Chapter 5

Under the same conditions of temperature and pressure, why does \(1 \mathrm{~L}\) of moist air weigh less than \(1 \mathrm{~L}\) of dry air? In weather forecasts, an oncoming lowpressure front usually means imminent rainfall. Explain.

Air entering the lungs ends up in tiny sacs called alveoli. It is from the alveoli that oxygen diffuses into the blood. The average radius of the alveoli is \(0.0050 \mathrm{~cm}\) and the air inside contains 14 percent oxygen. Assuming that the pressure in the alveoli is 1.0 atm and the temperature is \(37^{\circ} \mathrm{C},\) calculate the number of oxygen molecules in one of the alveoli.

A student breaks a thermometer and spills most of the mercury (Hg) onto the floor of a laboratory that measures \(15.2 \mathrm{~m}\) long, \(6.6 \mathrm{~m}\) wide, and \(2.4 \mathrm{~m}\) high. (a) Calculate the mass of mercury vapor (in grams) in the room at \(20^{\circ} \mathrm{C}\). The vapor pressure of mercury at \(20^{\circ} \mathrm{C}\) is \(1.7 \times 10^{-6} \mathrm{~atm} .\) (b) Does the concentration of mercury vapor exceed the air quality regulation of \(0.050 \mathrm{mg} \mathrm{Hg} / \mathrm{m}^{3}\) of air? (c) One way to treat small quantities of spilled mercury is to spray sulfur powder over the metal. Suggest a physical and a chemical reason for this action.

The Chemistry in Action essay "Super Cold Atoms" in Section \(5.7 .\) describes the cooling of rubidium vapor to \(5.0 \times 10^{-8} \mathrm{~K}\). Calculate the root-mean-square speed and average kinetic energy of a \(\mathrm{Rb}\) atom at this temperature.

The atmosphere on Mars is composed mainly of carbon dioxide. The surface temperature is \(220 \mathrm{~K}\) and the atmospheric pressure is about \(6.0 \mathrm{mmHg}\). Taking these values as Martian "STP," calculate the molar volume in liters of an ideal gas on Mars.

The atmosphere on Venus is composed of 96.5 percent \(\mathrm{CO}_{2}, 3.5\) percent \(\mathrm{N}_{2},\) and 0.015 percent \(\mathrm{SO}_{2}\) by volume. Its standard atmospheric pressure is \(9.0 \times 10^{6} \mathrm{~Pa}\). Calculate the partial pressures of the gases in pascals.

Apply your knowledge of the kinetic theory of gases to the following situations. (a) Two flasks of volumes \(V_{1}\) and \(V_{2}\left(V_{2}>V_{1}\right)\) contain the same number of helium atoms at the same temperature. (i) Compare the root-mean-square (rms) speeds and average kinetic energies of the helium (He) atoms in the flasks. (ii) Compare the frequency and the force with which the He atoms collide with the walls of their containers. (b) Equal numbers of He atoms are placed in two flasks of the same volume at temperatures \(T_{1}\) and \(T_{2}\left(T_{2}>T_{1}\right) .\) (i) Compare the rms speeds of the atoms in the two flasks. (ii) Compare the frequency and the force with which the He atoms collide with the walls of their containers. (c) Equal numbers of He and neon (Ne) atoms are placed in two flasks of the same volume, and the temperature of both gases is \(74^{\circ} \mathrm{C}\). Comment on the validity of the following statements: (i) The rms speed of He is equal to that of Ne. (ii) The average kinetic energies of the two gases are equal. (iii) The rms speed of each He atom is \(1.47 \times 10^{3} \mathrm{~m} / \mathrm{s}\)

At what temperature will He atoms have the same \(u_{\mathrm{rms}}\) value as \(\mathrm{N}_{2}\) molecules at \(25^{\circ} \mathrm{C} ?\)

Estimate the distance (in nanometers) between molecules of water vapor at \(100^{\circ} \mathrm{C}\) and 1.0 atm. Assume ideal behavior. Repeat the calculation for liquid water at \(100^{\circ} \mathrm{C}\), given that the density of water is \(0.96 \mathrm{~g} / \mathrm{cm}^{3}\) at that temperature. Comment on your results. (Assume water molecule to be a sphere with a diameter of \(0.3 \mathrm{nm} .\) ) (Hint: First calculate the number density of water molecules. Next, convert the number density to linear density, that is, number of molecules in one direction.)

Which of the noble gases would not behave ideally under any circumstance? Why?

