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Chapter 1: Q.1.1. (page 5)
The Fahrenheit temperature scale is defined so that ice melts at 320 F and water boils at 2120 F.
(a) Derive the formula for converting from Fahrenheit to Celsius and back
(b) What is absolute zero on the Fahrenheit scale?
(a)
(b) -459.67
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Estimate how long it should take to bring a cup of water to boiling temperature in a typical 600 -watt microwave oven, assuming that all the energy ends up in the water. (Assume any reasonable initial temperature for the water.) Explain why no heat is involved in this process.
In Problem 1.16 you calculated the pressure of earth's atmosphere as a function of altitude, assuming constant temperature. Ordinarily, however, the temperature of the bottom most 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 |dT/dz| 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 dT/dz 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 1.16b to find a formula for dT/dz in this case. The result should be a constant, independent of temperature and pressure, which evaluates to approximately -10oC/ km. This fundamental meteorological quantity is known as the dry adiabatic lapse rate.
Geologists measure conductive heat flow out of the earth by drilling holes (a few hundred meters deep) and measuring the temperature as a function of depth. Suppose that in a certain location the temperature increases byper kilometer of depth and the thermal conductivity of the rock is . What is the rate of heat conduction per square meter in this location? Assuming that this value is typical of other locations over all of the earth's surface, at approximately what rate is the earth losing heat via conduction? (The radius of the earth is .)
Put a few spoonfuls of water into a bottle with a tight lid. Make sure everything is at room temperature, measuring the temperature of the water with a thermometer to make sure. Now close the bottle and shake it as hard as you can for several minutes. When you're exhausted and ready to drop, shake it for several minutes more. Then measure the temperature again. Make a rough calculation of the expected temperature change, and compare.
A battery is connected in series to a resistor, which is immersed in water (to prepare a nice hot cup of tea). Would you classify the flow of energy from the battery to the resistor as "heat" or "work"? What about the flow of energy from the resistor to the water?
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