Solved: (a) Compute the specific heat at constant volume

A 20.0-L tank contains of helium at The molar mass of helium is (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 2.60 L, under a pressure of 0.180 atm and at a temperature of \(41.0^{\circ} \mathrm{C}\), 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.

A cylindrical tank has a tight-fitting piston that allows the volume of the tank to be changed. The tank originally contains of air at a pressure of 0.355 atm. The piston is slowly pulled out until the volume of the gas is increased to 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 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 atmosphere), Venus (with an average temperature of 730 K and pressure of 92 atm, with a atmosphere), and Saturns moon Titan (where the pressure is 1.5 atm and the temperature is with a atmosphere). (b) Compare each of these densities with that of the earths atmosphere, which is 1.20 kg m3 . Consult the periodic chart in 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 temperature of \(22.0^{\circ} \mathrm{C}\) 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 of air at atmospheric pressure and a temperature of At the end of the stroke, the air has been compressed to a volume of and the gauge pressure has increased to Compute the final temperature.

A welder using a tank of volume \(0.0750\mathrm{\ m}^3\) fills it with oxygen (molar mass 32.0 g/mol) at a gauge pressure of \(3.00\times10^5\mathrm{\ Pa}\) and temperature of \(37.0^{\circ} \mathrm{C}\). The tank has a small leak, and in time some of the oxygen leaks out. On a day when the temperature is \(22.0^{\circ} \mathrm{C}\), the gauge pressure of the oxygen in the tank is \(1.80\times10^5\mathrm{\ Pa}\). 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 \(0.480 \mathrm{~m}^{3}\) and the temperature is increased to \(157^{\circ} \mathrm{C}\).

An empty cylindrical canister 1.50 m long and 90.0 cm in diameter is to be filled with pure oxygen at \(22.0^{\circ} \mathrm{C}\) to store in a space station. To hold as much gas as possible, the absolute pressure of the oxygen will be 21.0 atm. The molar mass of oxygen is 32.0 g/mol. (a) How many moles of oxygen does this canister hold? (b) For someone lifting this canister, by how many kilograms does this gas increase the mass to be lifted?

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 a temperature of \(19.0^{\circ} \mathrm{C}\). What is the volume of the balloon if you cool it to the boiling point of liquid nitrogen (77.3 K)?

Deviations from the Ideal-Gas Equation. For carbon dioxide gas the constants in the van der Waals equation are and (a) If 1.00 mol of gas at 350 K is confined to a volume of find the pressure of the gas using the ideal-gas equation and the van der Waals equation. (b) Which equation gives a lower pressure? Why? What is the percentage difference of the van der Waals equation result from the ideal-gas equation result? (c) The gas is kept at the same temperature as it expands to a volume of Repeat the calculations of parts (a) and (b). (d) Explain how your calculations show that the van der Waals equation is equivalent to the ideal-gas equation if is small.

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 1003C 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 and the temperature at the surface is (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 a temperature of to what temperature can the gas be warmed before the tank ruptures? You can 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 0.200 m. (a) What is the force that the gas exerts on each of the six sides of the box when the gas temperature is \(20.0^{\circ} \mathrm{C}\)? (b) What is the force when the temperature of the gas is increased to \(100.0^{\circ} \mathrm{C}\)?

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

Make the same assumptions as in Example 18.4 (Section 18.1). How does the percentage decrease in air pressure in going from sea level to an altitude of 100 m compare to that when going from sea level to an altitude of 1000 m? If your second answer is not 10 times your first answer, explain why

(a) Calculate the mass of nitrogen present in a volume of \(3000\mathrm{\ cm}^3\) if the temperature of the gas is \(22.0^{\circ} \mathrm{C}\) and the absolute pressure of \(2.00\times10^{-13}\mathrm{\ atm}\) is a partial vacuum easily obtained in laboratories. (b) What is the density (in \(\mathrm{kg} / \mathrm{m}^{3}\)) of the \(N_{2}\)?

With the assumption that the air temperature is a uniform 0.0C (as in Example 18.4), what is the density of the air at an altitude of 1.00 km as a percentage of the density at the surface?

At an altitude of 11,000 m (a typical cruising altitude for a jet airliner), the air temperature is and the air density is What is the pressure of the atmosphere at that altitude? (Note: The temperature at this altitude is not the same as at the surface of the earth, so the calculation of Example 18.4 in Section 18.1 doesnt apply.)

A large organic molecule has a mass of \(1.41\times 10^{-21}\mathrm{\ kg}.\)What is the molar mass of this compound?

Suppose you inherit 3.00 mol of gold from your uncle (an eccentric chemist) at a time when this metal is selling for $14.75 per gram. Consult the periodic table in Appendix D and Table 12.1. (a) To the nearest dollar, what is this gold worth? (b) If you have your gold formed into a spherical nugget, what is its diameter?

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 \(9.00 \times 10^{-14}\) atm and an ordinary temperature of 300.0 K, how many molecules are present in a volume of \(1.00 \mathrm{\ cm}^3\)? (b) How many molecules would be present at the same temperature but at 1.00 atm instead?

The Lagoon Nebula (Fig. E18.25) 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 18.24. (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 people)?

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.

How Close Together Are Gas Molecules? Consider an ideal gas at and 1.00 atm pressure. 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?

Consider 5.00 mol of liquid water. (a) What volume is occupied by this amount of water? The molar mass of water is (b) Imagine the molecules to be, on average, uniformly spaced, with each molecule at the center of a small cube. What is the length of an edge of each small cube if adjacent cubes touch but dont overlap? (c) How does this distance compare with the diameter of a molecule?

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: The periodic table in Appendix D shows the molar mass (in ) of each element under the chemical symbol for that element.)

Gaseous Diffusion of Uranium. (a) A process called gaseous diffusion is often used to separate isotopes of uranium that is, atoms of the elements that have different masses, such as \({ }^{235} \mathrm{U}\) and \({ }^{238} \mathrm{U}\). The only gaseous compound of uranium at ordinary temperatures is uranium hexafluoride, \(\mathrm{UF}_{6}\). Speculate on how \({ }^{235} \mathrm{UF}_{6}\) and \({ }^{238} \mathrm{UF}_{6}\) molecules might be separated by diffusion. (b) The molar masses for \({ }^{235} \mathrm{UF}_{6}\) and \({ }^{238} \mathrm{UF}_{6}\) molecules are 0.349 kg/mol and 0.352 kg/mol, respectively. If uranium hexa- fluoride acts as an ideal gas, what is the ratio of the root-mean-square speed of \({ }^{235} \mathrm{UF}_{6}\) molecules to that of \({ }^{238} \mathrm{UF}_{6}\) molecules if the temperature is uniform?

