The Barometric Formula

If the atmospheric pressure at ground level is P0 = mmHg = inHg = kPa and the temperature is uniform at K = °C then the pressure at an altitude of h = m = ft is Ph = mmHg = inHg = kPa This calculation uses m = amu M = kg/mol.

Note that the model calculation assumes a uniform temperature, and is therefore not a realistic model of the atmosphere. The temperature tends to decrease with height, so the model calculation will overestimate the pressure at a given height.

The Barometric Formula

Starting at some point in midair, the change in pressure associated with a small change in height can be found in terms of the weight of the air. The change in pressure depends on density, but ρ depends on the pressure as follows: The solution for the change from the ground ( P0) to height h ( Ph ) gives Calculation Details of solution

The development of the barometric formula makes use of a number of concepts from kinetic theory, such as the ideal gas law and the associated molecular constants. In the exponential, the two terms have the units of energy. The numerator mgh is gravitational potential energy and the term kT is thermal energy.

The Barometric Formula

Having shown that the rate of change of pressure with height has the form it is necessary to take the limit as the change in height approaches zero, putting it in the form of a derivative. This type of equation can be solved for P by making a substitution of the type fitting the boundary conditions gives

The Barometric Formula

The equation for the variation of barometric pressure with height has the form which has the formal solution Substituting the solution gives Since this equation must be valid for all values of h, forcing the solution to fit the physical boundary conditions yields:

Atmospheric Pressure, Selected Heights

Compared to the standard Atmospheric pressure at sea level, 760 mmHg = 760 torr, the pressure at some heights above the earth are:

These pressures are considerably below those predicted by the barometric formula, which can be used to calculate variations in barometric pressure with height near the earth. (The predicted pressures above used temperature 300K and pressure 760 mmHg.) Though the pressure decreases rapidly, even at 200 km (124 mi) there is enough residual atmospheric pressure to gradually slow a satellite, limiting its lifetime.

Constituents of the Air

The constituents of dry air can be expressed as volume percentages, which will translate to the partial pressures out of the total atmospheric pressure