Chapter 14

Hemoglobin has a molecular mass of \(64500 \mathrm{u}\). Find the mass (in \(\mathrm{kg}\) ) of one molecule of hemoglobin.

The active ingredient in the allergy medication Claritin contains carbon (C), hydrogen (H), chlorine (Cl), nitrogen (N), and oxygen (O). Its molecular formula is \(\mathrm{C}_{22} \mathrm{H}_{23} \mathrm{ClN}_{2} \mathrm{O}_{2} .\) The standard adult dosage utilizes \(1.572 \times 10^{19}\) molecules of this species. Determine the mass (in grams) of the active ingredient in the standard dosage.

The artificial sweetener NutraSweet is a chemical called aspartame \(\left(\mathrm{C}_{14} \mathrm{H}_{18} \mathrm{~N}_{2} \mathrm{O}_{5}\right)\). What is (a) its molecular mass (in atomic mass units) and (b) the mass (in \(\mathrm{kg}\) ) of an aspartame molecule?

Manufacturers of headache remedies routinely claim that their own brands are more potent pain relievers than the competing brands. Their way of making the comparison is to compare the number of molecules in the standard dosage. Tylenol uses \(325 \mathrm{mg}\) of acetaminophen \(\left(\mathrm{C}_{8} \mathrm{H}_{9} \mathrm{NO}_{2}\right)\) as the standard dose, while Advil uses \(2.00 \times 10^{2} \mathrm{mg}\) of ibuprofen \(\left(\mathrm{C}_{13} \mathrm{H}_{18} \mathrm{O}_{2}\right)\). Find the number of molecules of pain reliever in the standard doses of (a) Tylenol and (b) Advil.

A mass of \(135 \mathrm{~g}\) of a certain element is known to contain \(30.1 \times 10^{23}\) atoms. What is the element?

A runner weighs \(580 \mathrm{~N}\) (about \(130 \mathrm{lb}\) ), and \(71 \%\) of this weight is water. (a) How many moles of water are in the runner's body? (b) How many water molecules \(\left(\mathrm{H}_{2} \mathrm{O}\right)\) are there?

A cylindrical glass of water \(\left(\mathrm{H}_{2} \mathrm{O}\right)\) has a radius of \(4.50 \mathrm{~cm}\) and a height of \(12.0 \mathrm{~cm} .\) The density of water is \(1.00 \mathrm{~g} / \mathrm{cm}^{3}\). How many moles of water molecules are contained in the glass?

A bicycle tire whose volume is \(4.1 \times 10^{-4} \mathrm{~m}^{3}\) has a temperature of \(296 \mathrm{~K}\) and an absolute pressure of \(4.8 \times 10^{5} \mathrm{~Pa}\). A cyclist brings the pressure up to \(6.2 \times 10^{5} \mathrm{~Pa}\) without changing the temperature or volume. How many moles of air must have been pumped into the tire?

It takes \(0.16 \mathrm{~g}\) of helium (He) to fill a balloon. How many grams of nitrogen \(\left(\mathrm{N}_{2}\right)\) would be required to fill the balloon to the same pressure, volume, and temperature?

An ideal gas at \(15.5^{\circ} \mathrm{C}\) and a pressure of \(1.72 \times 10^{5} \mathrm{~Pa}\) occupies a volume of \(2.81 \mathrm{~m}^{3}\). (a) How many moles of gas are present? (b) If the volume is raised to \(4.16 \mathrm{~m}^{3}\) and the temperature raised to \(28.2{ }^{\circ} \mathrm{C}\), what will be the pressure of the gas?

\(\mathbf{1}\) An ideal gas at \(15.5^{\circ} \mathrm{C}\) and a pressure of \(1.72 \times 10^{5} \mathrm{~Pa}\) occupies a volume of \(2.81 \mathrm{~m}^{3}\) (a) How many moles of gas are present? (b) If the volume is raised to \(4.16 \mathrm{~m}^{3}\) and the temperature raised to \(28.2{ }^{\circ} \mathrm{C},\) what will be the pressure of the gas?

Oxygen for hospital patients is kept in special tanks, where the oxygen has a pressure of 65.0 atmospheres and a temperature of \(288 \mathrm{~K}\). The tanks are stored in a separate room, and the oxygen is pumped to the patient's room, where it is administered at a pressure of 1.00 atmosphere and a temperature of \(297 \mathrm{~K}\). What volume does \(1.00 \mathrm{~m}^{3}\) of oxygen in the tanks occupy at the conditions in the patient's room?