The ideas of average and root-mean-square value can be applied to any distribution. A class of 150 students had the following scores on a 100-point quiz: (a) Find the average score for the class. (b) Find the root-meansquare score for the class.

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 a temperature of while the gas in box B is at This is all we know about the gas in the boxes. Which of the following statements must be true? Which could be true? (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. Explain the reasoning behind your answers.

A container with volume 1.48 L is initially evacuated. Then it is filled with 0.226 g of . 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 \(\left(c=3.0 \times 10^8 \mathrm{\ m}/\mathrm{s}\right)\)? (b) What would the temperature of the plasma be if the deuterons had an rms speed equal to 0.10c? Text Transcription: _1^2 H (c = 3.0 times 10^8 m/s)

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 \(-100.0^{\circ} \mathrm{C}\) 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} / \mathrm{m}^{3})) of the atmosphere?

(a) Oxygen has a molar mass of What is the average translational kinetic energy of an oxygen molecule at a temperature of 300 K? (b) What is the average value of the square of its speed? (c) What is the root-mean-square speed? (d) What is 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 actually 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 a pressure of and a temperature of 300 K. (This pressure is readily attainable in the laboratory; see Exercise 18.24.) As in Example 18.8, model the air molecules as spheres of radius

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: The periodic table in 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 \(10^{-16}\mathrm{\ kg}\). The Brownian motion (rapid, irregular movement) of these particles, resulting from collisions with air molecules, can be observed with a microscope. (a) Find the root-mean-square speed of Brownian motion for a particle with a mass of \(3.00\times10^{-16}\mathrm{\ kg}\) in air at 300 K. (b) Would the root-mean-square speed be different if the particle were in hydrogen gas at the same temperature? Explain.

(a) How much heat does it take to increase the temperature of 2.50 mol of a diatomic ideal gas by 50.0 K near room temperature if the gas is held at constant volume? (b) What is the answer to the question in part (a) if the gas is monatomic rather than diatomic?

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

(a) Compute the specific heat at constant volume of nitrogen gas, and compare it with the specific heat of liquid water. The molar mass of is (b) You warm 1.00 kg of water at a constant volume of 1.00 L from to in a kettle. For the same amount of heat, how many kilograms of air would you be able to warm to What volume (in liters) would this air occupy at and a pressure of 1.00 atm? Make the simplifying assumption that air is 100%

(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 (b) The actual specific heat of water vapor at low pressures is about Compare this with your calculation and comment on the actual role of vibrational motion.

(a) Use Eq. 18.28 to calculate the specific heat at constant volume of aluminum in units of Consult the periodic table in Appendix D. (b) Compare the answer in part (a) with the value given in Table 17.3. Try to explain any disagreement between these two values.

For a gas of nitrogen molecules what must the temperature be if 94.7% of all the molecules have speeds less than (a) (b) (c) Use Table 18.2. The molar mass of is

For diatomic carbon dioxide gas (\(\mathrm{CO}_{2}\), molar mass 44.0 g/mol) at T = 300 K, calculate (a) the most probable speed \(v_{mp}\); (b) the average speed \(v_{av}\); (c) the root-mean-square speed \(v_{rms}\).

CALC Prove that as given by Eq. (18.33) is maximum for Use this result to obtain Eq. (18.34).

Solid water (ice) is slowly warmed from a very low temperature. (a) What minimum external pressure 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 (b) Above a certain maximum pressure no boiling transition is observed. What is this pressure? Describe the sequence of phase transitions that occur if

Puffy cumulus clouds, which are made of water droplets, occur at lower altitudes in the atmosphere. Wispy cirrus clouds, which are made of ice crystals, occur only at higher altitudes. Find the altitude y (measured from sea level) above which only cirrus clouds can occur. On a typical day and at altitudes less than the temperature at an altitude y is given by where and

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

A physics lecture room has a volume of . (a) For a pressure of 1.00 atm and a temperature of , use the ideal-gas law to estimate the number of air molecules in the room. Assume all the air is . (b) Calculate the particle densitythat is, the number of molecules per cubic centimeter. (c) Calculate the mass of the air in the room.

CP BIO 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 respectively, at the bottom of the mountain. (Note that 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 in this problem.) (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 bubble that occupied 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 only to the changing water pressure, not to any temperature difference, an assumption that 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 \(500.0 \mathrm{~m}^{3}\) and the surrounding air is at \(15.0^{\circ} \mathrm{C}\), 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 \(1.23 \mathrm{~kg} / \mathrm{m}^{3}\).

(a) Use Eq. (18.1) to estimate the change in the volume of a solid steel sphere of volume 11 L when the temperature and pressure increase from \(21^{\circ} \mathrm{C}\) and \(1.013 \times 10^{5} \mathrm{~Pa}\) to \(42^{\circ} \mathrm{C}\) and \(2.10 \times 10^{7} \mathrm{~Pa}\). (Hint: Consult Chapters 11 and 17 to determine the values of \(\beta \text { and } k\).) (b) In Example 18.3 the change in volume of an 11-L steel scuba tank was ignored. Was this a good approximation? Explain.

A cylinder 1.00 m tall with inside diameter 0.120 m is used to hold propane gas (molar mass ) for use in a barbecue. It is initially filled with gas until the gauge pressure is and the temperature is The temperature of the gas remains constant as it is partially emptied out of the tank, until the gauge pressure is 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 The temperature at the surface was and at the bottom it was The density of seawater is (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: You may 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 (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 Which body has denser atmosphere?

Pressure on Venus. At the surface of Venus the average temperature is a balmy \(460^{\circ} \mathrm{C}\) due to the greenhouse effect (global warming!), the pressure is 92 earth-atmospheres, and the acceleration due to gravity is \(0.894 g_{\text {earth }}\). The atmosphere is nearly all \(\mathrm{CO}_{2}\) (molar mass 44.0 g/mol) and the temperature remains remarkably constant. We shall assume that the temperature does not change at all 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 \(\mathrm{CO}_{2}\) molecules at the surface of Venus and at an altitude of 1.00 km?

