Barometric Formula - Derivation

Derivation

The barometric formula can be derived fairly easily using the ideal gas law:

When density is known:

And assuming that all pressure is hydrostatic:

Dividing the by the expression we get:

Integrating this expression from the surface to the altitude z we get:

Assuming constant temperature, molar mass, and gravitational acceleration, we get the barometric formula:

In this formulation, is the gas constant, and the term gives the scale height (approximately equal to 8.4 km for the troposphere).

(For exact results, it should be remembered that atmospheres containing water do not behave as an ideal gas. See real gas or perfect gas or gas for further understanding)