Solution: Modern vacuum pumps make it easy to attain

Section 18.1 states that ordinarily, pressure, volume, and temperature cannot change individually without one affecting the others. Yet when a liquid evaporates, its volume changes, even though its pressure and temperature are constant. Is this inconsistent? Why or why not?

In the ideal-gas equation, could an equivalent Celsius temperature be used instead of the Kelvin one if an appropriate numerical value of the constant R is used? Why or why not?

When a car is driven some distance, the air pressure in the tires increases. Why? Should you let out some air to reduce the pressure? Why or why not?

The coolant in an automobile radiator is kept at a pressure higher than atmospheric pressure. Why is this desirable? The radiator cap will release coolant when the gauge pressure of the coolant reaches a certain value, typically 15 lb>in.2 or so. Why not just seal the system completely?

Unwrapped food placed in a freezer experiences dehydration, known as freezer burn. Why?

A group of students drove from their university (near sea level) up into the mountains for a skiing weekend. Upon arriving at the slopes, they discovered that the bags of potato chips they had brought for snacks had all burst open. What caused this to happen?

The derivation of the ideal-gas equation included the assumption that the number of molecules is very large, so that we could compute the average force due to many collisions. However, the ideal-gas equation holds accurately only at low pressures, where the molecules are few and far between. Is this inconsistent? Why or why not?

A rigid, perfectly insulated container has a membrane dividing its volume in half. One side contains a gas at an absolute temperature \(T_0\) and pressure \(p_0\), while the other half is completely empty. Suddenly a small hole develops in the membrane, allowing the gas to leak out into the other half until it eventually occupies twice its original volume. In terms of \(T_0\) and \(p_0\), what will be the new temperature and pressure of the gas when it is distributed equally in both halves of the container? Explain your reasoning.

(a) Which has more atoms: a kilogram of hydrogen or a kilogram of lead? Which has more mass? (b) Which has more atoms: a mole of hydrogen or a mole of lead? Which has more mass? Explain your reasoning

Use the concepts of the kinetic-molecular model to explain: (a) why the pressure of a gas in a rigid container increases as heat is added to the gas and (b) why the pressure of a gas increases as we compress it, even if we do not change its temperature

The proportions of various gases in the earths atmosphere change somewhat with altitude. Would you expect the proportion of oxygen at high altitude to be greater or less than at sea level compared to the proportion of nitrogen? Why?

Comment on the following statement: When two gases are mixed, if they are to be in thermal equilibrium, they must have the same average molecular speed. Is the statement correct? Why or why not?

The kinetic-molecular model contains a hidden assumption about the temperature of the container walls. What is this assumption? What would happen if this assumption were not valid?

The temperature of an ideal gas is directly proportional to the average kinetic energy of its molecules. If a container of ideal gas is moving past you at 2000 m>s, is the temperature of the gas higher than if the container was at rest? Explain your reasoning

If the pressure of an ideal monatomic gas is increased while the number of moles is kept constant, what happens to the average translational kinetic energy of one atom of the gas? Is it possible to change both the volume and the pressure of an ideal gas and keep the average translational kinetic energy of the atoms constant? Explain.

In deriving the ideal-gas equation from the kineticmolecular model, we ignored potential energy due to the earths gravity. Is this omission justified? Why or why not?

Imagine a special air filter placed in a window of a house. The tiny holes in the filter allow only air molecules moving faster than a certain speed to exit the house, and allow only air molecules moving slower than that speed to enter the house from outside. What effect would this filter have on the temperature inside the house? (It turns out that the second law of thermodynamicswhich we will discuss in Chapter 20tells us that such a wonderful air filter would be impossible to make.)

A gas storage tank has a small leak. The pressure in the tank drops more quickly if the gas is hydrogen or helium than if it is oxygen. Why?

Consider two specimens of ideal gas at the same temperature. Specimen A has the same total mass as specimen B, but the molecules in specimen A have greater molar mass than they do in specimen B. In which specimen is the total kinetic energy of the gas greater? Does your answer depend on the molecular structure of the gases? Why or why not?

The temperature of an ideal monatomic gas is increased from 25C to 50C. Does the average translational kinetic energy of each gas atom double? Explain. If your answer is no, what would the final temperature be if the average translational kinetic energy was doubled?

If the root-mean-square speed of the atoms of an ideal gas is to be doubled, by what factor must the Kelvin temperature of the gas be increased? Explain.

(a) If you apply the same amount of heat to 1.00 mol of an ideal monatomic gas and 1.00 mol of an ideal diatomic gas, which one (if any) will increase more in temperature? (b) Physically, why do diatomic gases have a greater molar heat capacity than monatomic gases?

The discussion in Section 18.4 concluded that all ideal monatomic gases have the same heat capacity \(C_V\). Does this mean that it takes the same amount of heat to raise the temperature of 1.0 g of each one by 1.0 K? Explain your reasoning.

In a gas that contains N molecules, is it accurate to say that the number of molecules with speed v is equal to f 1v2? Is it accurate to say that this number is given by Nf 1v2? Explain your answers.

The atmosphere of the planet Mars is 95.3% carbon dioxide 1CO22 and about 0.03% water vapor. The atmospheric pressure is only about 600 Pa, and the surface temperature varies from -30C to -100C. The polar ice caps contain both CO2 ice and water ice. Could there be liquid CO2 on the surface of Mars? Could there be liquid water? Why or why not?

A beaker of water at room temperature is placed in an enclosure, and the air pressure in the enclosure is slowly reduced. When the air pressure is reduced sufficiently, the water begins to boil. The temperature of the water does not rise when it boils; in fact, the temperature drops slightly. Explain these phenomena.

Ice is slippery to walk on, and especially slippery if you wear ice skates. What does this tell you about how the melting temperature of ice depends on pressure? Explain.