An automobile tire has a volume of \(0.0150 \mathrm {\ m}^3\) on a cold day when the temperature of the air in the tire is \(5.0^{\circ} \mathrm{C}\) and atmospheric pressure is 1.02 atm. Under these conditions the gauge pressure is measured to be 1.70 atm \(\left(\text{about }25\mathrm{lb}/\text{in.}^2\right)\). After the car is driven on the highway for 30 min, the temperature of the air in the tires has risen to \(45.0^{\circ} \mathrm{C}\) and the volume has risen to \(0.0159 \mathrm{\ m}^3\). 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 \(30.1\mathrm{\ g}/\mathrm{mol}\). The system is warmed to a temperature of 490 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 whose volume is is to be filled with hydrogen at atmospheric pressure (a) If the hydrogen is stored in cylinders with volumes of at a gauge pressure of 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 the gas in the balloon and the surrounding air are both at The molar mass of hydrogen is The density of air at and atmospheric pressure is See Chapter 12 for a discussion of buoyancy. (c) What weight could be supported if the balloon were filled with helium (molar mass ) instead of hydrogen, again at

A vertical cylindrical tank contains 1.80 mol of an ideal gas under a pressure of 0.500 atm at 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, as shown in Fig. P18.65. 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 Assume that the air above the water expands at constant temperature, and take the atmospheric pressure to be (a) What is the speed with which water flows out of the hose when (b) As water flows out of the tank, h decreases. Calculate the speed of flow for and for (c) At what value of h does the flow stop?

A person at rest inhales 0.50 L of air with each breath at a pressure of 1.00 atm and a temperature of \(20.0^{\circ} \mathrm{C}\). 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 \(20.0^{\circ} \mathrm{C}\). 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

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

The size of an oxygen molecule is about Make a rough estimate of the pressure at which the finite volume of the molecules should cause noticeable deviations from ideal-gas behavior at ordinary temperatures

You have two identical containers, one containing gas A and the other gas B. The masses of these molecules are and Both gases are under the same pressure and are at (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?

Insect Collisions. A cubical cage 1.25 m on each side contains 2500 angry bees, each flying randomly at 1.10 m/s. We can model these insects as spheres 1.50 cm in diameter. On the average, (a) how far does a typical bee travel between collisions, (b) what is the average time between collisions, and (c) how many collisions per second does a bee make?

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 Assume that all the gas is , of molar mass (a) Find the mass of a single molecule. (b) How much translational kinetic energy does an average molecule have? (c) How many 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 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 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

CP, CALC 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: where r is the distance between the centers of the molecules and and are positive constants. The corresponding force is given in Eq. (14.26). (a) Graph and versus r. (b) Let be the value of r at which and let be the value of r at which Show the locations of and on your graphs of and Which of these values represents the equilibrium separation between the molecules? (c) Find the values of and in terms of and find the ratio (d) If the molecules are located a distance apart [as calculated in part (c)], how much work must be done to pull them apart so that

(a) What is the total random translational kinetic energy of 5.00 L of hydrogen gas (molar mass ) with pressure and temperature 300 K? (Hint: Use the procedure of Problem 18.71 as a guide.) (b) If the tank containing the gas is placed on a swift jet moving at 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.

The speed of propagation of a sound wave in air at \(27^{\circ} \mathrm{C}\) is about 350 m/s. Calculate, for comparison, (a) \(v_{rms}\) for nitrogen molecules and (b) the rms value of \(v_{x}\) at this temperature. The molar mass of nitrogen \(\left(\mathrm{N}_{2}\right)\) is 28.0 g/mol.

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 ) (b) The escape speed for a particle to leave the gravitational influence of the sun is given by where M is the suns mass, R its radius, and G the gravitational constant (see Example 13.5 of Section 13.3). Use the data in Appendix F to calculate this escape speed. (c) Can appreciable quantities of hydrogen escape from the sun? Can any hydrogen escape? Explain

CP (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 where g is the acceleration due to gravity at the planets surface and is the planets radius. Ignore air resistance. (See Problem 18.76.) (b) If the planet in question is the earth, at what temperature does the average translational kinetic energy of a nitrogen molecule (molar mass ) equal that required to escape? What about a hydrogen molecule (molar mass (c) Repeat part (b) for the moon, for which and (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 and as a fraction of the escape speed from the respective planet (see Problem 18.76). (b) Hydrogen gas is a rare element in the earths atmosphere. In the atmosphere of Jupiter, by contrast, 89% of all molecules are Explain why, using your results from part (a). (c) Suppose an astronomer claims to have discovered an oxygen 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 and a surface temperature of about 200 K

(a) For what mass of molecule or particle is equal to at 300 K? (b) If the particle is an ice crystal, how many molecules does it contain? The molar mass of water is (c) Calculate the diameter of the particle if it is a spherical piece of ice. Would it be visible to the naked eye?

In describing the heat capacities of solids in Section 18.4, we stated that the potential energy of a harmonic oscillator averaged over one period of the motion is equal to the kinetic energy averaged over one period. Prove this result using Eqs. (14.13) and (14.15) for the position and velocity of a simple harmonic oscillator. For simplicity, assume that the initial position and velocity make the phase angle equal to zero. (Hint: Use the trigonometric identities and What is the average value of over one period?)

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 (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

(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 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 The molar mass of oxygen atoms is (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

For each polyatomic gas in Table 18.1, compute the value of the molar heat capacity at constant volume, \(C_{V}\), on the assumption that there is no vibrational energy. Compare with the measured values in the table, and compute the fraction of the total heat capacity that is due to vibration for each of the three gases. (Note: \(\mathrm{CO}_{2}\) is linear; \(\mathrm{SO}_{2}\) and \(\mathrm{H}_{2} \mathrm{S}\) are not. Recall that a linear polyatomic molecule has two rotational degrees of freedom, and a nonlinear molecule has three.)

(a) Show that \(\int_0^{\infty}f(v)\ dv=1\) where is \(f(v)\) the MaxwellBoltzmann distribution of Eq. (18.32). (b) In terms of the physical definition of \(f(v)\) explain why the integral in part (a) must have this value.

CALC Calculate the integral in Eq. (18.31), and compare this result to as given by Eq. (18.16). (Hint: You may use the tabulated integral where n is a positive integer and is a positive constant.)

CALC Calculate the integral in Eq. (18.30), and compare this result to as given by Eq. (18.35). (Hint: Make the change of variable and use the tabulated integral where n is a positive integer and is a positive constant.)

CALC (a) Explain why in a gas of N molecules, the number of molecules having speeds in the finite interval to is (b) If is small, then is approximately constant over the interval and For oxygen gas at use this approximation to calculate the number of molecules with speeds within of Express your answer as a multiple of N. (c) Repeat part (b) for speeds within of (d) Repeat parts (b) and (c) for a temperature of (e) Repeat parts (b) and (c) for a temperature of (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?

