Chapter 19, Problem 56P

(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

$\frac{dP}{dy}=-\rho g$

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, PV^γ = constant, show that for an ideal gas undergoing an adiabatic process, P^1−γT^γ = constant. Then show that the parcel’s pressure and temperature are related by

$(1-\gamma )\frac{dP}{dy}+\gamma \frac{P}{T}\frac{dT}{dy}=0$

and thus

$(1-\gamma )(-\rho g)+\gamma \frac{P}{T}\frac{dT}{dy}=0.$

(b) Use the ideal gas law with the result from part (a) to show that the change in the parcel’s temperature with change in altitude is given by

$\frac{dT}{dy}=\frac{1-\gamma }{\gamma }\frac{mg}{k}$

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?