Chapter 18

A \(20.0-\mathrm{L}\) tank contains 0.225 \(\mathrm{kg}\) of helium at \(18.0^{\circ} \mathrm{C}\) . The molar mass of helium is 4.00 \(\mathrm{g} / \mathrm{inol}\) (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 \(\mathrm{L}\) , under a pressure of 1.30 atm and at a temperanure 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 \(\mathrm{g} / \mathrm{mol.}\)

A cylindrical tank has a tight-fitting piston that allows the volume of the tank to be changed. The tank originally contains 0.110 \(\mathrm{m}^{3}\) of air at pressure of 3.40 atm. The piston is slowly pulled out until the volume of the gas is increased to 0.390 \(\mathrm{m}^{3} .\) If the temperature remains constant, what is the final value of the pressure?

A \(3.00-\mathrm{L}\) tank contains air at 3.00 \(\mathrm{atm}\) and \(20.0^{\circ} \mathrm{C} .\) The tank is sealed and cooled until the pressure is 1.00 atm. (a) What is the temperature then in degrees Celsius? Assume 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 vohme when the pressure again becomes 3.00 \(\mathrm{atm} ?\)

You have several identical balloons. You experimentally determine that a balloon will break if its volume exceeds 0.900 \(\mathrm{L}\). The pressure of the gas inside the balloon equals air pressure \((1.00 \mathrm{atm} \). (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 499 \(\mathrm{cm}^{3}\) of air at atmospheric pressure \(\left(1.01 \times 10^{5} \mathrm{Pa}\right)\) and a temperature of \(27.0^{\circ} \mathrm{C}\) . At the end of the stroke, the air has been compressed to a volume of 46.2 \(\mathrm{cm}^{3}\) and the gauge pressure has increased to \(2.72 \times 10^{5}\) Pa. Compute the final temperature.

A welder using a tank of volume 0.0750 \(\mathrm{m}^{3}\) fills it with oxygen \((\text { molar mass } 32.0 \mathrm{g} / \mathrm{mol})\) at a gauge pressure of \(3.00 \times 10^{5} \mathrm{Pa}\) and tenperature 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 \(1.80 \times 10^{5}\) 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 in \(^{3}\) of nitrogen gas at \(27^{\circ} \mathrm{C}\) and \(1.50 \times 10^{5} \mathrm{Pa}\) (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 \(\mathrm{m}\) long and 90.0 \(\mathrm{cm}\) in diameter is to be filled with pure oxygen at \(22.0^{\circ} \mathrm{C}\) to sture in a space station. To hold as thuch gas as possible, the absolute pressure of the oxygen will be 21.0 atm. The molar mass of oxygen is 32.0 \(\mathrm{g} / \mathrm{mol}\) . (a) How many tholes of oxygen does this canister hold? (b) For someone lifting this canister, by how many kilo-grams does this gas increase the mass to be lifted?

The gas inside a balloon will always have a pressure nearly equal to atnospheric 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 \(\mathrm{L}\) at a tenperature of \(19.0^{\circ} \mathrm{C} .\) What is the volume of the balloon if you cool it to the boiling point of liguid nitrogen \((77.3 \mathrm{K}) ?\)

For carbon dioxide gas \(\left(\mathrm{CO}_{2}\right),\) the constants in the van der Waals equation are \(a=0.364 \mathrm{J} \cdot \mathrm{m}^{3} / \mathrm{mol}^{2}\) and \(b=4.27 \times 10^{-5} \mathrm{m}^{3} / \mathrm{mol} .\) (a) If 1.00 \(\mathrm{mol}\) of \(\mathrm{CO}_{2} \mathrm{gas}\) at 350 \(\mathrm{K}\) is confined to a volume of 400 \(\mathrm{cm}^{3}\) , 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 ideat-gas equation result? (c) The gas is kept at the same temperature as it expands to a volume of 4000 \(\mathrm{cm}^{3} .\) Repeat the calculations of parts (a) and (b). (d) Explain how your calculations show that the van der Waals equation is equivalent to the ideat-gas equation if \(n / V\) is small.