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 is If the air temperature is 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 of air? (The molar mass of water is Assume that water vapor can be treated as an ideal gas.)

The Dew Point. The vapor pressure of water (see Problem 18.88) decreases as the temperature decreases. If the amount of water vapor in the air is kept constant as the air is cooled, a temperature is reached, called the dew point, at which the partial pressure and vapor pressure coincide and the vapor is saturated. If the air is cooled further, vapor condenses to liquid until the partial pressure again equals the vapor pressure at that temperature. The temperature in a room is \(30.0^{\circ} \mathrm{C}\). A meteorologist cools a metal can by gradually adding cold water. When the can temperature reaches \(16.0^{\circ} \mathrm{C}\), water droplets form on its outside surface. What is the relative humidity of the \(30.0^{\circ} \mathrm{C}\) air in the room? The table lists the vapor pressure of water at various temperatures:

Altitude at Which Clouds Form. On a spring day in the midwestern United States, the air temperature at the surface is Puffy cumulus clouds form at an altitude where the air temperature equals the dew point (see Problem 18.89). If the air temperature decreases with altitude at a rate of at approximately what height above the ground will clouds form if the relative humidity at the surface is 35% and 80%? (Hint: Use the table in Problem 18.89.)

CP Dark Nebulae and the Interstellar Medium. The dark area in Fig. P18.91 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 ) at a temperature of about 20 K. (A light-year is the distance light travels in vacuum in one year and is equal to ) (a) Estimate the mean free path for a hydrogen atom in a dark nebula. The radius of a hydrogen atom is (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 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 and the temperatures of dark nebulae and the ISM must be related by (f) In the vicinity of the sun, the ISM contains about 1 hydrogen atom per 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?

CALC Earths Atmosphere. In the troposphere, the part of the atmosphere that extends from earths surface to an altitude of about the temperature is not uniform but decreases with increasing elevation. (a) Show that if the temperature variation is approximated by the linear relationship where is the temperature at the earths surface and T is the temperature at height y, the pressure p at height y is given by where is the pressure at the earths surface and M is the molar mass for air. The coefficient is called the lapse rate of temperature. It varies with atmospheric conditions, but an average value is about (b) Show that the above result reduces to the result of Example 18.4 (Section 18.1) in the limit that (c) With calculate p for and compare your answer to the result of Example 18.4. Take and p0 = 1.00 atm.

In Example 18.7 (Section 18.3) we saw that It is not difficult to show that this is always the case. (The only exception is when the particles have the same speed, in which case (a) For two particles with speeds and show that regardless of the numerical values of and Then show that if (b) Suppose that for a collection of N particles you know that Another particle, with speed u, is added to the collection of particles. If the new rms and average speeds are denoted as and show that (c) Use the expressions in part (b) to show that regardless of the numerical value of u. (d) Explain why your results for (a) and (c) together show that for any collection of particles if the particles do not all have the same speed.

Problem 1DQ 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?

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

Problem 2DQ 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?

Helium gas with a volume of 2.60 L, under a pressure of 0.180 atm and at a temperature of 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.

Problem 3DQ On a chilly morning you can “see your breath.” Can you really? What are you actually seeing? Does this phenomenon depend on the temperature of the air. the humidity, or both? Explain.

Problem 3E 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?

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?

A 3.00-L tank contains air at 3.00 atm and 20.0\(^\circ\)C. 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?

Problem 5DQ 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?

Problem 5E 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 CO2 atmosphere), Venus (with an average temperature of 730 K and pressure of 92 atm, with a CO2 atmosphere), and Saturn’s moon Titan (where the pressure is 1.5 atm and the temperature is – 178o C, with a N2 atmosphere). (b) Compare each of these densities with that of the earth’s atmosphere, which is 1.20 kg/m3. Consult Appendix D to determine molar masses.

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

Problem 6E 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 22.0o C 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.

Problem 7DQ “Freeze-drying” food involves the same process as “freezer burn,” referred to in Discussion Question For freeze-drying, the food is usually frozen first, and then placed in a vacuum chamber and irradiated with infrared radiation. What is the purpose of the vacuum? The radiation? What advantages might freeze-drying have in comparison to ordinary drying? Question: Unwrapped food placed in a freezer experiences dehydration, known as “freezer burn” Why?

Problem 7E A Jaguar XK8 convertible has an eight-cylinder engine. At the beginning of its compression stroke, one of the cylinders contains 499 cm 3 of air at atmospheric pressure (1.01 X 105 Pa) and a temperature of 27.0o C. At the end of the stroke, the air has been compressed to a volume of 46.2 cm 3 and the gauge pressure has increased to 2.72 X 106 Pa. Compute the final temperature.

Problem 8DQ 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?

Problem 8E A welder using a tank of volume 0.0750 m3 fills it with oxygen (molar mass 32.0 g/mol) at a gauge pressure of 3.00 X 105 Pa and temperature of 37.0o C. The tank has a small leak, and in time some of the oxygen leaks out. On a day when the temperature is 22.0o C, the gauge pressure of the oxygen in the tank is 1.80 X 105 Pa. Find (a) the initial mass of oxygen and (b) the mass of oxygen that has leaked out.

How does evaporation of perspiration from your skin cool your body?

A large cylindrical tank contains \(0.750 \mathrm{~m}^{3}\) of nitrogen gas at \(27^{\circ} \mathrm{C} \text { and } 7.50 \times 10^{3}\) 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 and the temperature is increased to \(157^{\circ} \mathrm{C}\)?

Problem 10DQ A rigid, perfectly insulated container has a membrane dividing its volume in half. One side contains a gas at an absolute temperature T0 and pressure p0, 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 T0 and p0, 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.

An empty cylindrical canister 1.50 m long and 90.0 cm in diameter is to be filled with pure oxygen at \(22.0^{\circ} \mathrm{C}\) to store in a space station. To hold as much gas as possible, the absolute pressure of the oxygen will be 21.0 atm. The molar mass of oxygen is 32.0 g/mol. (a) How many moles of oxygen does this canister hold? (b) For someone lifting this canister, by how many kilograms does this gas increase the mass to be lifted?

Problem 11DQ (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.

Problem 11E 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 19.0o C. What is the volume of the balloon if you cool it to the boiling point of liquid nitrogen (77.3 K)?

Problem 12DQ 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.

