Barometric Pressure with Altitude

Stack a column of air on itself and each layer has to carry the weight of everything above it. Write that as a hydrostatic balance, substitute the ideal gas law for the density, hold the temperature constant, and the integration gives an exponential: $P = P_0 e^{-Mgz/RT}$. The group $RT/Mg$ has units of length and is called the scale height, the climb over which pressure falls by a factor of e. At 15 °C it works out to $8.31446 \times 288.15 / (0.0289644 \times 9.80665) = 2395.8/0.28404 = 8435$ m. So a climb of 1000 m from sea level is 0.1186 scale heights, the exponential factor is $e^{-0.1186} = 0.8882$, and the pressure falls from 101.325 kPa to 89.997 kPa.

The molar mass in that expression, 0.0289644 kg/mol, is the value the US Standard Atmosphere assigns to dry air, and it is a weighted average over nitrogen, oxygen, argon and carbon dioxide rather than a property of any single substance. Humid air is lighter than dry air, because a water molecule at 18 g/mol displaces a nitrogen molecule at 28, so a saturated tropical atmosphere has a slightly larger scale height than this constant admits. The effect is under one percent and is usually swamped by the temperature assumption, which is the real weakness here.

That assumption is worth being blunt about. The atmosphere is not isothermal, and this equation knows nothing about the lapse rate. Compare it against the standard atmosphere, which integrates a 6.5 °C/km gradient properly and gives 89.875 kPa at 1000 m: the isothermal answer of 89.997 kPa is high by 0.12 kPa, or about a tenth of a percent, which nobody cares about. Go to 5000 m and the isothermal form using the sea-level temperature returns 56.0 kPa against the standard atmosphere's 54.05 kPa, an error near four percent. The fix is free and is the whole reason the variable here is called the LAYER temperature: feed it the mean temperature of the layer rather than the temperature at the bottom. Using the average of 15 °C and the 17.5 °C below zero found at 5000 m gives 54.06 kPa, which is right to a few hundredths of a percent.

In dispersion work this equation earns its place three ways. It supplies the ambient pressure that Holland's plume-rise equation needs. It sets the air density that converts a stack's volumetric flow to a mass flow, and a plant at 1500 m elevation is moving air about fifteen percent less dense than a plant at sea level, which changes both the exit velocity and the buoyancy flux. And it is the correction that puts a measured emission rate onto the standard conditions a permit is written against. The recurring errors are the obvious two: temperatures in Celsius rather than kelvin, which makes the exponent nonsense, and using the equation above the tropopause, where the real atmosphere switches to an isothermal and then a warming regime and a single-layer model has nothing left to say.