A relation known as the barometric formula is useful for estimating the change in atmospheric pressure with altitude. The formula is given by \(P=P_{0} e^{-g .1 h / R T},\) where \(P\) and \(P_{0}\) are the pressures at height \(h\) and sea level, respectively; \(g\) is the acceleration due to gravity \(\left(9.8 \mathrm{~m} / \mathrm{s}^{2}\right) ; \mathscr{M}\) is the average molar mass of air \((29.0 \mathrm{~g} / \mathrm{mol}) ;\) and \(R\) is the gas constant. Calculate the atmospheric pressure in atm at a height of \(5.0 \mathrm{~km}\), assuming the temperature is constant at \(5^{\circ} \mathrm{C}\) and \(P_{0}=1.0 \mathrm{~atm} .\)

An equimolar mixture of \(\mathrm{H}_{2}\) and \(\mathrm{D}_{2}\) effuses through an orifice (small hole) at a certain temperature. Calculate the composition (in mole fractions) of the gases that pass through the orifice. The molar mass of \(\mathrm{D}_{2}\) is \(2.014 \mathrm{~g} / \mathrm{mol}\)

A mixture of calcium carbonate \(\left(\mathrm{CaCO}_{3}\right)\) and magnesium carbonate \(\left(\mathrm{MgCO}_{3}\right)\) of mass \(6.26 \mathrm{~g}\) reacts completely with hydrochloric acid (HCl) to generate \(1.73 \mathrm{~L}\) of \(\mathrm{CO}_{2}\) at \(48^{\circ} \mathrm{C}\) and 1.12 atm. Calculate the mass percentages of \(\mathrm{CaCO}_{3}\) and \(\mathrm{MgCO}_{3}\) in the mixture.

A 6.11 -g sample of a Cu-Zn alloy reacts with HCl acid to produce hydrogen gas. If the hydrogen gas has a volume of \(1.26 \mathrm{~L}\) at \(22^{\circ} \mathrm{C}\) and \(728 \mathrm{mmHg}\), what is the percent of Zn in the alloy?

A stockroom supervisor measured the contents of a partially filled 25.0 -gallon acetone drum on a day when the temperature was \(18.0^{\circ} \mathrm{C}\) and atmospheric pressure was \(750 \mathrm{mmHg}\), and found that 15.4 gallons of the solvent remained. After tightly sealing the drum, an assistant dropped the drum while carrying it upstairs to the organic laboratory. The drum was dented and its internal volume was decreased to 20.4 gallons. What is the total pressure inside the drum after the accident? The vapor pressure of acetone at \(18.0^{\circ} \mathrm{C}\) is \(400 \mathrm{mmHg} .\)

In 2.00 min, \(29.7 \mathrm{~mL}\) of He effuse through a small hole. Under the same conditions of pressure and temperature, \(10.0 \mathrm{~mL}\) of a mixture of \(\mathrm{CO}\) and \(\mathrm{CO}_{2}\) effuse through the hole in the same amount of time. Calculate the percent composition by volume of the mixture.

Use the kinetic theory of gases to explain why hot air rises.

One way to gain a physical understanding of \(b\) in the van der Waals equation is to calculate the "excluded volume." Assume that the distance of closest approach between two similar atoms is the sum of their radii \((2 r) .\) (a) Calculate the volume around each atom into which the center of another atom cannot penetrate. (b) From your result in (a), calculate the excluded volume for 1 mole of the atoms, which is the constant \(b\). How does this volume compare with the sum of the volumes of 1 mole of the atoms?

Identify the gas whose root-mean-square speed is 2.82 times that of hydrogen iodide (HI) at the same temperature.

A gaseous hydrocarbon (containing C and H atoms) in a container of volume \(20.2 \mathrm{~L}\) at \(350 \mathrm{~K}\) and 6.63 atm reacts with an excess of oxygen to form \(205.1 \mathrm{~g}\) of \(\mathrm{CO}_{2}\) and \(168.0 \mathrm{~g}\) of \(\mathrm{H}_{2} \mathrm{O} .\) What is the molecular formula of the hydrocarbon?

(a) Show that the pressure exerted by a fluid \(P\) (in pascals) is given by \(P=h d g,\) where \(h\) is the column of the fluid in meters, \(d\) is the density in \(\mathrm{kg} / \mathrm{m}^{3},\) and \(g\) is the acceleration due to gravity \(\left(9.81 \mathrm{~m} / \mathrm{s}^{2}\right)\) (Hint: See Appendix 1.) (b) The volume of an air bubble that starts at the bottom of a lake at \(5.24^{\circ} \mathrm{C}\) increases by a factor of 6 as it rises to the surface of water where the temperature is \(18.73^{\circ} \mathrm{C}\) and the air pressure is 0.973 atm. The density of the lake water is \(1.02 \mathrm{~g} / \mathrm{cm}^{3}\). Use the equation in (a) to determine the depth of the lake in meters.

In 2012 , Felix Baumgartner jumped from a balloon roughly \(24 \mathrm{mi}\) above Earth, breaking the record for the highest skydive. He reached speeds of more than 700 miles per hour and became the first skydiver to exceed the speed of sound during free fall. The helium-filled plastic balloon used to carry Baumgartner to the edge of space was designed to expand to \(8.5 \times 10^{8} \mathrm{~L}\) in order to accommodate the low pressures at the altitude required to break the record. (a) Calculate the mass of helium in the balloon from the conditions at the time of the jump \((8.5 \times\) \(\left.10^{8} \mathrm{~L},-67.8^{\circ} \mathrm{C}, 0.027 \mathrm{mmHg}\right) .\) (b) Determine the volume of the helium in the balloon just before it was released, assuming a pressure of 1.0 atm and a temperature of \(23^{\circ} \mathrm{C}\).

Which of the following has a greater mass: a sample of air of volume \(V\) at a certain temperature \(T\) and pressure \(P\) or a sample of air plus water vapor having the same volume and at the same temperature and pressure?