Problem 12E Deviations from the Ideal-Gas Equation. For carbondioxide gas (CO2), the constants in the van der Waals equation are a = 0.364 J. m3/mol2 and b = 4.27 × 10?5 m3/mol. (a) If 1.00 mol of CO2 gas at 350 K is confined to a volume of 400 cm3, find the pressure of the gas using the ideal-gas equation and the vander Waals equation. (b) Which equation gives a lower pressure? Why? What is the percentage difference of the vander Waals equation result from the ideal-gas equation result? (c) The gas is kept at the same temperature as it expands to a volume of 4000 cm3. Repeat the calculations of parts (a) and (b). (d) Explain how your calculations show that the van der Waals equation is equivalent to the ideal-gas equation if n/V is small.

The proportions of various gases in the earth’s 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?

Problem 13E 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?

Problem 14DQ 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?

Problem 14E 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 4.0o C, and the temperature at the surface is 23.0o 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?

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?

Problem 15E 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 23.0o 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.

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.

Three moles of an ideal gas are in a rigid cubical box with sides of length 0.200 m. (a) What is the force that the gas exerts on each of the six sides of the box when the gas temperature is 20.0°C? (b) What is the force when the temperature of the gas is increased to 100.0°C?

Problem 17DQ 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.

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

Problem 18DQ In deriving the ideal-gas equation from the kinetic- molecular model, we ignored potential energy due to the earth’s gravity. Is this omission justified? Why or why not?

Make the same assumptions as in Example 18.4 (Section 18.1). How does the percentage decrease in air pressure in going from sea level to an altitude of 100 m compare to that when going from sea level to an altitude of 1000 m? If your second answer is not 10 times your first answer, explain why.

Problem 19DQ 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?

Problem 19E (a) Calculate the mass of nitrogen present in a volume of 3000 cm3 if the gas is at 22.0°C and the absolute pressure of 2.00 X 10-13 atm is a partial vacuum easily obtained in laboratories. (b) What is the density (in kg/m3) of the N2?

Problem 20DQ 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?

With the assumption that the air temperature is a uniform \(0.0^{\circ} \mathrm{C}\) (as in Example 18.4), what is the density of the air at an altitude of 1.00 km as a percentage of the density at the surface? Equation Transcription: Text Transcription: 0.0degC

Problem 21DQ 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?

At an altitude of 11,000 m (a typical cruising altitude for a jet airliner), the air temperature is \(-56.5^{\circ} \mathrm{C}\) and the air density is 0.364 \(\mathrm{kg} / \mathrm{m}^{3}\). What is the pressure of the atmosphere at that altitude? (Note: The temperature at this altitude is not the same as at the surface of the earth, so the calculation of Example 18.4 in Section 18.1 doesn’t apply.) Equation Transcription: Text Transcription: -56.5degC kg/m^3

Problem 22DQ The temperature of an ideal monatomic gas is increased from 25o C to 50o C. 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?

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

Problem 23DQ 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.

Suppose you inherit 3.00 mol of gold from your uncle (an eccentric chemist) at a time when this metal is selling for $14.75 per gram. Consult the periodic table in Appendix D and Table 12.1. (a) To the nearest dollar, what is this gold worth? (b) If you have your gold formed into a spherical nugget, what is its diameter?

Problem 24DQ (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?

Problem 24E 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 X 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 discussion in Section 18.4 concluded that all ideal monatomic gases have the same heat capacity CV. 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.

The Lagoon Nebula (Fig. E18.25) 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 18.24. (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?

Problem 26DQ In a gas that contains N molecules, is it accurate to say that the number of molecules with speed v is equal to f(v)? Is it accurate to say that this number is given by Nf(v)? Explain your answers.

Problem 26E 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 6 × 109 people)?

Problem 27DQ 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 thermodynamics—which we will discuss in Chapter 20—tells us that such a wonderful air filter would be impossible to make.)

Problem 27E 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.

Problem 28DQ 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.

How Close Together Are Gas Molecules? Consider an ideal gas at \(27^{\circ} \mathrm{\ C}\) and 1.00 atm pressure. 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? Text Transcription: 27^circ C

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.

Problem 29E Consider 5.00 mol of liquid water. (a) What volume is occupied by this amount of water? The molar mass of water is 18.0 g/mol (b) Imagine the molecules to be, on average, uniformly spaced, with each molecule at the center of a small cube. What is the length of an edge of each small cube if adjacent cubes touch but don’t overlap? (c) How does this distance compare with the diameter of a molecule’?

Problem 30DQ 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 279o C. Despite its high temperature, the water doesn’t boil. Why not?

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.)

Problem 31DQ The dark areas on the moon’s 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 moon’s surface?

Gaseous Diffusion of Uranium. (a) A process called gaseous diffusion is often used to separate isotopes of uranium-that is, atoms of the elements that have different masses, such as 235U and 238U. The only gaseous compound of uranium at ordinary temperatures is uranium hexafluoride, 235UF6 Speculate on how 235UF6 and 238UF6 molecules might be separated by diffusion. (b) The molar masses for 235UF6, and 238UF6 molecules are 0.349 kg/mol and 0.352 kg/mol, respectively. If uranium hexafluoride acts as an ideal gas, what is the ratio of the root-mean-square speed of molecules to that of 238UF6 molecules if the temperature is uniform?

Problem 32DQ 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?

Problem 32E The ideas of average and root-mean-square value can be applied to any distribution. A class of 150 students had the following scores on a 100-point quiz: Score Number of Students 10 11 20 12 30 24 40 15 50 19 60 10 70 12 80 20 90 17 100 10 (a) Find the average score for the class. (b) Find the root-means square score for the class.

Problem 33E 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 50o C while the gas in box B is at 10o C. 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.

Problem 34E A container with volume 1.48 L is initially Evaluated 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, \({ }_{1}^{2} H\), 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 \(\left(c=3.0 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)\)? (b) What would the temperature of the plasma be if the deuterons had an rms speed equal to \(0.10 c\)? Equation Transcription: Text Transcription: ^2_1H (c=3.0x10^8 m/s) 0.10c

Problem 36E Martian Climate. The atmosphere of Mars is mostly CO2 (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 0.0o C in summer to – 100o C in winter. Over the course of a Martian year, what are the ranges of (a) the rms speeds of the CO2 molecules and (b) the density (in mol/m3) of the atmosphere?

Problem 37E Oxygen (O2) 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 molecule’s 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 a pressure of \(3.50 \times 10^{-13}\) and a temperature of 300 K. (This pressure is readily attainable in the laboratory; see Exercise 18.24.) As in Example 18.8, model the air molecules as spheres of radius \(2.0 \times 10^{-10}\) m. Equation Transcription: Text Transcription: 3.50x10^-13 2.0x10^-10

Problem 39E At what temperature is the root-mean-square speed of nitrogen molecules equal to the root-mean-square speed of hydrogen molecules at 20.0o 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 H2 is twice the molar mass of hydrogen atoms, and similarly for N2.)