The total lung volume for a typical physics student is 6.00 \(\mathrm{L}\) . A physics student fills her lungs with air an absolute pressure of 1.00 atm. Then, holding her breath, she compresses her chest cavity, decreasing her lung volume to 5.70 \(\mathrm{L}\) . What is the pressure of the air in her lungs then? Assume that temperature of the air remains constant.

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

A metal tank with volume 3.10 L will burst if the absolute pressure of the gas it contains exceeds 100 atm. (a) If 11.0 mol of an ideal gas is put into the tank at a temperature of \(23.0^{\circ} \mathrm{C},\) 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 \(\mathrm{m}\) . (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 \(100.0^{\circ} \mathrm{C}\) ?

Modern vacuum pumps make it easy to attain pressures of the order of \(10^{-13}\) atm in the laboratory. (a) At a pressure of \(9.00 \times 10^{-14}\) am and an ordinary temperature of \(300.0 \mathrm{K},\) how many inolecules 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?

How many moles are in a \(1.00-\mathrm{kg}\) bottle of water? How many nolecules? The inolar mass of water is 18.0 \(\mathrm{g} / \mathrm{mol}\).

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 \(\mathrm{g} / \mathrm{mol}\) (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?

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 g/mol) of each element under the chemical symbol for that element)

(a) A process called gaseous diffusion is often used to separate isotopes of uranium-that is, atoms of the elements that have different inasses, 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 \(^{238} \mathrm{UF}_{6}\) inolecules might be separated by diffusion. (b) The molar masses for \(^{235} \mathrm{UF}_{6}\) and \(^{238} \mathrm{UF}_{6}\) molecules are 0.349 \(\mathrm{kg} / \mathrm{mol}\) and \(0.352 \mathrm{kg} / \mathrm{nol},\) respectively. If uranium hexafluoride 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?

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 \(50^{\circ} \mathrm{C}\) while the gas in box \(B\) is at \(10^{\circ} \mathrm{C}\) . 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 \(B\) . There are more molecules in \(A\) than in \(B .(\mathrm{c}) A\) and \(B\) cannot 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.

The conditions of standard temperature and pressure (STP) are a temperature of \(0.00^{\circ} \mathrm{C}\) and a pressure of 1.00 \(\mathrm{atm}\) . (a) How many liters does 1.00 \(\mathrm{mol}\) of any ideal gas occupy at STP? (b) For a scientist on Venus, an absolute pressure of 1 Venusian-atmosphere is 92 Earth- atmospheres. Of course she would use the Venusian-atmosphere to define STP. Assuming she kept the same temperature, how many liters would 1 mole of ideal gas occupy on Venus?

(a) A deuteron, \(^{2}_{1} \mathrm{H},\) 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 \(\mathrm{K}\). What is the ms 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 ms speed equal to 0.10\(c ?\)

The atmosphere of Mars is mostly \(\mathrm{CO}_{2}\) (molar mass 44.0 \(\mathrm{g} / \mathrm{mol} )\) under a pressure of 650 \(\mathrm{Pa}\) , which we shall assume remains constant. In many places the temperature varies from \(0.0^{\circ} \mathrm{C}\) in summer to \(-100^{\circ} \mathrm{C}\) in winter. Over the course of a martian year, what are the ranges of \((\mathrm{a})\) the rms speeds of the \(\mathrm{CO}_{2}\) molecules, and (b) the density (in mollm') of the atmosphere?

(a) Oxygen \(\left(\mathrm{O}_{2}\right)\) has a molar mass of 32.0 \(\mathrm{g} / \mathrm{mol}\) . What is the average translational kinetic energy of an oxygen molecule at a temperature of 300 \(\mathrm{K} ?\) (b) What is the average value of the square at a 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 \(\mathrm{m}\) 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 aver- age force per unit area? (g) How many oxygen molecules traveling at this speed are necessary to produce an average pressure of 1 \(\mathrm{atm} ?\) (h) Compute the number of oxygen molecules that are actually contained in a vessel of this size at 300 \(\mathrm{K}\) 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 \(3.50 \times 10^{-13}\) atm and a temperature of 300 \(\mathrm{K}\) . (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 \(2.0 \times 10^{-10} \mathrm{m}\).