A Goodyear blimp typically contains \(5400 \mathrm{~m}^{3}\) of helium (He) at an absolute pressure of \(1.1 \times 10^{5} \mathrm{~Pa}\). The temperature of the helium is \(280 \mathrm{~K}\). What is the mass (in \(\mathrm{kg}\) ) of the helium in the blimp?

A clown at a birthday party has brought along a helium cylinder, with which he intends to fill balloons. When full, each balloon contains \(0.034 \mathrm{~m}^{3}\) of helium at an absolute pressure of \(1.2 \times 10^{5} \mathrm{~Pa}\). The cylinder contains helium at an absolute pressure of \(1.6 \times 10^{7} \mathrm{~Pa}\) and has a volume of \(0.0031 \mathrm{~m}^{3}\). The temperature of the helium in the tank and in the balloons is the same and remains constant. What is the maximum number of balloons that can be filled?

In a diesel engine, the piston compresses air at \(305 \mathrm{~K}\) to a volume that is onesixteenth of the original volume and a pressure that is 48.5 times the original pressure. What is the temperature of the air after the compression?

A \(0.030-\mathrm{m}^{3}\) container is initially evacuated. Then, \(4.0 \mathrm{~g}\) of water is placed in the container and, after some time, all the water evaporates. If the temperature of the water vapor is \(388 \mathrm{~K}\), what is its pressure?

On the sunlit surface of Venus, the atmospheric pressure is \(9.0 \times 10^{6} \mathrm{~Pa}\), and the temperature is \(740 \mathrm{~K}\). On the earth's surface the atmospheric pressure is \(1.0 \times 10^{5} \mathrm{~Pa}\), while the surface temperature can reach \(320 \mathrm{~K}\). These data imply that Venus has a "thicker" atmosphere at its surface than does the earth, which means that the number of molecules per unit volume \((N / V)\) is greater on the surface of Venus than on the earth. Find the ratio \((N / V)_{\text {Venus }} /(N / V)_{\text {Earth }}\).

What is the density (in \(\mathrm{kg} / \mathrm{m}^{3}\) ) of nitrogen gas (molecular mass \(=28 \mathrm{u}\) ) at a pressure of 2.0 atmospheres and a temperature of \(310 \mathrm{~K} ?\)

Multiple-Concept Example 4 reviews the principles that play roles in this problem. A primitive diving bell consists of a cylindrical tank with one end open and one end closed. The tank is lowered into a freshwater lake, open end downward. Water rises into the tank, compressing the trapped air, whose temperature remains constant during the descent. The tank is brought to a halt when the distance between the surface of the water in the tank and the surface of the lake is \(40.0 \mathrm{~m}\). Atmospheric pressure at the surface of the lake is \(1.01 \times 10^{5} \mathrm{~Pa}\). Find the fraction of the tank's volume that is filled with air.

Refer to Interactive Solution \(\underline{14.21}\) at for help with problems like this one. An apartment has a living room whose dimensions are \(2.5 \mathrm{~m} \times 4.0 \mathrm{~m} \times 5.0 \mathrm{~m}\). Assume that the air in the room is composed of \(79 \%\) nitrogen \(\left(\mathrm{N}_{2}\right)\) and \(21 \%\) oxygen \(\left(\mathrm{O}_{2}\right)\). At a temperature of \(22{ }^{\circ} \mathrm{C}\) and a pressure of \(1.01 \times 10^{5} \mathrm{~Pa}\), what is the mass (in grams) of the air?

Refer to Interactive Solution 14.21 at for help with problems like this one. An apartment has a living room whose dimensions are \(2.5 \mathrm{~m} \times 4.0 \mathrm{~m} \times 5.0 \mathrm{~m}\). Assume that the air in the room is composed of \(79 \%\) nitrogen \(\left(\mathrm{N}_{2}\right)\) and \(21 \%\) oxygen \(\left(\mathrm{O}_{2}\right) .\) At a temperature of \(22^{\circ} \mathrm{C}\) and a pressure of \(1.01 \times 10^{5} \mathrm{~Pa}\), what is the mass (in grams) of the air?

Multiple-Concept Example 4 deals with the concepts used in this problem. Conceptual Example 3 is also pertinent. A bubble, located \(0.200 \mathrm{~m}\) beneath the surface in a glass of beer, rises to the top. The air pressure at the top is \(1.01 \times 10^{5} \mathrm{~Pa}\). Assume that the density of beer is the same as that of fresh water. If the temperature and number of moles of \(\mathrm{CO} 2\) remain constant as the bubble rises, find the ratio of its volume at the top to that at the bottom.

