Chapter 18

A 20.0 -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?

Helium gas with a volume of \(2.60 \mathrm{L},\) 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 \(\mathrm{g} / \mathrm{mol} .\)

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 a pressure of 0.355 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 \(3.00-\) L tank contains air at 3.00 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 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 \(\mathrm{Pa}\) and the temperature is typically \(253 \mathrm{K},\) with a \(\mathrm{CO}_{2}\) atmosphere), Venus (with an average temperature of 730 \(\mathrm{K}\) and pressure of 92 atm, with a \(\mathrm{CO}_{2}\) atmosphere), and Saturn's moon Titan (where the pressure is 1.5 atm and the temperature is \(-178^{\circ} \mathrm{C},\) with a \(\mathrm{N}_{2}\) atmosphere). (b) Compare each of these densities with that of the earth's atmosphere, which is 1.20 \(\mathrm{kg} / \mathrm{m}^{3} .\) 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 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 \(2.72 \times 10^{6}\) Pa. Compute the final temperature.

A welder using a tank of volume 0.0750 \(\mathrm{m}^{3}\) fills it with oxygen (molar mass 32.0 \(\mathrm{g} / \mathrm{mol} )\) at a gauge pressure of 3.00 \(\mathrm{x}\) \(10^{5} \mathrm{Pa}\) 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 \(1.80 \times 10^{5} \mathrm{Pa} .\) 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 \(7.50 \times 10^{3}\) 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 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 \(\mathrm{g} / \mathrm{mol} .\) (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 \((77.3 \mathrm{K}) ?\)

Deviations from the Ideal-Gas Equation. 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 mol of \(\mathrm{CO}_{2}\) 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 ideal-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 ideal-gas equation if \(n / V\) 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 \(1003^{\circ} \mathrm{C}\) and the pressure is 92 atm?

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

A metal tank with volume 3.10 \(\mathrm{L}\) 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 \(100.0^{\circ} \mathrm{C} ?\)

At an altitude of \(11,000 \mathrm{m}\) (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.)

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 \(300.0 \mathrm{K},\) 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?

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

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 \(\mathrm{nm}\) apart?

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.)

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, 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 \(\mathrm{kg} / \mathrm{mol}\) and \(0.352 \mathrm{kg} / \mathrm{mol},\) respectively. If uranium hexafluoride acts as an ideal gas, what is the ratio of the root-meansquare 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 \(B\) . (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 \(\mathrm{L}\) is initially evacuated. Then it is filled with 0.226 \(\mathrm{g}\) of \(\mathrm{N}_{2} .\) Assume that the pressure of the gas is low enough for the gas to obey the ideal-gas law to high degree of accuracy. If the root-mean-square speed of the gas molecules is \(182 \mathrm{m} / \mathrm{s},\) 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 \(\mathrm{K} .\) 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\(c ?\)

(a) Oxygen (O) 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 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 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 \(\mathrm{K}\) 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 \(\mathrm{g} / \mathrm{mol}\) ) 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 \(3.00 \times 10^{-16} \mathrm{kg}\) in air at 300 \(\mathrm{K}\) . (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 \(\mathrm{K}\) 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?

(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 \(\mathrm{g} / \mathrm{mol} .\) (b) The actual specific heat 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.

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

CP Blo 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.) If you took a \(0.50-\) 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?

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 \(\mathrm{m}\) in Lake Michigan (which is fresh water), what will be the volume at the surface of an \(\mathrm{N}_{2}\) bubble that occupied 1.0 \(\mathrm{mm}^{3}\) 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 cylinder 1.00 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 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?

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 (about 25 \(\mathrm{lb} / \mathrm{in.} .\) ). 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 \(1.50 \mathrm{L},\) provided with a stop cock, 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 \(490 \mathrm{K},\) with the stop cock 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 hydrogen \(\left(\mathrm{H}_{2}\right)\) 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 12 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 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 \(3.50 \mathrm{m},\) the absolute pressure \(p\) 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 \(h=3.50 \mathrm{m} ?\) (b) As water flows out of the tank, \(h\) decreases. Calculate the speed of flow for \(h=3.00 \mathrm{m}\) and for \(h=2.00 \mathrm{m} .\) (c) At what value of \(h\) does the flow stop?

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 2000 \(\mathrm{m}\) 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.

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

The size of an oxygen molecule is about 2.0 \(\times 10^{-10} \mathrm{m}\) 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 \((T=300 \mathrm{K})\) .

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 \((A\) 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 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 \(\mathrm{m}\) 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?

(a) Compute the increase in gravitational potential energy for a nitrogen molecule (molar mass 28.0 \(\mathrm{g} / \mathrm{mol} )\) for an increase in elevation of 400 \(\mathrm{m}\) 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 \(T=15.0^{\circ} \mathrm{C}\) could rise to an altitude of 400 \(\mathrm{m} ?\) Is it likely that it could do so without hitting any other molecules along the way? Explain.

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) \(v_{\mathrm{rms}}\) 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 \(\mathrm{g} / \mathrm{mol} .\)

Hydrogen on the Sun. 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 \(1.67 \times 10^{-27}\) kg. (b) The escape speed for a particle to leave the gravitational influence of the sun is given by \((2 G M / R)^{1 / 2},\) where \(M\) is the sun's mass, \(R\) its radius, and \(G\) the gravitational constant \((\) see Example 13.5 of Section 13.3\() .\) Use 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.

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

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 \(\mathrm{J} / \mathrm{mol} \cdot \mathrm{K}\) ) 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.