Problem 40E Smoke particles in the air typically have masses of the order of 10-10 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 root-mean-square speed of Brownian motion for a particle with a mass of 3.00 X 10-16 kg in air at 300 K. (b) Would the root-mean-square speed be different if the particle were in hydrogen gas at the same temperature? Explain.

Problem 41E (a)How much heat does it take to increase the temperature of 2.50 mol of a diatomic ideal gas by 50.0 K near room temperature if the gas is held at constant volume? (b) What is the answer to the question in part (a) if the gas is monatomic rather than diatomic?

Problem 42E Perfectly rigid containers each hold n moles of ideal gas, one being hydrogen (H2) and the other being neon (Ne). If it takes 300 J of heat to increase the temperature of the hydrogen by 2.50o C, by how many degrees will the same amount of heat raise the temperature of the neon?

Problem 43E (a) Compute the specific heat at constant volume of nitrogen (N2) gas, and compare it with the specific heat of liquid water. The molar mass of N2 is 28.0 g/mol. (b) You warm 1.00 kg of water at a constant volume of 1.00 L from 20.0o C to 30.0o C in a kettle. For the same amount of heat, how many kilograms of 20.0o C air would you be able to warm to 30.0o C? What volume (in liters) would this air occupy at 20.0o C and a pressure of 1.00 atm? Make the simplifying assumption that air is 100% N2.

Problem 44E (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 J / kg ? K. Compare this with your calculation and on the actual role of vibrational motion.

(a) Use Eq. 18.28 to calculate the specific heat at constant volume of aluminum in units of \(J / \mathrm{kg} \cdot \mathrm{K}\). Consult the periodic table in Appendix D. (b) Compare the answer in part (a) with the value given in Table 17.3. Try to explain any disagreement between these two values. Equation Transcription: Text Transcription: J/kg.K

For a gas of nitrogen molecules \(\left(N_{2}\right)\), what must the temperature be if of all the molecules have speeds less than (a) 1500 \(\mathrm{m} / \mathrm{s}\); (c) 500 \(\mathrm{m} / \mathrm{s}\)? Use Table . The molar mass of \(N_{2}\) is 28.0 \(\mathrm{g} / \mathrm{mol}\). Equation Transcription: Text Transcription: (N_2) m/s m/s N_2 g/mol

Problem 47E 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 rms.

Problem 49E 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 < 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 < p < p2.

Problem 50E Puffy cumulus clouds, which are made of water droplets, occur at lower altitudes in the atmosphere. Wispy cirrus clouds, which are made of ice crystals, occur only at higher altitudes. Find the altitude y (measured from sea level) above which only cirrus clouds can occur. On a typical day and at altitudes less than 11 km, the temperature at an altitude y is given by T = T0 ? ?y, where T0 = 15.0°C and ? = 6.0 C0/1000 m.

Problem 51E The atmosphere of the planet Mars is 95.3% carbon dioxide (CO?) and about 0.03% water vapour. The atmospheric pressure is only about 600 Pa, and the surface temperature varies from?30°C to ?100°C. 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 physics lecture room at 1.00 atm and 27.0\(^\circ\)C has a volume of \(216 m^{3}\). (a) Use the ideal-gas law to estimate the number of air molecules in the room. Assume that all of the air is \(N_{2}\). Calculate (b) the particle density-that is, the number of \(N_{2}\) molecules per cubic centimeter-and (c) the mass of the air in the room.

CP BIO 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 \(22^{\circ} \mathrm{C}\) and 1.2 \(\mathrm{kg} / \mathrm{m}^{3}\), respectively, at the bottom of the mountain. (Note that the result of Example 18.4 doesn’t apply, since the expression derived in that example accounts for the variation of air density with altitude and we are told to ignore that in this problem.) (b) If you took a \(\text { 0.50-L }\) 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? Equation Transcription: Text Transcription: 22degC kg/m^3 0.50-L

Problem 54P CP BIO 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.)

Problem 55P CP 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 sur-rounding air is at 15.0°C, 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 15.0°C and atmospheric pressure is 1.23 kg/m3.

(a) Use Eq. (18.1) to estimate the change in the volume of a solid steel sphere of volume when the temperature and pressure increase from \(21^{\circ} \mathrm{C}\) and \(1.013 \times 10^{5}\) Pa to \(42^{\circ} \mathrm{C}\) and \(2.10 \times 10^{7}\) Pa. (Hint: Consult Chapters 11 and 17 to determine the values of \(\beta\) and ) (b) In Example the change in volume of an \(\text { 11-L }\) steel scuba tank was ignored. Was this a good approximation? Explain. Equation Transcription: Text Transcription: 21degC 1.013x10^5 42degC 2.10x10^7 Beta 11-L

Problem 57P 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 and the temperature is 22.0°C. The temperature of the gas remains constant as it is partially emptied out of the tank, until the gauge pressure is 2.50 × 105 Pa. Calculate the mass of propane that has been used.

CP 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 27.0\(^\circ\)C at the surface and 7.0\(^\circ\)C at the bottom. The density of seawater is \(1030 kg/m^{3}\). (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?

Problem 59P 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 oC? (b) Calculate the surface density in Titan’s atmosphere in molecules per cubic meter. (c) Compare the density of Titan’s surface atmosphere to the density of earth’s atmosphere at 22o C. Which body has denser atmosphere?

Problem 60P Pressure on Venus. At the surface of Venus the aver-age temperature is a balmy 460o C due to the greenhouse effect (global warming!), the pressure is 92 earth-atmospheres, and the acceleration due to gravity is 0.894 gearth. 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?

Problem 61P An automobile tire has a volume of 0.0150 m3 on a cold day when the temperature of the air in the tire is 5.0o C 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 45.0o C and the volume has risen to 0.0159 m3. What then is the gauge pressure?

Problem 62P A flask with a volume of 1.50 L, provided with a stop-cock, contains ethane gas (C2H6) at 300 K and atmospheric pressure (1.013 X 105 Pa). 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?

Problem 63P CP A balloon of volume 750 m3 is to be filled with hydrogen at atmospheric pressure (1.01 X 105 Pa). (a) If the hydrogen is stored in cylinders with volumes of 1.90 m3 at a gauge pressure of 1.20 X 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 15.0o C? The molar mass of hydro-gen (H2) is 2.02 g/mol. The density of air at 15.0o C 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 15.0o C?

Problem 64P A vertical cylindrical tank contains 1.80 mol of an ideal gas under a pressure of 0.500 atm at 20.0°C. 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?