Interactive Solution 14.24 at offers one approach to this problem. One assumption of the ideal gas law is that the atoms or molecules themselves occupy a negligible volume. Verify that this assumption is reasonable by considering gaseous xenon (Xe). Xenon has an atomic radius of \(2.0 \times 10^{-10} \mathrm{~m}\). For STP conditions, calculate the percentage of the total volume occupied by the atoms.

The mass of a hot-air balloon and its occupants is \(320 \mathrm{~kg}\) (excluding the hot air inside the balloon). The air outside the balloon has a pressure of \(1.01 \times 10^{5} \mathrm{~Pa}\) and a density of \(1.29 \mathrm{~kg} / \mathrm{m}^{3}\). To lift off, the air inside the balloon is heated. The volume of the heated balloon is \(650 \mathrm{~m}^{3}\). The pressure of the heated air remains the same as that of the outside air. To what temperature (in kelvins) must the air be heated so that the balloon just lifts off? The molecular mass of air is \(29 \mathrm{u}\).

A gas fills the right portion of a horizontal cylinder whose radius is \(5.00 \mathrm{~cm} .\) The initial pressure of the gas is \(1.01 \times 10^{5} \mathrm{~Pa}\). A frictionless movable piston separates the gas from the left portion of the cylinder, which is evacuated and contains an ideal spring, as the drawing shows. The piston is initially held in place by a pin. The spring is initially unstrained, and the length of the gas-filled portion is \(20.0 \mathrm{~cm}\). When the pin is removed and the gas is allowed to expand, the length of the gas-filled chamber doubles. The initial and final temperatures are equal. Determine the spring constant of the spring.

A cylindrical glass beaker of height \(1.520 \mathrm{~m}\) rests on a table. The bottom half of the beaker is filled with a gas, and the top half is filled with liquid mercury that is exposed to the atmosphere. The gas and mercury do not mix because they are separated by a frictionless, movable piston of negligible mass and thickness. The initial temperature is \(273 \mathrm{~K}\). The temperature is increased until a value is reached when one-half of the mercury has spilled out. Ignore the thermal expansion of the glass and mercury, and find this temperature.

The average value of the squared speed \(\overline{v^{2}}\) does not equal the square of the average speed \((\bar{v})^{2}\). To verify this fact, consider three particles with the following speeds: \(v_{1}=3.0 \mathrm{~m} / \mathrm{s}, v_{2}=7.0 \mathrm{~m} / \mathrm{s}\), and \(v_{3}=9.0 \mathrm{~m} / \mathrm{s} .\) Calculate (a) \(\overline{v^{2}}=\frac{1}{3}\left(v_{1}^{2}+v_{2}^{2}+v_{3}^{2}\right)\) and (b) \((\bar{v})^{2}=\left[\frac{1}{3}\left(v_{1}+v_{2}+v_{3}\right)\right]^{2}\).

The average value of the squared speed \(v^{2}\) does not equal the square of the average speed \((\bar{v})^{2}\). To verify this fact, consider three particles with the following speeds: \(v_{1}=3.0 \mathrm{~m} / \mathrm{s}, v_{2}=7.0 \mathrm{~m} / \mathrm{s},\) and \(v_{3}=9.0 \mathrm{~m} / \mathrm{s} .\) Calculate \((\mathrm{a}) \bar{v}^{\overline{2}}=\frac{1}{3}\left(v_{1}^{2}+v_{2}^{2}+v_{3}^{2}\right)\) and (b) \((\bar{v})^{2}=\left[\frac{1}{3}\left(v_{1}+v_{2}+v_{3}\right)\right]^{2}\).

If the translational rms speed of the water vapor molecules \(\left(\mathrm{H}_{2} \mathrm{O}\right)\) in air is \(648 \mathrm{~m} / \mathrm{s},\) what is the translational rms speed of the carbon dioxide molecules \(\left(\mathrm{CO}_{2}\right)\) in the same air? Both gases are at the same temperature.

If the translational rms speed of the water vapor molecules \(\left(\mathrm{H}_{2} \mathrm{O}\right)\) in air is \(648 \mathrm{~m} / \mathrm{s}\), what is the translational rms speed of the carbon dioxide molecules \(\left(\mathrm{CO}_{2}\right)\) in the same air? Both gases are at the same temperature.

