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Chapter 1: Q.1.24 (page 17)
Calculate the total thermal energy in a gram of lead at room temperature, assuming that none of the degrees of freedom are "frozen out" (this happens to be a good assumption in this case).
The total thermal energy.
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Calculate the heat capacity of liquid water per molecule, in terms of K . Suppose (incorrectly) that all the thermal energy of water is stored in quadratic degrees of freedom. How many degrees of freedom would each molecule have to have?
According to a standard reference table, the R value of a 3.5 inch-thick vertical air space (within a wall) is 1.0 R in English units), while the R value of a 3.5-inch thickness of fiberglass batting is 10.9. Calculate the R value of a 3.5-inch thickness of still air, then discuss whether these two numbers are reasonable. (Hint: These reference values include the effects of convection.)
Consider a uniform rod of material whose temperature varies only along its length, in the direction. By considering the heat flowing from both directions into a small segment of length
derive the heat equation,
where , is the specific heat of the material, and is its density. (Assume that the only motion of energy is heat conduction within the rod; no energy enters or leaves along the sides.) Assuming that is independent of temperature, show that a solution of the heat equation is
where is a constant background temperature and is any constant. Sketch (or use a computer to plot) this solution as a function of , for several values of . Interpret this solution physically, and discuss in some detail how energy spreads through the rod as time passes.
In Problem 1.16 you calculated the pressure of the earth’s atmosphere as a function of altitude, assuming constant temperature. Ordinarily, however, the temperature of the bottommost 10-15 km of the atmosphere (called the troposphere) decreases with increasing altitude, due to heating from the ground (which is warmed by sunlight). If the temperature gradient exceeds a certain critical value, convection will occur: Warm, low-density air will rise, while cool, high-density air sinks. The decrease of pressure with altitude causes a rising air mass to expand adiabatically and thus to cool. The condition for convection to occur is that the rising air mass must remain warmer than the surrounding air despite this adiabatic cooling.
a. Show that when an ideal gas expands adiabatically, the temperature and pressure are related by the differential equation
b. Assume that is just at the critical value for convection to begin so that the vertical forces on a convecting air mass are always approximately in balance. Use the result of Problem 1.16(b) to find a formula for in this case. The result should be a constant, independent of temperature and pressure, which evaluates to approximately . This fundamental meteorological quantity is known as the dry adiabatic lapse rate.
In analogy with the thermal conductivity, derive an approximate formula for the viscosity of an ideal gas in terms of its density, mean free path, and average thermal speed. Show explicitly that the viscosity is independent of pressure and proportional to the square root of the temperature. Evaluate your formula numerically for air at room temperature and compare to the experimental value quoted in the text.
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