Hydrothermal vents are openings in the ocean floor that discharge very hot water. The water emerging from one such vent off the Oregon coast, 2400 m below the surface, is at 279°C. Despite its high temperature, the water doesn’t boil. Why not?

The dark areas on the moons surface are called maria, Latin for seas, and were once thought to be bodies of water. In fact, the maria are not seas at all, but plains of solidified lava. Given that there is no atmosphere on the moon, how can you explain the absence of liquid water on the moons surface?

In addition to the normal cooking directions printed on the back of a box of rice, there are also high-altitude directions. The only difference is that the high-altitude directions suggest increasing the cooking time and using a greater volume of boiling water in which to cook the rice. Why should the directions depend on the altitude in this way?

. A 20.0-L tank contains 4.86 * 10-4 kg of helium at 18.0C. The molar mass of helium is 4.00 g>mol. (a) How many moles of helium are in the tank? (b) What is the pressure in the tank, in pascals and in atmospheres?

Helium gas with a volume of 3.20 L, under a pressure of 0.180 atm and at 41.0C, is warmed until both pressure and volume are doubled. (a) What is the final temperature? (b) How many grams of helium are there? The molar mass of helium is 4.00 g>mol

A cylindrical tank has a tight-fitting piston that allows the volume of the tank to be changed. The tank originally contains 0.110 m3 of air at a pressure of 0.355 atm. The piston is slowly pulled out until the volume of the gas is increased to 0.390 m3 . If the temperature remains constant, what is the final value of the pressure

A 3.00-L tank contains air at 3.00 atm and 20.0C. The tank is sealed and cooled until the pressure is 1.00 atm. (a) What is the temperature then in degrees Celsius? Assume that the volume of the tank is constant. (b) If the temperature is kept at the value found in part (a) and the gas is compressed, what is the volume when the pressure again becomes 3.00 atm?

Planetary Atmospheres. (a) Calculate the density of the atmosphere at the surface of Mars (where the pressure is 650 Pa and the temperature is typically 253 K, with a \(\mathrm{CO}_2\) atmosphere), Venus (with an average temperature of 730 K and pressure of 92 atm, with a \(\mathrm{CO}_2\) atmosphere), and Saturn’s moon Titan (where the pressure is 1.5 atm and the temperature is \(-178^\circ \mathrm{C}\), with a \(\mathrm N_2\) atmosphere). (b) Compare each of these densities with that of the earth’s atmosphere, which is \(1.20 \ \mathrm {kg/m}^3\) . Consult Appendix D to determine molar masses.

You have several identical balloons. You experimentally determine that a balloon will break if its volume exceeds 0.900 L. The pressure of the gas inside the balloon equals air pressure (1.00 atm). (a) If the air inside the balloon is at a constant 22.0C and behaves as an ideal gas, what mass of air can you blow into one of the balloons before it bursts? (b) Repeat part (a) if the gas is helium rather than air

A Jaguar XK8 convertible has an eight-cylinder engine. At the beginning of its compression stroke, one of the cylinders contains 499 cm3 of air at atmospheric pressure 11.01 * 105 Pa2 and a temperature of 27.0C. At the end of the stroke, the air has been compressed to a volume of 46.2 cm3 and the gauge pressure has increased to 2.72 * 106 Pa. Compute the final temperature.

A welder using a tank of volume 0.0750 m3 fills it with oxygen 1molar mass 32.0 g>mol2 at a gauge pressure of 3.00 * 105 Pa and temperature of 37.0C. The tank has a small leak, and in time some of the oxygen leaks out. On a day when the temperature is 22.0C, the gauge pressure of the oxygen in the tank is 1.80 * 105 Pa. Find (a) the initial mass of oxygen and (b) the mass of oxygen that has leaked out.

A large cylindrical tank contains \(0.750 \ \mathrm m^3\) of nitrogen gas at \(27^\circ \mathrm{C}\) and \(7.50 \times 10^3 \ \mathrm{Pa}\) (absolute pressure). The tank has a tight-fitting piston that allows the volume to be changed. What will be the pressure if the volume is decreased to \(0.410 \ \mathrm m^3\) and the temperature is increased to \(157^\circ \mathrm C\)?

A 20.0-L tank contains \(4.86 \times 10^{-4}\) kg of helium at 18.0°C. The molar mass of helium is 4.00 g/mol. (a) How many moles of helium are in the tank? (b) What is the pressure in the tank, in pascals and in atmospheres?

The gas inside a balloon will always have a pressure nearly equal to atmospheric pressure, since that is the pressure applied to the outside of the balloon. You fill a balloon with helium (a nearly ideal gas) to a volume of 0.600 L at 19.0C. What is the volume of the balloon if you cool it to the boiling point of liquid nitrogen (77.3 K)?

An ideal gas has a density of 1.33 * 10-6 g>cm3 at 1.00 * 10-3 atm and 20.0C. Identify the gas.

If a certain amount of ideal gas occupies a volume V at STP on earth, what would be its volume (in terms of V) on Venus, where the temperature is 1003°C and the pressure is 92 atm?

A diver observes a bubble of air rising from the bottom of a lake (where the absolute pressure is 3.50 atm) to the surface (where the pressure is 1.00 atm). The temperature at the bottom is \(4.0^\circ \mathrm{C}\), and the temperature at the surface is \(23.0^\circ \mathrm{C}\). (a) What is the ratio of the volume of the bubble as it reaches the surface to its volume at the bottom? (b) Would it be safe for the diver to hold his breath while ascending from the bottom of the lake to the surface? Why or why not?

A metal tank with volume 3.10 L will burst if the absolute pressure of the gas it contains exceeds 100 atm. (a) If 11.0 mol of an ideal gas is put into the tank at \(23.0^\circ \mathrm C\), to what temperature can the gas be warmed before the tank ruptures? Ignore the thermal expansion of the tank. (b) Based on your answer to part (a), is it reasonable to ignore the thermal expansion of the tank? Explain.