Two gas cylinders are identical. One contains the monatomic gas argon (Ar), and the other contains an equal mass of the monatomic gas krypton \((\mathrm{Kr}) .\) The pressures in the cylinders are the same, but the temperatures are different. Determine the ratio \(\frac{\overline{\mathrm{KE}}_{\mathrm{Krypton}}}{\overline{\mathrm{KE}}_{\mathrm{Argon}}}\) of the average kinetic energy of a krypton atom to the average kinetic energy of an argon atom.

Consult Interactive Solution \(14.33\) at to see how this problem can be solved. Very fine smoke particles are suspended in air. The translational rms speed of a smoke particle is \(2.8 \times 10^{-3} \mathrm{~m} / \mathrm{s}\), and the temperature is \(301 \mathrm{~K}\). Find the mass of a particle.

Consult Interactive Solution \(\underline{14.33}\) at to see how this problem can be solved. Very fine smoke particles are suspended in air. The translational rms speed of a smoke particle is \(2.8 \times 10^{-3} \mathrm{~m} / \mathrm{s}\), and the temperature is \(301 \mathrm{~K}\). Find the mass of a particle.

Each molecule in a gas has an average kinetic energy. What is the total average kinetic energy of all the molecules in \(3.0 \mathrm{~mol}\) of a gas whose temperature is \(320 \mathrm{~K} ?\)

Compressed air can be pumped underground into huge caverns as a form of energy storage. The volume of a cavern is \(5.6 \times 10^{5} \mathrm{~m}^{3},\) and the pressure of the air in it is \(7.7 \times 10^{6} \mathrm{~Pa}\). Assume that air is a diatomic ideal gas whose internal energy \(U\) is given by \(U=\frac{5}{2} n R T\) If one home uses \(30.0 \mathrm{~kW} \cdot \mathrm{h}\) of energy per day, how many homes could this internal energy serve for one day?

Helium (He), a monatomic gas, fills a \(0.010-\mathrm{m}^{3}\) container. The pressure of the gas is \(6.2 \times 10^{5} \mathrm{~Pa}\). How long would a 0.25 -hp engine have to run \((1 \mathrm{hp}=746 \mathrm{~W})\) to produce an amount of energy equal to the internal energy of this gas?

In a TV, electrons with a speed of \(8.4 \times 10^{7} \mathrm{~m} / \mathrm{s}\) strike the screen from behind, causing it to glow. The electrons come to a halt after striking the screen. Each electron has a mass of \(9.11 \times 10^{-31} \mathrm{~kg}\), and there are \(6.2 \times 10^{16}\) electrons per second hitting the screen over an area of \(\mathrm{m}^{2}\). What is the pressure that the electrons exert on the screen?

In \(10.0\) s, 200 bullets strike and embed themselves in a wall. The bullets strike the wall perpendicularly. Each bullet has a mass of \(5.0 \times 10^{-3} \mathrm{~kg}\) and a speed of \(1200 \mathrm{~m} / \mathrm{s}\). (a) What is the average change in momentum per second for the bullets? (b) Determine the average force exerted on the wall. (c) Assuming the bullets are spread out over an area of \(3.0 \times 10^{-4} \mathrm{~m}^{2}\), obtain the average pressure they exert on this region of the wall.

In \(10.0 \mathrm{~s}, 200\) bullets strike and embed themselves in a wall. The bullets strike the wall perpendicularly. Each bullet has a mass of \(5.0 \times 10^{-3} \mathrm{~kg}\) and a speed of \(1200 \mathrm{~m} / \mathrm{s}\). (a) What is the average change in momentum per second for the bullets? (b) Determine the average force exerted on the wall. (c) Assuming the bullets are spread out over an area of \(3.0 \times 10^{-4} \mathrm{~m}^{2},\) obtain the average pressure they exert on this region of the wall.

Instead, they have a system of tiny tubes, called tracheae, through which oxygen diffuses into their bodies. The tracheae begin at the surface of the insect's body and penetrate into the interior. Suppose that a trachea is \(1.9 \mathrm{~mm}\) long with a cross-sectional area of \(2.1 \times 10^{-9} \mathrm{~m}^{2} .\) The concentration of oxygen in the air outside the insect is \(0.28 \mathrm{~kg} / \mathrm{m}^{3}\), and the diffusion constant is \(1.1 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\). If the mass per second of oxygen diffusing through a trachea is \(1.7 \times 10^{-12} \mathrm{~kg} / \mathrm{s}\), find the oxygen concentration at the interior end of the tube.