CP A large tank of water has a hose connected to it, as shown in Fig. P18.65. The tank is sealed at the top and has compressed air between the water surface and the top. When the water height has the value , the absolute pressure of the compressed air is \(4.20 \times 10^{5}\) Pa. Assume that the air above the water expands at constant temperature, and take the atmospheric pressure to be \(1.00 \times 10^{5}\) 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, decreases. Calculate the speed of flow for \(h=3.00\) m and for \(h=2.00\) m. (c) At what value of does the flow stop? Equation Transcription: Text Transcription: 4.20x10^5 1.00x10^5 h=3.50 h=3.00 h=2.00

Problem 66P BIO A person at rest inhales 0.50 L of air with each breath at a pressure of 1.00 atm and a temperature of 20.0o C. 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 20.0o C. 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.

Problem 67P BIO 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.

Problem 68P The size of an oxygen molecule is about 2.0 X 10-10 m. Make a rough estimate of the pressure at which the finite volume of the molecules should cause noticeable deviations from ideal-gas behavior at ordinary temperatures (T = 300 K).

Problem 69P You have two identical containers, one containing gas A and the other gas B. The masses of these molecules are mA = 3.34 X 10-27 kg and mB = 5.34 X 10-26 kg. Both gases are under the same pressure and are at 10.0o C. (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?

Problem 70P Insect Collisions. A cubical cage 1.25 m on each side contains 2500 angry bees, each flying randomly at 1.10 m/s. We can model these insects as spheres 1.50 cm in diameter. On the average, (a) how far does a typical bee travel between collisions, (b) what is the average time between collisions, and (c) how many collisions per second does a bee make?

Problem 71P 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 22.0o C. 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?

Problem 72P CP (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 earth’s 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 = 15.0o C 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.

CP, CALC The Lennard-Jones Potential. A commonly used potential-energy function for the interaction of two molecules (see Fig. ) is the Lennard-Jones potential: \(U(r)=U_{0}\left[\left(\frac{R_{0}}{r}\right)^{12}-2\left(\frac{R_{0}}{r}\right)^{6}\right]\) where is the distance between the centers of the molecules and \(U_{0}\) and \(R_{0}\) are positive constants. The corresponding force \(F(r)\) is given in Eq. (14.26). (a) Graph \(U(r)\) and \(F(r)\) versus . (b) Let \(r_{1}\) be the value of at which \(U(r)=0\), and let \(r_{2}\) be the value of at which \(F(r)=0\). Show the locations of \(r_{1}\) and \(r_{2}\) on your graphs of \(U(r)\) and \(F(r)\). Which of these values represents the equilibrium separation between the molecules? (c) Find the values of \(r_{1}\) and \(r_{2}\) in terms of \(R_{0}\), and find the ratio \(r_{1} / r_{2}\). (d) If the molecules are located a distance \(r_{2}\) apart [as calculated in part (c)], how much work must be done to pull them apart so that \(r \rightarrow \infty\)? Equation Transcription: Text Transcription: U(r)=U_0[(R_0/r)^12-2(R_0/r)^6] U_0 R_0 F(r) U(r) F(r) r_1 U(r)=0 r_2 F(r)=0 r_1 r_2 U(r) F(r) r_1 r_2 R_0 r_1/r_2 r_2 r>infinity

Problem 74P (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 X 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

Problem 75P The speed of propagation of a sound wave in air at 27°C is about 350 m/s. Calculate, for comparison, (a) ?rms for nitrogen molecules and (b) the rms value of ?x, at this temperature. The molar mass of nitrogen (N2) is 28.0 g/mol.

Hydrogen on the Sun. The surface of the sun has a temperature of about 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 \(1.67 \times 10^{-27}\) 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 is the sun's mass, its radius, and the gravitational constant (see Example of Section 13.3). Use the data in Appendix to calculate this escape speed. (c) Can appreciable quantities of hydrogen escape from the sun? Can any hydrogen escape? Explain. Equation Transcription: Text Transcription: 1.67x10^-27 (2GM/R)^1/2

CP (a) Show that a projectile with mass can "escape" from the surface of a planet if it is launched vertically upward with a kinetic energy greater than \(m g R_{p}\), where is the acceleration due to gravity at the planet's surface and \(R_{p}\) is the planet's radius. Ignore air resistance. (See Problem 18.76.) (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 \(\mathrm{g} / \mathrm{mol}\)) equal that required to escape? What about a hydrogen molecule (molar mass 2.02 \(\mathrm{g} / \mathrm{mol}\)? (c) Repeat part (b) for the moon, for which \(g=1.63 \mathrm{~m} / \mathrm{s}^{2}\) and \(R_{p}\) = 1740 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. Equation Transcription: Text Transcription: mgR_p R_p g/mol g/mol g=1.63 m/s^2 R_p

Planetary Atmospheres. (a) The temperature near the top of Jupiter’s multicolored cloud layer is about 140 K. The temperature at the top of the earth’s 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 18.76). (b) Hydrogen gas is \(\left(H_{2}\right)\) a rare element in the earth’s atmosphere. In the atmosphere of Jupiter, by contrast, 89% of all molecules are \(\mathrm{H}_{2}\). Explain why, using your results from part (a). (c) Suppose an astronomer claims to have discovered an oxygen \(\left(\mathrm{O}_{2}\right)\) 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 2,400 \(\mathrm{kg} / \mathrm{m}^{3}\), and a surface temperature of about 200 K. Equation Transcription: Text Transcription: (H_2) H_2 (O_2) kg/m^3

Problem 79P (a) For what mass of molecule or particle is ?rms equal to 1.00 mm/s at 300 K? (b) If the particle is an ice crystal, how many molecules does it contain? The molar mass of water is 18.0 g/mol (c) Calculate the diameter of the particle if it is a spherical piece of ice. Would it be visible to the naked eye?

Problem 80P In describing the heat capacities of solids in Section 18.4, we stated that the potential energy a harmonic oscillator averaged over one period of the motion is equal to the kinetic energy averaged over one period. Prove this result using Eqs. (14.13) and (14.15) for the position and velocity of a simple harmonic oscillator. For simplicity, assume that the initial position rind velocity make the phase angle ? up equal to zero. (Hint: Use the trigonometric identities cos2(?) = [1 + cos(2?))/2 and sin2(?) = [1 ? cos(2?))/2. What is the average value of cos(2?t) overone period?) Displacement of simple harmonic Oscillator: Velocity at a Displacement x:

Problem 81P 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.

