Barometric formula

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The barometric formula, sometimes called the exponential atmosphere or isothermal atmosphere, is a formula used to model how the pressure (or density) of the air changes with altitude. It is based on the simplifying (not very realistic) assumption that the temperature does not depend on altitude. However, this formula agrees reasonably well with the actual pressure and density variations above the earth's surface up to a height of about 450,000 ft (140 km).

$\rho = \rho_0 e^{- z / h} \,$

$P = P_0 e^{- M g_0 z / (RT)}$

where h is the scale height, ρ (rho) is density, P is pressure, $P 0$ is pressure at ground level (mean sea level pressure is 1013.25 hPa), M = 0.029 kg mol^-1 (the mass of 1 mole of air), R = 8.314 J K^-1 mol^-1 is the gas constant, T is temperature, $g 0$ is the acceleration due to gravity (about 9.8 m s^-2 depending on your location, see g) and z is the vertical height above the earth's surface.

Using the same principles, the above equation can be solved for altitude as a function of pressure. This formulation is known as the hypsometric equation.

As a rule of thumb, the pressure decreases by about 1% for every 80 metres increase in altitude.

An alternative rule of thumb, density decreases by half every 20,000 feet (6000 m) below the tropopause, and every 15,000 (4500 m) feet above the tropopause to the stratopause.

[edit] Derivation

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

$\rho = \frac{MP}{RT}$

And assuming that all pressure is hydrostatic:

$dP = - \rho g\,dz\,$

Substituting the first expression into the second we get:

$\frac{dP}{P} = - \frac{M g\,dz}{RT}$

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

$P = P_0 e^{-M g z/RT}\,$

In this formulation, R is the gas constant, and the term $R T / M g$ gives the scale height (approximately equal to 7.4 km for the troposphere).

[edit] Chemical distributions

The barometric formula can also be used as an approximation for the distribution of different chemical species in the atmosphere. Below the turbopause, the relative chemical composition of the atmosphere remains constant, thus the scale height is identical for all chemical species. Above the turbopause the chemical composition begins to vary with each chemical species displaying a different scale height.