which gives us a physical interpretation for α. The constant 1/α is a length scale that characterizes how pressure varies with height and is often referred to as the pressure scale height.