Halliday, Resnick & Walker — The Kinetic Theory of Gases: Problem 12

12 Adiabatic wind. The normal airflow over the Rocky Mountains is west to east. The air loses much of its moisture content and is chilled as it climbs the western side of the mountains. When it descends on the eastern side, the increase in pressure toward lower altitudes causes the temperature to increase. The flow, then called a chinook wind, can rapidly raise the air temperature at the base of the mountains. Assume that the air pressure \(p\) depends on altitude \(y\) according to \(p = p_0 \exp(-ay)\), where \(p_0 = 1.00 \text{ atm}\) and \(a = 1.16 \times 10^{-4} \text{ m}^{-1}\). Also assume that the ratio of the molar specific heats is \(\gamma = 4/3\). A parcel of air with an initial temperature of \(-13.0^\circ\text{C}\) descends adiabatically from \(y_1 = 4267 \text{ m}\) to \(y = 1567 \text{ m}\). What is its temperature at the end of the descent?