Three moles of an ideal gas are in a rigid cubical box with sides of length 0.300 m. (a) What is the force that the gas exerts on each of the six sides of the box when the gas temperature is 20.0C? (b) What is the force when the temperature of the gas is increased to 100.0C?

With the assumptions of Example 18.4 (Section 18.1), at what elevation above sea level is air pressure 90% of the pressure at sea level?

With the assumption that the air temperature is a uniform 0.0C, what is the density of the air at an altitude of 1.00 km as a percentage of the density at the surface?

. (a) Calculate the mass of nitrogen present in a volume of 3000 cm3 if the gas is at 22.0C and the absolute pressure of 2.00 * 10-13 atm is a partial vacuum easily obtained in laboratories. (b) What is the density (in kg>m3 ) of the N2?

Helium gas with a volume of 3.20 L, under a pressure of 0.180 atm and at 41.0°C, is warmed until both pressure and volume are doubled. (a) What is the final temperature? (b) How many grams of helium are there? The molar mass of helium is 4.00 g/mol.

How many moles are in a 1.00-kg bottle of water? How many molecules? The molar mass of water is 18.0 g>mol.

A large organic molecule has a mass of 1.41 * 10-21 kg. What is the molar mass of this compound?

Modern vacuum pumps make it easy to attain pressures of the order of 10-13 atm in the laboratory. Consider a volume of air and treat the air as an ideal gas. (a) At a pressure of 9.00 * 10-14 atm and an ordinary temperature of 300.0 K, how many molecules are present in a volume of 1.00 cm3 ? (b) How many molecules would be present at the same temperature but at 1.00 atm instead?

The Lagoon Nebula (Fig. E18.24) is a cloud of hydrogen gas located 3900 light-years from the earth. The cloud is about 45 light-years in diameter and glows because of its high temperature of 7500 K. (The gas is raised to this temperature by the stars that lie within the nebula.) The cloud is also very thin; there are only 80 molecules per cubic centimeter. (a) Find the gas pressure (in atmospheres) in the Lagoon Nebula. Compare it to the laboratory pressure referred to in Exercise 18.23. (b) Science-fiction films sometimes show starships being buffeted by turbulence as they fly through gas clouds such as the Lagoon Nebula. Does this seem realistic? Why or why not?

In a gas at standard conditions, what is the length of the side of a cube that contains a number of molecules equal to the population of the earth (about 7 * 109 people)?

How Close Together Are Gas Molecules? Consider an ideal gas at 27C and 1.00 atm. To get some idea how close these molecules are to each other, on the average, imagine them to be uniformly spaced, with each molecule at the center of a small cube. (a) What is the length of an edge of each cube if adjacent cubes touch but do not overlap? (b) How does this distance compare with the diameter of a typical molecule? (c) How does their separation compare with the spacing of atoms in solids, which typically are about 0.3 nm apart?

(a) What is the total translational kinetic energy of the air in an empty room that has dimensions 8.00 m * 12.00 m * 4.00 m if the air is treated as an ideal gas at 1.00 atm? (b) What is the speed of a 2000-kg automobile if its kinetic energy equals the translational kinetic energy calculated in part (a)?

A flask contains a mixture of neon (Ne), krypton (Kr), and radon (Rn) gases. Compare (a) the average kinetic energies of the three types of atoms and (b) the root-mean-square speeds. (Hint: Appendix D shows the molar mass (in g>mol) of each element under the chemical symbol for that element.)

We have two equal-size boxes, A and B. Each box contains gas that behaves as an ideal gas. We insert a thermometer into each box and find that the gas in box A is at 50C while the gas in box B is at 10C. This is all we know about the gas in the boxes. Which of the following statements must be true? Which could be true? Explain your reasoning. (a) The pressure in A is higher than in B. (b) There are more molecules in A than in B. (c) A and B do not contain the same type of gas. (d) The molecules in A have more average kinetic energy per molecule than those in B. (e) The molecules in A are moving faster than those in B.

A container with volume 1.64 L is initially evacuated. Then it is filled with 0.226 g of N2. Assume that the pressure of the gas is low enough for the gas to obey the ideal-gas law to a high degree of accuracy. If the root-mean-square speed of the gas molecules is 182 m>s, what is the pressure of the gas?

(a) A deuteron, \(_{1}^{2} \mathrm{H}\), is the nucleus of a hydrogen isotope and consists of one proton and one neutron. The plasma of deuterons in a nuclear fusion reactor must be heated to about 300 million K. What is the rms speed of the deuterons? Is this a significant fraction of the speed of light in vacuum \((c=\left.3.0 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)\)? (b) What would the temperature of the plasma be if the deuterons had an rms speed equal to 0.10c?

Martian Climate. The atmosphere of Mars is mostly \(\mathrm{CO}_2\) (molar mass 44.0 g/mol) under a pressure of 650 Pa, which we shall assume remains constant. In many places the temperature varies from \(0.0^\circ \mathrm C\) in summer to \(-100^\circ \mathrm C\) in winter. Over the course of a Martian year, what are the ranges of (a) the rms speeds of the \(\mathrm{CO}_2\) molecules and (b) the density (in \(\mathrm{mol/m}^3\) ) of the atmosphere?

Oxygen 1O22 has a molar mass of 32.0 g>mol. What is (a) the average translational kinetic energy of an oxygen molecule at a temperature of 300 K; (b) the average value of the square of its speed; (c) the root-mean-square speed; (d) the momentum of an oxygen molecule traveling at this speed? (e) Suppose an oxygen molecule traveling at this speed bounces back and forth between opposite sides of a cubical vessel 0.10 m on a side. What is the average force the molecule exerts on one of the walls of the container? (Assume that the molecules velocity is perpendicular to the two sides that it strikes.) (f) What is the average force per unit area? (g) How many oxygen molecules traveling at this speed are necessary to produce an average pressure of 1 atm? (h) Compute the number of oxygen molecules that are contained in a vessel of this size at 300 K and atmospheric pressure. (i) Your answer for part (h) should be three times as large as the answer for part (g). Where does this discrepancy arise?

