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:

where,

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.

The earth is enveloped by layers of air, known as the atmosphere. It is pulled down toward the surface by gravity, and exerts a pressure known as atmospheric pressure.

Atmospheric pressure changes with height and can be given in terms of the weight or density of air. However, from the ideal gas law, density also depends on pressure.

Assuming that the air temperature and the acceleration due to gravity remain constant with height and by substituting the expression for density and rearranging the equation, the rate of change of pressure with height can be obtained.

Here, the constant quantities inside the parentheses are termed alpha.

The equation is then integrated from sea level and solved to obtain the relation for pressure with height—the Barometric formula.

By considering nitrogen molecules at 300 kelvin, one can obtain an approximate value of one over alpha, a characteristic length known as pressure scale height. 

The atmospheric pressure drops exponentially by a factor of one over e for every 8,800 meters above sea level.