Problem 5.45 (Schroeder’s Intro to Thermal Physics)

Problem 5.45

In Problem 1.40 you calculated the atmospheric temperature gradient required for unsaturated air to spontaneously undergo convection. When a rising air mass becomes saturated, however, the condensing water droplets will give up energy, thus slowing the adiabatic cooling process.

(a) Use the first law of thermodynamics to show that, as condensation forms during adiabatic expansion, the temperature of an air mass changes by

where is the number of moles of water vapor present, is the latent heat of vaporization per mole, and I’ve set for air. You may assume that the makes up only a small fraction of the air mass.

(b) Assuming that the air is always saturated during this process, the ratio is a known function of temperature and pressure. Carefully express in terms of , , and the vapor pressure Use the Clausius-Clapeyron relation to eliminate

(c) Combine the results of parts (a) and (b) to obtain a formula relating the temperature gradient, , to the pressure gradient, Eliminate the latter using the “barometric equation” from Problem 1.16. You should finally obtain

where is the mass of a mole of air. The prefactor is just the dry adiabatic lapse rate calculated in Problem 1.40, while the rest of the expression gives the correction due to heating from the condensing water vapor. The whole result is called the wet adiabatic lapse rate; it is the critical temperature gradient above which saturated air will spontaneously convect.

(d) Calculate the wet adiabatic lapse rate at atmospheric pressure (1 bar) and 25°C, then at atmospheric pressure and 0°C. Explain why the results are different, and discuss their implications. What happens at higher altitudes, where the pressure is lower?

Solution:

Problem 5.45 Solution (Download)

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