At what temperature is the root-mean-square speed of nitrogen molecules equal to the root-mean-square speed of hydrogen molecules at \(20.0^{\circ} \mathrm{C} ?\) (Hint 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 \(\mathrm{N}_{2}\) .)

(a) Compute the specific heat capacity at constant volume of nitrogen \(\left(\mathrm{N}_{2}\right)\) gas, and compare with the specific heat capacity of liquid water. The molar mass of \(\mathrm{N}_{2}\) is 28.0 \(\mathrm{g} / \mathrm{mol}\) . (b) You warm 1.00 \(\mathrm{kg}\) of water at a constant volume of 1.00 \(\mathrm{L}\) from \(20.0^{\circ} \mathrm{C}\) to \(30.0^{\circ} \mathrm{C}\) in a kettle. For the same amount of beat, how many kilograms of \(20.0^{\circ} \mathrm{C}\) air would you be able to warm to \(30.0^{\circ} \mathrm{C} ?\) What volume (in liters) would this air occupy at \(20.0^{\circ} \mathrm{C}\) and a pressure of 1.00 atm? Make the simplifying assumption that air is 100\(\% \mathrm{N}_{2}\) .

(a) Calculate the specific heat capacity 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 capacity of water vapor at low pressures is about 2000 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) . Compare this with your calculation and comment on the actual role of vibrational motion.

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 \(\mathrm{km}\), the temperature at an altitude \(y\) is given by \(T=T_{0}-\alpha y,\) where \(T_{0}=15.0^{\circ} \mathrm{C}\) and \(\alpha=6.0 \mathrm{C}^{\circ} / 1000 \mathrm{m} .\)

A physicist places a piece of ice at \(0.00^{\circ} \mathrm{C}\) and a beaker of water at \(0.00^{\circ} \mathrm{C}\) inside a glass box and closes the lid of the box. All the air is then removed from the box. If the ice, water, and beaker are all maintained at a temperature of \(0.00^{\circ} \mathrm{C},\) describe the final equilibrium state inside the box.

A cylinder 1.00 \(\mathrm{m}\) tall with inside diameter 0.120 \(\mathrm{m}\) is used to hold propane gas (molar mass 44.1 \(\mathrm{g} / \mathrm{mol}\) ) for use in a barbecue. It is initially filled with gas until the gauge pressure is \(1.30 \times 10^{6} \mathrm{Pa}\) and the temperature is \(22.0^{\circ} \mathrm{C} .\) The temperature of the gas remains constant as it is partially emptied out of the tank, until the gauge pressure is \(2.50 \times 10^{5}\) Pa. Calculate the mass of propane that has been used.

During a test dive in \(1939,\) prior to being accepted by the U.S. Navy, the submarine Squalus sank at a point where the depth of water was 73.0 \(\mathrm{m}\) . The temperature at the surface was \(27.0^{\circ} \mathrm{C},\) and at the bottom it was \(7.0^{\circ} \mathrm{C}\) . The density of seawater is 1030 \(\mathrm{kg} / \mathrm{m}^{3} .\) (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 \(\mathrm{m}\) 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 \(\mathrm{K}\) . (a) What is the surface temperature in \(^{\circ} \mathrm{C}\) ? (b) Calculate the surface density in Titar's atmosphere in molecules per cubic meter (c) Compare the density of Titan's surface atmosphere to the density of Earth's atmosphere at \(22^{\circ} \mathrm{C}\) . Which body has denser atmosphere?