Calculate the mean free path of air molecules at 3.50 * 10-13 atm and 300 K. (This pressure is readily attainable in the laboratory; see Exercise 18.23.) As in Example 18.8, model the air molecules as spheres of radius 2.0 * 10-10 m.

At what temperature is the root-mean-square speed of nitrogen molecules equal to the root-mean-square speed of hydrogen molecules at \(20.0^\circ \mathrm C\)? (Hint: Appendix D shows the molar mass (in g/mol) of each element under the chemical symbol for that element. The molar mass of \(\mathrm H_2\) is twice the molar mass of hydrogen atoms, and similarly for \(\mathrm N_2\).)

Smoke particles in the air typically have masses of the order of 10-16 kg. The Brownian motion (rapid, irregular movement) of these particles, resulting from collisions with air molecules, can be observed with a microscope. (a) Find the rootmean-square speed of Brownian motion for a particle with a mass of 3.00 * 10-16 kg in air at 300 K. (b) Would the root-meansquare speed be different if the particle were in hydrogen gas at the same temperature? Explain.

How much heat does it take to increase the temperature of 1.80 mol of an ideal gas by 50.0 K near room temperature if the gas is held at constant volume and is (a) diatomic; (b) monatomic?

Perfectly rigid containers each hold n moles of ideal gas, one being hydrogen 1H22 and the other being neon 1Ne2. If it takes 300 J of heat to increase the temperature of the hydrogen by 2.50C, by how many degrees will the same amount of heat raise the temperature of the neon?

Perfectly rigid containers each hold n moles of ideal gas, one being hydrogen 1H22 and the other being neon 1Ne2. If it takes 300 J of heat to increase the temperature of the hydrogen by 2.50C, by how many degrees will the same amount of heat raise the temperature of the neon?

(a) Calculate the specific heat at constant volume of water vapor, assuming the nonlinear triatomic molecule has three translational and three rotational degrees of freedom and that vibrational motion does not contribute. The molar mass of water is 18.0 g/mol. (b) The actual specific heat of water vapor at low pressures is about \(2000 \ \mathrm {J/kg} \cdot \ \mathrm K\). Compare this with your calculation and comment on the actual role of vibrational motion.

For diatomic carbon dioxide gas (CO2, molar mass 44.0 g>mol) at T = 300 K, calculate (a) the most probable speed vmp; (b) the average speed vav; (c) the root-mean-square speed vrms.

For a gas of nitrogen molecules 1N22, what must the temperature be if 94.7% of all the molecules have speeds less than (a) 1500 m>s; (b) 1000 m>s; (c) 500 m>s? Use Table 18.2. The molar mass of N2 is 28.0 g>mol.

Solid water (ice) is slowly warmed from a very low temperature. (a) What minimum external pressure p1 must be applied to the solid if a melting phase transition is to be observed? Describe the sequence of phase transitions that occur if the applied pressure p is such that p 6 p1. (b) Above a certain maximum pressure p2, no boiling transition is observed. What is this pressure? Describe the sequence of phase transitions that occur if p1 6 p 6 p2.

Meteorology. The vapor pressure is the pressure of the vapor phase of a substance when it is in equilibrium with the solid or liquid phase of the substance. The relative humidity is the partial pressure of water vapor in the air divided by the vapor pressure of water at that same temperature, expressed as a percentage. The air is saturated when the humidity is 100%. (a) The vapor pressure of water at 20.0C is 2.34 * 103 Pa. If the air temperature is 20.0C and the relative humidity is 60%, what is the partial pressure of water vapor in the atmosphere (that is, the pressure due to water vapor alone)? (b) Under the conditions of part (a), what is the mass of water in 1.00 m3 of air? (The molar mass of water is 18.0 g>mol. Assume that water vapor can be treated as an ideal gas.)

Calculate the volume of 1.00 mol of liquid water at 20°C (at which its density is \(998 \mathrm{\ kg} / \mathrm{m}^{3}\)), and compare that with the volume occupied by 1.00 mol of water at the critical point, which is \(56 \times 10^{-6} \mathrm{\ m}^{3}\). Water has a molar mass of 18.0 g/mol.

A physics lecture room at 1.00 atm and 27.0C has a volume of 216 m3 . (a) Use the ideal-gas law to estimate the number of air molecules in the room. Assume that all of the air is N2. Calculate (b) the particle densitythat is, the number of N2 molecules per cubic centimeterand (c) the mass of the air in the room

The Effect of Altitude on the Lungs. (a) Calculate the change in air pressure you will experience if you climb a 1000-m mountain, assuming that the temperature and air density do not change over this distance and that they were 22C and 1.2 kg>m3 , respectively, at the bottom of the mountain. (Note: The result of Example 18.4 doesnt apply, since the expression derived in that example accounts for the variation of air density with altitude and we are told to ignore that here.) (b) If you took a 0.50-L breath at the foot of the mountain and managed to hold it until you reached the top, what would be the volume of this breath when you exhaled it there?

The Bends. If deep-sea divers rise to the surface too quickly, nitrogen bubbles in their blood can expand and prove fatal. This phenomenon is known as the bends. If a scuba diver rises quickly from a depth of 25 m in Lake Michigan (which is fresh water), what will be the volume at the surface of an N2 bubble that occupied 1.0 mm3 in his blood at the lower depth? Does it seem that this difference is large enough to be a problem? (Assume that the pressure difference is due to only the changing water pressure, not to any temperature difference. This assumption is reasonable, since we are warm-blooded creatures.)