Insects do not have lungs as we do, nor do they breathe through their mouths. Instead, they have a system of tiny tubes, called tracheae, through which oxygen diffuses into their bodies. The tracheae begin at the surface of the insect's body and penetrate into the interior. Suppose that a trachea is \(1.9 \mathrm{~mm}\) long with a cross-sectional area of \(2.1 \times 10^{-9} \mathrm{~m}^{2}\). The concentration of oxygen in the air outside the insect is \(0.28 \mathrm{~kg} / \mathrm{m}^{3}\) and the diffusion constant is \(1.1 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\). If the mass per second of oxygen diffusing through a trachea is \(1.7 \times 10^{-12} \mathrm{~kg} / \mathrm{s}\), find the oxygen concentration at the interior end of the tube.

The diffusion constant of for the alcohol ethanol in water is \(12.4 \times 10^{-10} \mathrm{~m}^{2} / \mathrm{s} . \mathrm{A}\) cylinder has a cross- sectional area of \(4.00 \mathrm{~cm}^{2}\) and a length of \(2.00 \mathrm{~cm}\). A difference in ethanol concentration of \(1.50 \mathrm{~kg} / \mathrm{m}^{3}\) is maintained between the ends of the cylinder. In one hour, what mass of ethanol diffuses through the cylinder?

The diffusion constant for the amino acid glycine in water is \(1.06 \times 10^{-9} \mathrm{~m}^{2} / \mathrm{s}\). In a 2.0 \(\mathrm{cm}\) -long tube with a cross-sectional area of \(1.5 \times 10^{-4} \mathrm{~m}^{2}\), the mass rate of diffusion is \(m / t=4.2 \times 10^{-14} \mathrm{~kg} / \mathrm{s}\), because the glycine concentration is maintained at a value of \(8.3 \times 10^{-3} \mathrm{~kg} / \mathrm{m}^{3}\) at one end of the tube and at a lower value at the other end. What is the lower concentration?

At the start of a trip, a driver adjusts the absolute pressure in her tires to be \(2.81 \times 10^{5} \mathrm{~Pa}\) when the outdoor temperature is \(284 \mathrm{~K}\). At the end of the trip she measures the pressure to be \(3.01 \times 10^{5} \mathrm{~Pa}\). Ignoring the expansion of the tires, find the air temperature inside the tires at the end of the trip.

A young male adult takes in about \(5.0 \times 10^{-4} \mathrm{~m}^{3}\) of fresh air during a normal breath. Fresh air contains approximately \(21 \%\) oxygen. Assuming that the pressure in the lungs is \(1.0 \times 10^{5} \mathrm{~Pa}\) and air is an ideal gas at a temperature of \(310 \mathrm{~K}\), find the number of oxygen molecules in a normal breath.

A frictionless gas-filled cylinder is fitted with a movable piston, as the drawing shows. The block resting on the top of the piston determines the constant pressure that the gas has. The height \(h\) is \(0.120 \mathrm{~m}\) when the temperature is \(273 \mathrm{~K}\) and increases as the temperature increases. What is the value of \(h\) when the temperature reaches \(318 \mathrm{~K}\) ?

Initially, the translational rms speed of a molecule of an ideal gas is \(463 \mathrm{~m} / \mathrm{s}\). The pressure and volume of this gas are kept constant, while the number of molecules is doubled. What is the final translational rms speed of the molecules?

Suppose that a tank contains \(680 \mathrm{~m}^{3}\) of neon at an absolute pressure of \(1.01 \times 10^{5} \mathrm{~Pa}\). The temperature is changed from 293.2 to \(294.3 \mathrm{~K}\). What is the increase in the internal energy of the neon?

A tank contains \(0.85 \mathrm{~mol}\) of molecular nitrogen \(\left(\mathrm{N}_{2}\right)\). Determine the mass (in grams) of nitrogen that must be removed from the tank in order to lower the pressure from 38 to 25 atm. Assume that the volume and temperature of the nitrogen in the tank do not change.

Estimate the spacing between the centers of neighboring atoms in a piece of solid aluminum, based on a knowledge of the density \(\left(2700 \mathrm{~kg} / \mathrm{m}^{3}\right)\) and atomic mass (26.9815 u) of aluminum.