Atmospheric pressure - Altitude variation

Atmospheric pressure or barometric pressure is the pressure within the atmosphere of Earth. For most purposes it can be approximated by the hydrostatic pressure exerted by the weight of the air above a surface.

We can perform an approximate calculation of the atmospheric pressure at sea level assuming that, on average, the air above a 1 cm2 surface at sea level has roughly a mass m = 1.03 kg and thus its weight is 10.1 N (see figure below)

Therefore, using the definition of pressure:

More precise measurements yield a value of the atmospheric pressure of:

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Before the use of the SI units became widely accepted, the standard atmosphere (atm) was used as a unit of atmospheric pressure. It is approximately equal to the atmospheric pressure at sea level. Therefore:

Atmospheric pressure varies with the altitude; since the air mass above a certain surface is smaller as we climb up a mountain, air pressure on mountains is usually lower than air pressure at sea level. We can obtain an expression for the atmospheric pressure as a function of elevation. If we assume that the atmospheric circulation is not relevant in this situation, we can start from the differential expression of the hydrostatic pressure and integrate it.

In order to do so, let’s assume that the atmosphere is a gas which obeys the equation of state of an ideal gas:

We will take as reference point the atmospheric pressure at sea level p0. The density of the atmosphere at this altitude is ρ0.

If we apply the equation of state of an ideal gas to both the reference point and to a point at height h above sea level we have:

Isolating RT from both equations and equalizing:

Now we can substitute into the differential expression of the hydrostatic pressure” target=”_blank” rel=”noopener”> and perform the integral:

We can isolate the pressure as a function of the altitude h:

In the figure below a plot of this model is shown. We have assumed that the atmosphere is at t = 150C, and that its relative humidity is 0%. Atmospheric pressure at sea level is p0 = 101 325 Pa. Under those conditions the density of air is ρ0 = 1.225 kg/m3

The atmospheric pressure can be measured with a mercury barometer. It is a glass tube closed at the top and placed in an open container filled with mercury.

At equilibrium, the height of the mercury column inside the tube equals the atmospheric pressure. At sea level, this height is h = 760 mm. That’s the reason why the millimeter of mercury was formerly used as a unit of pressure.

If the barometer is placed at a higher altitude, the mercury drops to a lower level in the column.

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