Barometric Formula
(©2002, François G. Amar, All rights reserved)

The barometric formula:

P(h) = P(0)e^-Mgh/RT (pure gas or total pressure of mixture)

is one of the few equations in general chemistry that depends on the effect of gravity. This equation is derived assuming that the temperature of the atmosphere is the same at every altitude (clearly not always true). For this reason it is a model of an isothermal atmosphere.

The formula given above allows us to calculate the pressure, P(h), of a pure gas as a function of altitude, h, if we know:

• P(0) the pressure of the gas at sea-level (h=0)
• M the molar mass of the gas expressed in kg/mole
• g the acceleration due to gravity at the surface of the planet (on earth, g=9.81 m/s²)
• h the height above sea-level in meters
• T the temperature in K
• the gas constant (if the units given above are used, R=8.3145 J/molK is appropriate)

This formula also works for a gas mixture if M is interpreted as an average molar mass for the mixture,

M = SiMixi     (here the mole fraction, xi = ni/ntot = Pi/Ptot)
      

For example the average molar mass of dry air is (neglecting trace gases):

32.00 g/mol x 0.21 + 28.00 g/mol x 0.79 = 28.8 g/mole or 0.0288 kg/mole

If we add a subscript i to the barometric formula, it can be used to determine the partial pressure of a component in a mixture of gases:

P_i(h) = P_i (0)e^-Mi^gh/RT (partial pressure of component i in mixture)

 

In this formula and in the calculation of the average molar mass, M_i is the molar mass (in kg/mole)

of the i ^th component of the mixture.

Barometric pressure problem:

What is the partial pressure of oxygen at the top of Pike's Peak (14,500 feet)?

What is the total pressure (assuming that dry air has an average MW of 28.8 g/mol)?

Take the temperature to be 10°C. Answer here!

 

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