How to Calculate Altitude Pressure
You can determine altitude based on barometric pressure or vice versa using a formula derived from statistical properties of air molecules. You need only know the pressure of the air at a reference or base altitude beforehand, assuming that the temperature is the same at both altitudes.
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Instructions
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1
Measure or identify the barometric pressure, P0, at a reference altitude.
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2
Convert the altitude above the reference altitude to meters and denote it by the letter h. Denote h as negative if the altitude is lower than the reference altitude. The reference altitude itself need not be determined--merely the difference, h, between the two altitudes. You may be able to determine the difference with, for example, a GPS device.
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3
Calculate mgh, where g is the gravitational constant and m is the (average) mass of an air molecule. Use 9.8 meters per second-squared for g and 4.8156x10^-26 kilograms for m, where the caret ^ indicates exponentiation. m is equivalently 29 atomic mass units, in case you're familiar with molecular units.
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4
Divide -mgh by kT, where k is Boltzmann's constant and T is the temperature in degrees Kelvin. Note the negative sign in front of the "mgh." Use 1.3807x10^-23 meters-squared kilograms/seconds*Kelvins for Boltzmann's constant. Degrees Kelvin is degrees Celsius plus 273.
For example, for height 1,000 meters above where P0 was measured and temperature 10 degrees Celsius, you'd get -[4.8156 x 10^-26kg x 9.80m/s^2 x 1000m] / [1.3807x10^-23 m^2 kg/s^2K x 283K] = -0.1208. Note that the result is unitless, since it's a ratio of two energy measurements. Note also that exponents need to be unitless in physics.
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5
Make the result of Step 4 the exponent of e, the base of the natural logarithm. Scientific calculators all have an e^x key to calculate this. Multiply the result by P0. This is the air pressure at height h above the altitude at which P0 was measured.
Continuing the above example, P0 x exp[-0.1208] = P0 x 0.886. This means that going 1,000 meters up reduces the air pressure by just over 11 percent.
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Check your result against an online calculator. (See Resources.)
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Tips & Warnings
To understand the exponent in the above equation more intuitively, note that it's a ratio of gravitational energy to kinetic energy. If kinetic energy dominates, the pressure differential between the two altitudes is low. If gravitational energy dominates, the pressure differential is high. In terms of the molecules themselves, high kinetic energy means energetic molecules have enough kinetic energy (temperature) to bounce up to high altitudes, against gravity's pull, to keep high altitudes well populated.
The altitude problem assumes uniform temperature T--an unrealistic assumption. Temperature tends to drop with height, making the pressure calculated at height h in Step 5 an overestimate.
References
- Photo Credit old barometer image by Tomasz Nowicki from Fotolia.com
