where p0 is the pressure at sea level and H is a constant approximately equal to
8 km. More specifically, H = RT/Mg, where M is the average molar mass of air and T
is the temperature. This formula represents the outcome of the competition between
the potential energy of the molecules in the gravitational field of the Earth and the
stirring effects of thermal motion; it is derived on the basis of the Boltzmann
distribution
(Section F.5a). The barometric formula fits the observed pressure distribution
quite well even for regions well above the troposphere (Fig. 1.10). It implies that the
pressure of the air falls to half its sea-level value at h = H ln 2, or 6 km.
Local variations of pressure, temperature, and composition in the troposphere
are manifest as ‘weather’. A small region of air is termed a parcel. First, we note that
a parcel of warm air is less dense than the same parcel of cool air. As a parcel rises, it
expands adiabatically (that is, without transfer of heat from its surroundings), so it
cools. Cool air can absorb lower concentrations of water vapour than warm air, so the
moisture forms clouds. Cloudy skies can therefore be associated with rising air and
clear skies are often associated with descending air.
The motion of air in the upper altitudes may lead to an accumulation in some
regions and a loss of molecules from other regions. The former result in the formation
of regions of high pressure (‘highs’ or anticyclones) and the latter result in regions of
low pressure (‘lows’, depressions, or cyclones). On a weather map, such as that shown
in Fig. 1.11, the lines of constant pressure marked on it are called isobars. Elongated
regions of high and low pressure are known, respectively, as ridges and troughs.
Horizontal pressure differentials result in the flow of air that we call wind
(Fig. 1.12). Winds coming from the north in the Northern hemisphere and from the
south in the Southern hemisphere are deflected towards the west as they migrate from
a region where the Earth is rotating slowly (at the poles) to where it is rotating most
rapidly (at the equator). Winds travel nearly parallel to the isobars, with low pressure
to their left in the Northern hemisphere and to the right in the Southern hemisphere.
At the surface, where wind speeds are lower, the winds tend to travel perpendicular to
the isobars from high to low pressure. This differential motion results in a spiral outward
flow of air clockwise in the Northern hemisphere around a high and an inward
counterclockwise flow around a low.
The air lost from regions of high pressure is restored as an influx of air converges
into the region and descends. As we have seen, descending air is associated with clear
skies. It also becomes warmer by compression as it descends, so regions of high pressure
are associated with high surface temperatures. In winter, the cold surface air may
prevent the complete fall of air, and result in a temperature inversion, with a layer of
warm air over a layer of cold air. Geographical conditions may also trap cool air, as in
Los Angeles, and the photochemical pollutants we know as smog may be trapped
under the warm layer.
Real gases
Real gases do not obey the perfect gas law exactly except in the limit of p → 0.
Deviations from the law are particularly important at high pressures and low temperatures,
especially when a gas is on the point of condensing to liqu