A hot-air balloon stays aloft because hot air at atmospheric pressure is less dense than cooler air at the same pressure. If the volume of the balloon is 500.0 m3 and the surrounding air is at 15.0C, what must the temperature of the air in the balloon be for it to lift a total load of 290 kg (in addition to the mass of the hot air)? The density of air at 15.0C and atmospheric pressure is 1.23 kg>m3 .

In an evacuated enclosure, a vertical cylindrical tank of diameter D is sealed by a 3.00-kg circular disk that can move up and down without friction. Beneath the disk is a quantity of ideal gas at temperature T in the cylinder (Fig. P18.50). Initially the disk is at rest at a distance of h = 4.00 m above the bottom of the tank. When a lead brick of mass 9.00 kg is gently placed on the disk, the disk moves downward. If the temperature of the gas is kept constant and no gas escapes from the tank, what distance above the bottom of the tank is the disk when it again comes to rest?

A cylinder 1.00 m tall with inside diameter 0.120 m is used to hold propane gas (molar mass 44.1 g>mol) for use in a barbecue. It is initially filled with gas until the gauge pressure is 1.30 * 106 Pa at 22.0C. The temperature of the gas remains constant as it is partially emptied out of the tank, until the gauge pressure is 3.40 * 105 Pa. Calculate the mass of propane that has been used.

During a test dive in 1939, prior to being accepted by the U.S. Navy, the submarine Squalus sank at a point where the depth of water was 73.0 m. The temperature was 27.0C at the surface and 7.0C at the bottom. The density of seawater is 1030 kg>m3 . (a) A diving bell was used to rescue 33 trapped crewmen from the Squalus. The diving bell was in the form of a circular cylinder 2.30 m high, open at the bottom and closed at the top. When the diving bell was lowered to the bottom of the sea, to what height did water rise within the diving bell? (Hint: Ignore the relatively small variation in water pressure between the bottom of the bell and the surface of the water within the bell.) (b) At what gauge pressure must compressed air have been supplied to the bell while on the bottom to expel all the water from it?

Atmosphere of Titan. Titan, the largest satellite of Saturn, has a thick nitrogen atmosphere. At its surface, the pressure is 1.5 earth-atmospheres and the temperature is 94 K. (a) What is the surface temperature in C? (b) Calculate the surface density in Titans atmosphere in molecules per cubic meter. (c) Compare the density of Titans surface atmosphere to the density of earths atmosphere at 22C. Which body has denser atmosphere?

. Pressure on Venus. At the surface of Venus the average temperature is a balmy 460C due to the greenhouse effect (global warming!), the pressure is 92 earth-atmospheres, and the acceleration due to gravity is 0.894gearth. The atmosphere is nearly all CO2 (molar mass 44.0 g>mol), and the temperature remains remarkably constant. Assume that the temperature does not change with altitude. (a) What is the atmospheric pressure 1.00 km above the surface of Venus? Express your answer in Venus-atmospheres and earth-atmospheres. (b) What is the root-mean-square speed of the CO2 molecules at the surface of Venus and at an altitude of 1.00 km?

An automobile tire has a volume of 0.0150 m3 on a cold day when the temperature of the air in the tire is 5.0C and atmospheric pressure is 1.02 atm. Under these conditions the gauge pressure is measured to be 1.70 atm (about 25 lb>in.2 ). After the car is driven on the highway for 30 min, the temperature of the air in the tires has risen to 45.0C and the volume has risen to 0.0159 m3. What then is the gauge pressure?

A flask with a volume of 1.50 L, provided with a stopcock, contains ethane gas \(\left(\mathrm{C}_{2} \mathrm{H}_{6}\right)\) at 300 K and atmospheric pressure \(\left(1.013\times 10^5\mathrm{\ Pa}\right)\). The molar mass of ethane is 30.1 g/mol. The system is warmed to a temperature of 550 K, with the stopcock open to the atmosphere. The stopcock is then closed, and the flask is cooled to its original temperature. (a) What is the final pressure of the ethane in the flask? (b) How many grams of ethane remain in the flask?

A balloon of volume 750 m3 is to be filled with hydrogen at atmospheric pressure 11.01 * 105 Pa2. (a) If the hydrogen is stored in cylinders with volumes of 1.90 m3 at a gauge pressure of 1.20 * 106 Pa, how many cylinders are required? Assume that the temperature of the hydrogen remains constant. (b) What is the total weight (in addition to the weight of the gas) that can be supported by the balloon if both the gas in the balloon and the surrounding air are at 15.0C? The molar mass of hydrogen 1H22 is 2.02 g>mol. The density of air at 15.0C and atmospheric pressure is 1.23 kg>m3 . See Chapter 12 for a discussion of buoyancy. (c) What weight could be supported if the balloon were filled with helium (molar mass 4.00 g>mol) instead of hydrogen, again at 15.0C?

A vertical cylindrical tank contains 1.80 mol of an ideal gas under a pressure of 0.300 atm at 20.0C. The round part of the tank has a radius of 10.0 cm, and the gas is supporting a piston that can move up and down in the cylinder without friction. There is a vacuum above the piston. (a) What is the mass of this piston? (b) How tall is the column of gas that is supporting the piston?

A large tank of water has a hose connected to it (Fig. P18.59). The tank is sealed at the top and has compressed air between the water surface and the top. When the water height h has the value 3.50 m, the absolute pressure p of the compressed air is \(4.20 \times 10^5\mathrm{\ Pa}\). Assume that the air above the water expands at constant temperature, and take the atmospheric pressure to be \(1.00 \times 10^5\mathrm{\ Pa}\). (a) What is the speed with which water flows out of the hose when h = 3.50 m? (b) As water flows out of the tank, h decreases. Calculate the speed of flow for h = 3.00 m and for h = 2.00 m.

