Consider a parcel of air moving to a different altitude

Chapter 19, Problem 19.56

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Consider a parcel of air moving to a different altitude yin the Earths atmosphere (Fig. 19-33). As the parcelchanges altitude it acquires the pressure P of thesurrounding air. From Eq. 13-4 we havedPdy = ~ P gwhere p is the parcels altitude-dependent mass density.FIGURE 19-33 56.During this motion, the parcels volume will change and,because air is a poor heat conductor, we assume this expansionor contraction will take place adiabatically. (a) Starting withEq. 19-15, PV7 = constant, show that for an idealgas undergoing an adiabatic process, p l yT y = constant.Then show that the parcels pressure and temperature arerelated by,d P P d T^ - 7^ + y T ^ = and thusx/ x P d T (i - y)(-pg) + y = -(b) Use the ideal gas law with the result from part (a) toshow that the change in the parcels temperature withchange in altitude is given bydTdy1 - 7 mg7 kwhere m is the average mass of an air molecule and k isthe Boltzmann constant, (c) Given that air is a diatomicgas with an average molecular mass of 29, show thatdT/dy = 9.8C/km. This value is called the adiabaticlapse rate for dry air. (d) In California, the prevailing westerlywinds descend from one of the highest elevations (the4000-m Sierra Nevada mountains) to one of the lowestelevations (Death Valley, -100 m) in the continental UnitedStates. If a dry wind has a temperature of 5C at the top ofthe Sierra Nevada, what is the winds temperature after ithas descended to Death Valley?