The Barometric Formula

If the atmospheric pressure at ground level is
=mmHg =inHg
=kPa

and the temperature is uniform at
K = C

then the pressure at an altitude of
h = m = ft is

=mmHg =inHg
=kPa


This calculation uses m = amu
M = kg/mol.
Derivation of formulaPressure at selected heightsConstituents of air
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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 depends on the pressure as follows.
.
The solution for the change from the ground
( ) to height h ( ) gives


CalculationDetails of solution
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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

CalculationForm of solution
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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

CalculationFitting boundary conditions
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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:

Height
Measured pressure
Predicted pressure
30 km (19 mi) 9.5 torr 25 torr
60 km (37 mi)0.21 torr .8 torr
90 km (56 mi) 0.0019 torr .03 torr

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.

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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


Since a mole of any ideal gas occupies the same volume, it follows that the volume percentage is also the percentage by number. The average molecular mass can then be found by weighting the masses by their volume percentages above.

Atmospheric pressureBarometric formulaRespiration
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