A flask with a volume of 1.50 \(\mathrm{L}\) , provided with a stopock, contains ethane gas \(\left(\mathrm{C}_{2} \mathrm{H}_{6}\right)\) at 300 \(\mathrm{K}\) 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 \(380 \mathrm{K},\) 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 750 \(\mathrm{m}^{3}\) is to be filled with hydrogen at atmospheric pressure \(\left(1.01 \times 10^{5} \mathrm{Pa}\right) .\) (a) If the hydrogen is stored in cylinders with volumes of 1.90 \(\mathrm{m}^{3}\) at a gauge pressure of \(1.20 \times 10^{6} \mathrm{Pa},\) 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 the gas in the balloon and the surrounding air are both at \(15.0^{\circ} \mathrm{C}\) ? The molar mass of hydro\(\operatorname{gen}\left(\mathrm{H}_{2}\right)\) is 2.02 \(\mathrm{g} / \mathrm{mol}\) . The density of air at \(15.0^{\circ} \mathrm{C}\) and atmospheric pressure is 1.23 \(\mathrm{kg} / \mathrm{m}^{3}\) . See Chapter 14 for a discussion of buoyancy. (c) What weight could be supported if the balloon were filled with helium (molar mass 4.00 \(\mathrm{g} / \mathrm{mol}\) ) instead of hydrogen, again at \(15.0^{\circ} \mathrm{C} ?\)

A vertical cylindrical tank contains 1.80 \(\mathrm{mol}\) of an ideal gas under a pressure of 1.00 atm at \(20.0^{\circ} \mathrm{C}\) . The round part of the tank has a radius of \(10.0 \mathrm{cm},\) and the gas is supporting a piston that can move up and down in the cylinder without friction. (a) What is the mass of this piston? (b) How tall is the column of gas that is supporting the piston?

A person at rest inhales 0.50 \(\mathrm{L}\) 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 \(2,000 \mathrm{m}\) 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.

You have two identical containers, one containing gas \(A\) and the other gas \(B .\) The masses of these molecules are \(m_{A}=\) \(3.34 \times 10^{-27} \mathrm{kg}\) and \(m_{B}=5.34 \times 10^{-26} \mathrm{kg} .\) Both gases are under the same pressure and are at \(10.0^{\circ} \mathrm{C} .\) (a) Which molecules \((A \text { or } B)\) have greater translational kinetic energy per inolecule and \(\mathrm{ms}\) speeds? Now you want to raise the temperature of only one of these containers so that both gases will have the same rms speed. (b) 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?

A cubical cage 1.25 \(\mathrm{m}\) on each side contains 2500 angry bees, each flying randomly at 1.10 \(\mathrm{m} / \mathrm{s}\) . We can model these insects as spheres 1.50 \(\mathrm{cm}\) 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?

In the ideal-gas equation, the number of moles per volume \(n / V\) is simply equal to \(p / R T\) . In the van der Waals equation, solving for \(n / V\) in terms of the pressure \(p\) and temperature \(T\) is somewhat more involved. (a) Show the van der Waals equation can be written as $$\frac{n}{V}=\left(\frac{p+a n^{2} / V^{2}}{R T}\right)\left(1-\frac{b n}{V}\right)$$ (b) The van der Waals parameters for hydrogen sulfide gas \(\left(\mathrm{H}_{2} \mathrm{S}\right)\) are \(a=0.448 \mathrm{J} \cdot \mathrm{m}^{3} / \mathrm{mol}^{2}\) and \(b=4.29 \times 10^{-5} \mathrm{m}^{3} / \mathrm{mol}\) . Determine the number of moles per volume of \(\mathrm{H}_{2} \mathrm{S}\) gas at \(127^{\circ} \mathrm{C}\) and an absolute pressure of \(9.80 \times 10^{5} \mathrm{Pa}\) as follows: (i) Calculate a first approximation using the ideal-gas equation, \(n / V=p / R T\) . (ii) Substitute this approximation for \(n / V\) into the right-hand side of the equation in part (a). The result is a new, improved approximation for \(n / V\) . (iii) Substitute the new approximation for \(n / V\) into the right-hand side of the equation in (a). The result is a further improved approximation for \(n / V\) . (iv) Repeat step (iii) until successive approximations agree to the desired level of accuracy (in this case, to three significant figures). (c) Compare your final result in part (b) to the result \(p / R T\) obtained using the ideal-gas equation. Which result gives a larger value of \(n / V ?\)

