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(III) Consider a parcel of air moving to a different altitude y in the Earth’s atmosphere (Fig. 19–33). As the parcel changes altitude it acquires the pressure P of the surrounding air. From Eq. 13–4 we have
where ρ is the parcel’s altitude-dependent mass density.
FIGURE 19–33 Problem 56.
During this motion, the parcel’s volume will change and, because air is a poor heat conductor, we assume this expansion or contraction will take place adiabatically. (a) Starting with Eq. 19–15, PVγ = constant, show that for an ideal gas undergoing an adiabatic process, P1−γTγ = constant. Then show that the parcel’s pressure and temperature are related by
and thus
(b) Use the
where m is the average mass of an air molecule and k is the Boltzmann constant. (c) Given that air is a diatomic gas with an average molecular mass of 29, show that dT/dy = −9.8 C°/km. This value is called the adiabatic lapse rate for dry air. (d) In California, the prevailing westerly winds descend from one of the highest elevations (the 4000-m Sierra Nevada mountains) to one of the lowest elevations (Death Valley, −100 m) in the continental United States. If a dry wind has a temperature of −5°C at the top of the Sierra Nevada, what is the wind’s temperature after it has descended to Death Valley?
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