(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 \(\left(\mathrm{O}_{2}\right)\) for rotation about either the \(y-\) or \(z \text {-axis }\) shown in Fig. 18.18b. Treat the molecule as two massive points (representing the oxygen atoms) separated by a distance of \(1.21 \times 10^{-10}\) m. The molar mass of oxygen atoms is 16.0 \(\mathrm{g} / \mathrm{mol}\). (c) Find the rms angular velocity of rotation of an oxygen molecule about either the \(y-\) or \(z \text {-axis }\) shown in Fig. 18.18b. How does your answer compare to the angular velocity of a typical piece of rapidly rotating machinery (10,000 \(\mathrm{rev} / \mathrm{min}\))? Equation Transcription: Text Transcription: (O_2) y- z-axis 1.21x10^-10 g/mol y- z-axis rev/min

For each polyatomic gas in Table , compute the value of the molar heat capacity at constant volume, \(C_{V}\), on the assumption that there is no vibrational energy. Compare with the measured values in the table, and compute the fraction of the total heat capacity that is due to vibration for each of the three gases. (Note: \(\mathrm{CO}_{2}\) is linear; \(\mathrm{SO}_{2}\) and \(\mathrm{H}_{2} \mathrm{~S}\) are not. Recall that a linear polyatomic molecule has two rotational degrees of freedom, and a nonlinear molecule has three.) Equation Transcription: Text Transcription: C_V CO_2 SO_2 H_2S

CALC (a) Show that \(\int_{0}^{\infty} f(v) d v=1\), where \(f(v)\) is the Maxwell-Boltzmann distribution of Eq. (18.32). (b) In terms of the physical definition of \(f(v)\), explain why the integral in part (a) must have this value. Equation Transcription: Text Transcription: integ^infinity_0f(v)dv=1 f(v) f(v)

CALC Calculate the integral in Eq. (18.31), \(\int_{0}^{\infty} v^{2} f(v) d v\), and compare this result to \(\left(v^{2}\right)_{a v}\) as given by Eq. (18.16). (Hint: You may use the tabulated integral \(\int_{0}^{\infty} x^{2 n} e^{-\alpha x^{2}} d x=\frac{1 \cdot 3 \cdot 5 \cdots(2 n-1)}{2^{n+1} \alpha^{n}} \sqrt{\frac{\pi}{\alpha}}\) where \(\) is a positive integer and \(\alpha\) is a positive constant.) Equation Transcription: Text Transcription: integral_0^infinity v^2 f(v)dv (v^2)av integral_0^infinity x^2n e^-alpha x^2 dx=1.3.5...(2n-1)/2^n+1 alpha^n sqrt pi/alpha n alpha

Calculate the integral in Eq. (18.30), \(\int_{0}^{\infty} v f(v) d v\) and compare this result to \(v_{\mathrm{av}}\) as given by Eq. (18.35). (Hint: Make the change of variable \(v^{2}=x\) and use the tabulated integral \(\int_{0}^{\infty} x^{n} e^{-\alpha x} d x=\frac{n !}{\alpha^{n+1}}\) where n is a positive integer and \(\alpha\) is a positive constant.)

CALC (a) Explain why in a gas of \(N\) molecules, the number of molecules having speeds in the finite interval \(v\) to \(v+\Delta v\) is \(\Delta N=N \int_{v}^{v+\Delta v} f(v) d v\). (b) If \(\Delta v\) is small, then \(f(v)\) is approximately constant over the interval and \(\Delta N \approx N f(v) \Delta v\). For oxygen gas ( \(\mathrm{O}_{2}\), molar mass 32 \(\mathrm{gm} / \mathrm{mol}\) at \(T=300 \mathrm{~K}\), use this approximation to calculate the number of molecules with speeds within \(\Delta v=20 \mathrm{~m} / \mathrm{s}\) of \(v_{m p}). Express your answer as a multiple of (c) Repeat part (b) for speeds within \(\Delta v=20 \mathrm{~m} / \mathrm{s}\) of \(7 v_{m p}\) (d) Repeat parts (b) and (c) for a temperature of . (e) Repeat parts (b) and (c) for a temperature of . (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. ? Equation Transcription: Text Transcription: N v v+ delta v delta N=N integral _v^v+v f (v)dv delta v f(v) delta N approx N f(v) delta v O_2 T=300 K delta v=20 m/s v_mp delta v=20 m/s 7 v_mp gm/mol deltav=20 m/s v_mp

Problem 88P 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 20.0 ? C is 2.34 X 103 Pa. If the air temperature is 20.0o C 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 m 3 of air? (The molar mass of water is 18.0 g/mol. Assume that water vapor can be treated as an ideal gas.)

Problem 89P The Dew Point. The vapor pressure of water (see Problem) decreases as the temperature decreases. If the amount of water vapor in the air is kept constant as the air is cooled, a temperature is reached, called the dew point, at which the partial pressure and vapor pressure coincide and the vapor is saturated. If the air is cooled further, vapor condenses to liquid until the partial pressure again equals the vapor pressure at that temperature The temperature in a room is 30.0°C. A meteorologist cools a metal can by gradually adding cold water. When the can temperature reaches 16.0°C, water droplets form on its outside surface. What is the relative humidity at the 30.0°C air in the room? The table lists the vapor pressure of water at various temperatures: Temperature (°C) Vapour Pressure (Pa) 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 Problem: 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 20.0°C is 2.34 × 103 Pa If the air temperature is 20.0°C 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.)

Problem 90P Altitude at Which Clouds Form. On a spring day in the midwestern United States, the air temperature at the surface is 28.0°C. Puffy cumulus clouds form at an altitude where the air temperature equals the dew point (see Problem). 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 it? the relative humidity at the surface is 35% and 80%? (Hint: use the table in problem ) Problem 1: The Dew Point. The vapor pressure of water (see Problem) decreases as the temperature decreases. If the amount of water vapor in the air is kept constant as the air is cooled, a temperature is reached, called the dew point, at which the partial pressure and vapor pressure coincide and the vapor is saturated. If the air is cooled further, vapor condenses to liquid until the partial pressure again equals the vapor pressure at that temperature The temperature in a room is 30.0°C. A meteorologist cools a metal can by gradually adding cold water. When the can temperature reaches 16.0°C, water droplets form on its outside surface. What is the relative humidity at the 30.0°C air in the room? The table lists the vapor pressure of water at various temperatures: Temperature (°C) Vapour Pressure (Pa) 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

Prove that f(?) as given by Eq. (18.33) is maximum for ? = kT. Use this result to obtain Eq. (18.34).