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Chapter 5: 5.13 (page 159)
Use a Maxwell relation from the previous problem and the third law of thermodynamics to prove that the thermal expansion coefficient (defined in Problem 1.7) must be zero at T=0.
Coefficient of Expansion becomes Zero at T=0.
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Sketch qualitatively accurate graphs of G vs. P for the three phases of H20 (ice, water, and steam) at 0掳C. Put all three graphs on the same set of axes, and label the point corresponding to atmospheric pressure. How would |the graphs differ at slightly higher temperatures?
Derive the thermodynamic identity for (equation 5.23), and from it the three partial derivative relations 5.24.
A formula analogous to that for relates the isothermal and isentropic compressibilities of a material:
(Here is the reciprocal of the adiabatic bulk modulus considered in Problem 1.39.) Derive this formula. Also check that it is true for an ideal gas.
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 nw is the number of moles of water vapor present, L is the latent heat of vaporization per mole, and I've assumed for air.
(b) Assuming that the air is always saturated during this process, the ratio nw/n is a known function of temperature and pressure. Carefully express dnw/dz 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 Figure 5.18. Cumulus clouds form when rising air expands adiabatically and cools to the dew point (Problem 5.44); the onset of condensation slows the cooling, increasing the tendency of the air to rise further (Problem 5.45). These clouds began to form in late morning, in a sky that was clear only an hour before the photo was taken. By mid-afternoon they had developed into thunderstorms. the latter using the "barometric equation" from Problem 1.16. You should finally obtain
where " width="9">
(d) Calculate the wet adiabatic lapse rate at atmospheric pressure (I bar) and , then at atmospheric pressure and . Explain why the results are different, and discuss their implications. What happens at higher altitudes, where the pressure is lower?
Sketch qualitatively accurate graphs of vs.for the three phases of(ice, water, and steam) at atmospheric pressure. Put all three graphs on the same set of axes, and label the temperaturesand . How would the graphs differ at a pressure ofbar?
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