A canister of 1.20 mol of nitrogen gas \((28.0 \mathrm{g} / \mathrm{mol})\) at \(25.0^{\circ} \mathrm{C}\) is left on Jupiter's satellite after completion of a future space mission. Europa has no appreciable atmosphere, and the acceleration due to gravity at its surface is 1.30 \(\mathrm{m} / \mathrm{s}^{2}\) . After some time, the canister springs a small leak, allowing molecules to escape through a small hole. What is the maximum height (in \(\mathrm{km}\) ) above Europa's surface that a \(\mathrm{N}_{2}\) molecule having speed equal to the rms speed will reach if it is shot straight up out of the hole in the canister? Ignore the variation in \(g\) with altitude,

You blow up a spherical balloon to a diameter of 50.0 \(\mathrm{cm}\) until the absolute pressure inside is 1.25 atm and the temperature is \(22.0^{\circ} \mathrm{C}\) . Assume that all the gas in \(\mathrm{N}_{2}\) is of molar mass 28.0 \(\mathrm{g} / \mathrm{mol}\) . (a) Find the mass of a single \(\mathrm{N}_{2}\) molecule. (b) How much translational kinetic energy does an average \(\mathrm{N}_{2}\) molecule have? (c) How many \(\mathrm{N}_{2}\) molecules are in this balloon? (d) What is the total translational kinetic energy of all the inolecules in the balloon?

The speed of propagation of a sound wave in air at \(27^{\circ} \mathrm{C}\) is about 350 \(\mathrm{m} / \mathrm{s}\) . Calculate, for comparison, (a) \(v_{\mathrm{ms}}\) 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 \(28.0 \mathrm{g} / \mathrm{mol} .\)

The surface of the sun has a temperature of about 5800 \(\mathrm{K}\) and consists largely of hydrogen atoms. (a) Find the rms speed of a hydrogen atom at this temperature. (The mass of a single hydrogen atom is 1.67 \(\times 10^{-27} \mathrm{kg} . )\) (b) The escape speed for a particle to leave the gravitational influence of the sun is given by \((2 G M / R)^{1 / 2}\) , where \(M\) is the sun's mass, \(R\) its radius, and \(G\) the gravitational constant (see Example 12.5 of Section \(12.3 ) .\) Use the data in Appendix \(\mathrm{F}\) to calculate this escape speed. (c) Can appreciable quantities of hydrogen escape from the sun? Can any hydrogen escape? Explain.

It is possible to make crystalline solids that are only one layer of atoms thick. Such "two-dimensional" crystals can be created by depositing atoms on a very flat surface. (a) If the atoms in such a two-dimensional crystal can move only within the plane of the crystal, what will be its molar heat capacity near room temperature? Give your answer as a multiple of \(R\) and in \(\mathrm{J} / \mathrm{nol} \cdot \mathrm{K}\) . (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 \(\mathrm{K}\) . (b) Calculate the moment of inertia of an oxygen molecule \(\left(\mathrm{O}_{2}\right)\) for rotation about either the \(y\) - or \(z\) -axis shown in Fig. 18.18 . Treat the molecule as two massive points (representing the oxygen atoms) separated by a distance of \(1.21 \times 10^{-10} \mathrm{m}\) . 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 \(z\) -axis shown in Fig. 18.15. How does your answer compare to the angular velocity of a typical piece of rapidly rotating machinery \((10,000 \mathrm{rev} / \mathrm{min}) ?\)

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 terperature, expressed as a percentage. The air is saturated when the humidity is 100\(\%\) . (a) The vapor pressure of water at \(20.0^{\circ} \mathrm{C}\) is \(2.34 \times 10^{3} \mathrm{Pa}\) . If the air temperature is \(20.0^{\circ} \mathrm{C}\) 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 \(\mathrm{m}^{3}\) of air? (The molar mass of water is 18.0 \(\mathrm{g} / \mathrm{mol}\) . Assume that water vapor can be treated as an ideal gas.)