A light, plastic sphere with mass m = 9.00 g and density r = 4.00 kg>m3 is suspended in air by thread of negligible mass. (a) What is the tension T in the thread if the air is at 5.00o C and p = 1.00 atm? The molar mass of air is 28.8 g>mol. (b) How much does the tension in the thread change if the temperature of the gas is increased to 35.0o C? Ignore the change in volume of the plastic sphere when the temperature is changed.

How Many Atoms Are You? Estimate the number of atoms in the body of a 50-kg physics student. Note that the human body is mostly water, which has molar mass 18.0 g/mol, and that each water molecule contains three atoms.

A person at rest inhales 0.50 L of air with each breath at a pressure of 1.00 atm and a temperature of 20.0C. The inhaled air is 21.0% oxygen. (a) How many oxygen molecules does this person inhale with each breath? (b) Suppose this person is now resting at an elevation of 2000 m but the temperature is still 20.0C. Assuming that the oxygen percentage and volume per inhalation are the same as stated above, how many oxygen molecules does this person now inhale with each breath? (c) Given that the body still requires the same number of oxygen molecules per second as at sea level to maintain its functions, explain why some people report shortness of breath at high elevations

You have two identical containers, one containing gas A and the other gas B. The masses of these molecules are mA = 3.34 * 10-27 kg and mB = 5.34 * 10-26 kg. Both gases are under the same pressure and are at 10.0C. (a) Which molecules (A or B) have greater translational kinetic energy per molecule and rms speeds? (b) Now you want to raise the temperature of only one of these containers so that both gases will have the same rms speed. For which gas should you raise the temperature? (c) At what temperature will you accomplish your goal? (d) Once you have accomplished your goal, which molecules (A or B) now have greater average translational kinetic energy per molecule?

The size of an oxygen molecule is about 2.0 * 10-10 m. Make a rough estimate of the pressure at which the finite volume of the molecules should cause noticeable deviations from idealgas behavior at ordinary temperatures 1T = 300 K2.

A sealed box contains a monatomic ideal gas. The number of gas atoms per unit volume is 5.00 * 1020 atoms>cm3 , and the average translational kinetic energy of each atom is 1.80 * 10-23 J. (a) What is the gas pressure? (b) If the gas is neon (molar mass 20.18 g>mol), what is vrms for the gas atoms?

Helium gas is in a cylinder that has rigid walls. If the pressure of the gas is 2.00 atm, then the root-mean-square speed of the helium atoms is vrms = 176 m>s. By how much (in atmospheres) must the pressure be increased to increase the vrms of the He atoms by 100 m>s? Ignore any change in the volume of the cylinder.

You blow up a spherical balloon to a diameter of 50.0 cm until the absolute pressure inside is 1.25 atm and the temperature is 22.0C. Assume that all the gas is N2, of molar mass 28.0 g>mol. (a) Find the mass of a single N2 molecule. (b) How much translational kinetic energy does an average N2 molecule have? (c) How many N2 molecules are in this balloon? (d) What is the total translational kinetic energy of all the molecules in the balloon?

(a) Compute the increase in gravitational potential energy for a nitrogen molecule (molar mass 28.0 g>mol) for an increase in elevation of 400 m near the earths surface. (b) At what temperature is this equal to the average kinetic energy of a nitrogen molecule? (c) Is it possible that a nitrogen molecule near sea level where T = 15.0C could rise to an altitude of 400 m? Is it likely that it could do so without hitting any other molecules along the way? Explain

The Lennard-Jones Potential. A commonly used potential-energy function for the interaction of two molecules (see Fig. 18.8) is the Lennard-Jones 6-12 potential: U1r2 = U0 c aR0 r b 12 - 2a R0 r b 6 d where r is the distance between the centers of the molecules and U0 and R0 are positive constants. The corresponding force F1r2 is given in Eq. (14.26). (a) Graph U1r2 and F1r2 versus r. (b) Let r1 be the value of r at which U1r2 = 0, and let r2 be the value of r at which F1r2 = 0. Show the locations of r1 and r2 on your graphs of U1r2 and F1r2. Which of these values represents the equilibrium separation between the molecules? (c) Find the values of r1 and r2 in terms of R0, and find the ratio r1>r2. (d) If the molecules are located a distance r2 apart [as calculated in part (c)], how much work must be done to pull them apart so that r S q?

(a) What is the total random translational kinetic energy of 5.00 L of hydrogen gas (molar mass 2.016 g>mol) with pressure 1.01 * 105 Pa and temperature 300 K? (Hint: Use the procedure of Problem 18.67 as a guide.) (b) If the tank containing the gas is placed on a swift jet moving at 300.0 m>s, by what percentage is the total kinetic energy of the gas increased? (c) Since the kinetic energy of the gas molecules is greater when it is on the jet, does this mean that its temperature has gone up? Explain

It is possible to make crystalline solids that are only one layer of atoms thick. Such two-dimensional crystals can be created by depositing atoms on a very flat surface. (a) If the atoms in such a two-dimensional crystal can move only within the plane of the crystal, what will be its molar heat capacity near room temperature? Give your answer as a multiple of R and in J>mol # K. (b) At very low temperatures, will the molar heat capacity of a two-dimensional crystal be greater than, less than, or equal to the result you found in part (a)? Explain why

Hydrogen on the Sun. The surface of the sun has a temperature of about 5800 K and consists largely of hydrogen atoms. (a) Find the rms speed of a hydrogen atom at this temperature. (The mass of a single hydrogen atom is \(1.67 \times 10^{-27}\mathrm{\ kg}\).) (b) The escape speed for a particle to leave the gravitational influence of the sun is given by \((2 G M / R)^{1 / 2}\), where M is the sun’s mass, R its radius, and G the gravitational constant (see Example 13.5 of Section 13.3). Use Appendix F to calculate this escape speed. (c) Can appreciable quantities of hydrogen escape from the sun? Can any hydrogen escape? Explain.

