Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information....

Homework Problems

1 (a) Show that under conditions of standard pressure and temperature, the volume of a sample of an ideal gas depends only on the number of molecules in it.

(b) One mole is defined as 6.01023 atoms. Find the volume of one mole of an ideal gas, in units of liters, at standard temperature and pressure (0 C and 101 kPa).

2 A gas in a cylinder expands its volume by an amount ΔV , pushing out a piston. Show that the work done by the gas on the piston is given by ΔW = PΔV .
3 (a) A helium atom contains 2 protons, 2 electrons, and 2 neutrons. Find the mass of a helium atom.

(b) Find the number of atoms in 1 kg of helium.

(c) Helium gas is monoatomic. Find the amount of heat needed to raise the temperature of 1 kg of helium by 1 degree C. (This is known as helium's heat capacity at constant volume.)

4 Refrigerators, air conditioners, and heat pumps are heat engines that work in reverse. You put in mechanical work, and it the effect is to take heat out of a cooler reservoir and deposit heat in a warmer one: QL +W = QH. As with the heat engines discussed previously, the efficiency is defined as the energy transfer you want (QL for a refrigerator or air conditioner, QH for a heat pump) divided by the energy transfer you pay for (W).

Efficiencies are supposed to be unitless, but the efficiency of an air conditioner is normally given in terms of an EER rating (or a more complex version called an SEER). The EER is defined as QL/W, but expressed in the barbaric units of of Btu/watt-hour. A typical EER rating for a residential air conditioner is about 10 Btu/watt-hour, corresponding to an efficiency of about 3. The standard temperatures used for testing an air conditioner's efficiency are 80 F (27 C) inside and 95 F (35 C) outside.

(a) What would be the EER rating of a reversed Carnot engine used as an air conditioner?

(b) If you ran a 3-kW residential air conditioner, with an efficiency of 3, for one hour, what would be the effect on the total entropy of the universe? Is your answer consistent with the second law of thermodynamics?

5 (a) Estimate the pressure at the center of the Earth, assuming it is of constant density throughout. Use the technique of example 5 on page 140. Note that g is not constant with respect to depth - it equals Gmr/b3 for r, the distance from the center, less than b, the earth's radius 1). State your result in terms of G, m, and b.

(b) Show that your answer from part a has the right units for pressure.

(c) Evaluate the result numerically.

(d) Given that the earth's atmosphere is on the order of one thousandth the thickness of the earth's radius, and that the density of the earth is several thousand times greater than the density of the lower atmosphere, check that your result is of a reasonable order of magnitude.

1)Derivation: The shell theorem tells us that the gravitational field at r is the same as if all the mass existing at greater depths was concentrated at the earth's center. Since volume scales like the third power of distance, this constitutes a fraction (r/b)3 of the earth's mass, so the field is (Gm/r2)(r/b)3 = Gmr/b3.

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6 (a) Determine the ratio between the escape velocities from the surfaces of the earth and the moon.

(b) The temperature during the lunar daytime gets up to about 130 C. In the extremely thin (almost nonexistent) lunar atmosphere, estimate how the typical velocity of a molecule would compare with that of the same type of molecule in the earth's atmosphere. Assume that the earth's atmosphere has a temperature of 0 C.

(c) Suppose you were to go to the moon and release some fluorocarbon gas, with molecular formula CnF2n+2. Estimate what is the smallest fluorocarbon molecule (lowest n) whose typical velocity would be lower than that of an N2 molecule on earth in proportion to the moon's lower escape velocity. The moon would be able to retain an atmosphere made of these molecules.

7 Most of the atoms in the universe are in the form of gas that is not part of any star or galaxy: the intergalactic medium (IGM). The IGM consists of about 10-5 atoms per cubic centimeter, with a typical temperature of about 103 K. These are, in some sense, the density and temperature of the universe (not counting light, or the exotic particles known as dark matter). Calculate the pressure of the universe (or, speaking more carefully, the typical pressure due to the IGM).
8 A sample of gas is enclosed in a sealed chamber. The gas consists of molecules, which are then split in half through some process such as exposure to ultraviolet light, or passing an electric spark through the gas. The gas returns to thermal equilibrium with the surrounding room. How does its pressure now compare with its pressure before the molecules were split?

The figure shows a demonstration performed by Otto von Guericke for Emperor Ferdinand III, in which two teams of horses failed to pull apart a pair of hemispheres from which the air had been evacuated.

(a) What object makes the force that holds the hemispheres together?

(b) The hemispheres are in a museum in Berlin, and have a diameter of 65 cm. What is the amount of force holding them together? (Hint: The answer would be the same if they were cylinders or pie plates rather then hemispheres.)

Last Update: 2010-11-11