(a) Show that a projectile with mass m can “escape” from the surface of a planet if it is launched vertically upward with a kinetic energy greater than \(mgR_p\), where g is the acceleration due to gravity at the planet’s surface and \(R_p\) is the planet’s radius. Ignore air resistance. (See Problem 18.72.) (b) If the planet in question is the earth, at what temperature does the average translational kinetic energy of a nitrogen molecule (molar mass 28.0 g/mol) equal that required to escape? What about a hydrogen molecule (molar mass 2.02 g/mol?) (c) Repeat part (b) for the moon, for which \(g = 1.63 \ \mathrm{m/s}^2\) and \(R_p = 1740 \ \mathrm{km}\). (d) While the earth and the moon have similar average surface temperatures, the moon has essentially no atmosphere. Use your results from parts (b) and (c) to explain why.

Planetary Atmospheres. (a) The temperature near the top of Jupiters multicolored cloud layer is about 140 K. The temperature at the top of the earths troposphere, at an altitude of about 20 km, is about 220 K. Calculate the rms speed of hydrogen molecules in both these environments. Give your answers in m>s and as a fraction of the escape speed from the respective planet (see Problem 18.72). (b) Hydrogen gas 1H22 is a rare element in the earths atmosphere. In the atmosphere of Jupiter, by contrast, 89% of all molecules are H2. Explain why, using your results from part (a). (c) Suppose an astronomer claims to have discovered an oxygen 1O22 atmosphere on the asteroid Ceres. How likely is this? Ceres has a mass equal to 0.014 times the mass of the moon, a density of 2400 kg>m3 , and a surface temperature of about 200 K.

Calculate the integral in Eq. (18.31), 1 q 0 v2 f 1v2 dv, and compare this result to 1v2 2av as given by Eq. (18.16). (Hint: You may use the tabulated integral L q 0 x2n e-ax2 dx = 1 # 3 # 5 # # # 12n - 12 2n+1 an A p a where n is a positive integer and a is a positive constant.)

(a) Calculate the total rotational kinetic energy of the molecules in 1.00 mol of a diatomic gas at 300 K. (b) Calculate the moment of inertia of an oxygen molecule 1O22 for rotation about either the y- or z-axis shown in Fig. 18.18b. Treat the molecule as two massive points (representing the oxygen atoms) separated by a distance of 1.21 * 10-10 m. The molar mass of oxygen atoms is 16.0 g>mol. (c) Find the rms angular velocity of rotation of an oxygen molecule about either the y- or z-axis shown in Fig. 18.18b. How does your answer compare to the angular velocity of a typical piece of rapidly rotating machinery 110,000 rev>min2?

(a) Explain why in a gas of N molecules, the number of molecules having speeds in the finite interval v to v + v is N = N1 v+v v f 1v2 dv. (b) If v is small, then f 1v2 is approximately constant over the interval and N Nf 1v2v. For oxygen gas 1O2, molar mass 32.0 g>mol2 at T = 300 K, use this approximation to calculate the number of molecules with speeds within v = 20 m>s of vmp. Express your answer as a multiple of N. (c) Repeat part (b) for speeds within v = 20 m>s of 7vmp. (d) Repeat parts (b) and (c) for a temperature of 600 K. (e) Repeat parts (b) and (c) for a temperature of 150 K. (f) What do your results tell you about the shape of the distribution as a function of temperature? Do your conclusions agree with what is shown in Fig. 18.23?

(a) Explain why in a gas of N molecules, the number of molecules having speeds in the finite interval v to v + v is N = N1 v+v v f 1v2 dv. (b) If v is small, then f 1v2 is approximately constant over the interval and N Nf 1v2v. For oxygen gas 1O2, molar mass 32.0 g>mol2 at T = 300 K, use this approximation to calculate the number of molecules with speeds within v = 20 m>s of vmp. Express your answer as a multiple of N. (c) Repeat part (b) for speeds within v = 20 m>s of 7vmp. (d) Repeat parts (b) and (c) for a temperature of 600 K. (e) Repeat parts (b) and (c) for a temperature of 150 K. (f) What do your results tell you about the shape of the distribution as a function of temperature? Do your conclusions agree with what is shown in Fig. 18.23?

Oscillations of a Piston. A vertical cylinder of radius r contains an ideal gas and is fitted with a piston of mass m that is free to move (Fig. P18.79). The piston and the walls of the cylinder are frictionless, and the entire cylinder is placed in a constant-temperature bath. The outside air pressure is p0. In equilibrium, the piston sits at a height h above the bottom of the cylinder. (a) Find the absolute pressure of the gas trapped below the piston when in equilibrium. (b) The piston is pulled up by a small distance and released. Find the net force acting on the piston when its base is a distance h + y above the bottom of the cylinder, where y V h. (c) After the piston is displaced from equilibrium and released, it oscillates up and down. Find the frequency of these small oscillations. If the displacement is not small, are the oscillations simple harmonic? How can you tell?

A steel cylinder with rigid walls is evacuated to a high degree of vacuum; you then put a small amount of helium into the cylinder. The cylinder has a pressure gauge that measures the pressure of the gas inside the cylinder. You place the cylinder in various temperature environments, wait for thermal equilibrium to be established, and then measure the pressure of the gas. You obtain these results: T 1C2 p 1Pa2 Normal boiling point of nitrogen -195.8 254 Icewater mixture 0.0 890 Outdoors on a warm day 33.3 999 Normal boiling point of water 100.0 1214 Hot oven 232 1635 (a) Recall (Chapter 17) that absolute zero is the temperature at which the pressure of an ideal gas becomes zero. Use the data in the table to calculate the value of absolute zero in C. Assume that the pressure of the gas is low enough for it to be treated as an ideal gas, and ignore the change in volume of the cylinder as its temperature is changed. (b) Use the coefficient of volume expansion for steel in Table 17.2 to calculate the percentage change in the volume of the cylinder between the lowest and highest temperatures in the table. Is it accurate to ignore the volume change of the cylinder as the temperature changes? Justify your answer

The Dew Point and Clouds. The vapor pressure of water (see Exercise 18.44) decreases as the temperature decreases. The table lists the vapor pressure of water at various temperatures: Temperature 1C2 Vapor Pressure 1Pa2 10.0 1.23 * 103 12.0 1.40 * 103 14.0 1.60 * 103 16.0 1.81 * 103 18.0 2.06 * 103 20.0 2.34 * 103 22.0 2.65 * 103 24.0 2.99 * 103 26.0 3.36 * 103 28.0 3.78 * 103 30.0 4.25 * 103 If the amount of water vapor in the air is kept constant as the air is cooled, the dew point temperature is reached, at which the partial pressure and vapor pressure coincide and the vapor is saturated. If the air is cooled further, the vapor condenses to liquid until the partial pressure again equals the vapor pressure at that temperature. The temperature in a room is 30.0C. (a) A meteorologist cools a metal can by gradually adding cold water. When the cans temperature reaches 16.0C, water droplets form on its outside surface. What is the relative humidity of the 30.0C air in the room? On a spring day in the midwestern United States, the air temperature at the surface is 28.0C. Puffy cumulus clouds form at an altitude where the air temperature equals the dew point. If the air temperature decreases with altitude at a rate of 0.6 C>100 m, at approximately what height above the ground will clouds form if the relative humidity at the surface is (b) 35%; (c) 80%?

The statistical quantities average value and root-mean-square value can be applied to any distribution. Figure P18.82 shows the scores of a class of 150 students on a 100-point quiz. (a) Find the average score for the class. (b) Find the rms score for the class. (c) Which is higher: the average score or the rms score? Why?

Dark Nebulae and the Interstellar Medium. The dark area in Fig. P18.83 that appears devoid of stars is a dark nebula, a cold gas cloud in interstellar space that contains enough material to block out light from the stars behind it. A typical dark nebula is about 20 light-years in diameter and contains about 50 hydrogen atoms per cubic centimeter (monatomic hydrogen, not H2) at about 20 K. (A light-year is the distance light travels in vacuum in one year and is equal to 9.46 * 1015 m.) (a) Estimate the mean free path for a hydrogen atom in a dark nebula. The radius of a hydrogen atom is 5.0 * 10-11 m. (b) Estimate the rms speed of a hydrogen atom and the mean free time (the average time between collisions for a given atom). Based on this result, do you think that atomic collisions, such as those leading to H2 molecule formation, are very important in determining the composition of the nebula? (c) Estimate the pressure inside a dark nebula. (d) Compare the rms speed of a hydrogen atom to the escape speed at the surface of the nebula (assumed spherical). If the space around the nebula were a vacuum, would such a cloud be stable or would it tend to evaporate? (e) The stability of dark nebulae is explained by the presence of the interstellar medium (ISM), an even thinner gas that permeates space and in which the dark nebulae are embedded. Show that for dark nebulae to be in equilibrium with the ISM, the numbers of atoms per volume 1N>V2 and the temperatures 1T2 of dark nebulae and the ISM must be related by 1N>V2nebula 1N>V2ISM = TISM Tnebula (f) In the vicinity of the sun, the ISM contains about 1 hydrogen atom per 200 cm3 . Estimate the temperature of the ISM in the vicinity of the sun. Compare to the temperature of the suns surface, about 5800 K. Would a spacecraft coasting through interstellar space burn up? Why or why not?

Earth’s Atmosphere. In the troposphere, the part of the atmosphere that extends from earth’s surface to an altitude of about 11 km, the temperature is not uniform but decreases with increasing elevation. (a) Show that if the temperature variation is approximated by the linear relationship \(T=T_{0}-\alpha y\) where \(T_{0}\) is the temperature at the earth’s surface and T is the temperature at height y, the pressure p at height y is \(\ln \left(\frac{p}{p_{0}}\right)=\frac{M g}{R \alpha} \ln \left(\frac{T_{0}-\alpha y}{T_{0}}\right)\) where \(p_{0}\) is the pressure at the earth’s surface and M is the molar mass for air. The coefficient \(\alpha\) is called the lapse rate of temperature. It varies with atmospheric conditions, but an average value is about 0.6 C°/100 m. (b) Show that the above result reduces to the result of Example 18.4 (Section 18.1) in the limit that \(\alpha \rightarrow 0\). (c) With \(\alpha=0.6\mathrm{C}^{\circ}/100\mathrm{\ m}\), calculate p for y = 8863 m and compare your answer to the result of Example 18.4. Take \(T_0=288\mathrm{\ K}\text{ and }p_0=1.00\mathrm{\ atm}\) atm.

What is one reason the noble gases are preferable to air (which is mostly nitrogen and oxygen) as an insulating material? (a) Noble gases are monatomic, so no rotational modes contribute to their molar heat capacity; (b) noble gases are monatomic, so they have lower molecular masses than do nitrogen and oxygen; (c) molecular radii in noble gases are much larger than those of gases that consist of diatomic molecules; (d) because noble gases are monatomic, they have many more degrees of freedom than do diatomic molecules, and their molar heat capacity is reduced by the number of degrees of freedom.

Estimate the ratio of the thermal conductivity of Xe to that of He. (a) 0.015; (b) 0.061; (c) 0.10; (d) 0.17.

The rate of effusion—that is, leakage of a gas through tiny cracks—is proportional to \(v_\mathrm{rms}\). If tiny cracks exist in the material that’s used to seal the space between two glass panes, how many times greater is the rate of He leakage out of the space between the panes than the rate of Xe leakage at the same temperature? (a) 370 times; (b) 19 times; (c) 6 times; (d) no greater—the He leakage rate is the same as for Xe.