Exercises and Problems

Problems labeled INT integrate material from earlier chapters.

Exercises

Section 18.1 Solids, Liquids, and Gases

| What volume of water has the same mass as $100 {cm}^{3}$ of gold?
|| What is the diameter of a copper sphere that has the same mass as a $10 cm \times 10 cm \times 10 cm$ cube of aluminum?
|| A hollow aluminum sphere with outer diameter 10.0 cm has a mass of 690 g. What is the sphere’s inner diameter?
| The nucleus of a uranium atom has a diameter of $1.5 \times 10^{- 14} m$ and a mass of $4.0 \times 10^{- 25} kg$ . What is the density of the nucleus?

Section 18.2 Atoms and Moles

| How many atoms are in a $2.0 cm \times 2.0 cm \times 2.0 cm$ cube of aluminum?
| How many moles are in a $2.0 cm \times 2.0 cm \times 2.0 cm$ cube of copper?
| An element in its solid phase has mass density $1750 kg / m^{3}$ and number density $4.39 \times 10^{28} atoms / m^{3}$ . What is the element’s atomic mass number?
| What is the volume in ${cm}^{3}$ of 1.0 mol of ice?
|| A cube of lead contains $1.1 \times 10^{23}$ atoms. What is the length in cm of one side of the cube?
|| What volume of aluminum has the same number of atoms as $10 {cm}^{3}$ of mercury?

Section 18.3 Temperature

| At what temperature does the numerical value in $° F$ match the numerical value in $° C$ ?
| The lowest and highest natural temperatures ever recorded on earth are $- 129 ° F$ in Antarctica and $134 ° F$ in Death Valley. What are these temperatures in $° C$ and in K?
|| A demented scientist creates a new temperature scale, the “Z scale.” He decides to call the boiling point of nitrogen $0 ° Z$ and the melting point of iron $1000 ° Z$ .
1. What is the boiling point of water on the Z scale?
2. Convert $500 ° Z$ to degrees Celsius and to kelvins.

Section 18.4 Thermal Expansion

Section 18.5 Phase Changes

| At room temperature $(20 ° C)$ , a 5.0-cm-long brass rod is $20 μ m$ too long to fit into a slot. To what temperature should you cool the rod so that it just barely fits?
|| Two students each build a piece of scientific equipment that uses a 655-mm-long metal rod. One student uses a brass rod, the other an invar rod. If the temperature increases by 5.0 $°$ C, how much more does the brass rod expand than the invar rod?
|| A concrete bridge is built of 325-cm-long concrete slabs with an expansion joint between them. The slabs just touch on a $115 ° F$ day, the hottest day for which the bridge is designed. What is the gap between the slabs when the temperature is $0 ° F$ ?
|| A surveyor has a steel measuring tape that is calibrated to be 100.000 m long (i.e., accurate to $\pm 1 mm$ ) at $20 ° C$ . If she measures the distance between two stakes to be 65.175 m on a $3 ° C$ day, does she need to add or subtract a correction factor to get the true distance? How large, in mm, is the correction factor?
|| Common outdoor thermometers are filled with red-colored ethyl alcohol. One thermometer has a 0.40-mm-diameter capillary tube attached to a 9.0-mm-diameter spherical bulb. On a 0°C morning, the column of alcohol stands 30 mm above the bulb. What is the temperature in $° C$ when the column of alcohol stands 130 mm above the bulb? The expansion of the glass is much less than that of the alcohol and can be ignored.

Section 18.6 Ideal Gases

|| 3.0 mol of gas at a temperature of $- 120 ° C$ fills a 2.0 L container. What is the gas pressure?
| A cylinder contains nitrogen gas. A piston compresses the gas to half its initial volume. Afterward,
1. Has the mass density of the gas changed? If so, by what factor? If not, why not?
2. Has the number of moles of gas changed? If so, by what factor? If not, why not?
| A gas at $100 ° C$ fills volume $V_{0}$ . If the pressure is held constant, what is the volume if (a) the Celsius temperature is doubled and (b) the Kelvin temperature is doubled?
| A rigid container holds 2.0 mol of gas at a pressure of 1.0 atm and a temperature of $30 ° C$ .
1. What is the container’s volume?
2. What is the pressure if the temperature is raised to $130 ° C$ ?
|| BIO The total lung capacity of a typical adult is $5.0 L$ . Approximately $20 %$ of the air is oxygen. At sea level and at a body temperature of $37 ° C,$ how many oxygen molecules do the lungs contain at the end of a strong inhalation?
|| The solar corona is a very hot atmosphere surrounding the visible surface of the sun. X-ray emissions from the corona show that its temperature is about $2 \times 10^{6} K$ . The gas pressure in the corona is about 0.03 Pa. Estimate the number density of particles in the solar corona.
|| A 20-cm-diameter cylinder that is 40 cm long contains 50 g of oxygen gas at $20 ° C$ .
1. How many moles of oxygen are in the cylinder?
2. How many oxygen molecules are in the cylinder?
3. What is the number density of the oxygen?
4. What is the reading of a pressure gauge attached to the tank?
|| At the summit of Mt. Everest, the pressure is 34 kPa and the temperature during the climbing season is typically $- 20 ° C.$ What is the number density of oxygen molecules at the summit as a fraction of its value at 1 atm pressure, $20 ° C$ ?

Section 18.7 Ideal-Gas Processes

| A gas with initial state variables $p_{1}$ , $V_{1}$ , and $T_{1}$ is cooled in an isochoric process until $p_{2} = \frac{1}{3} p_{1}$ . What are (a) $V_{2}$ and (b) $T_{2}$ ?
| A gas with initial state variables $p_{1}$ , $V_{1}$ , and $T_{1}$ expands isothermally until $V_{2} = 2 V_{1}$ . What are (a) $T_{2}$ and (b) $p_{2} ?$
| A rigid container holds hydrogen gas at a pressure of 3.0 atm and a temperature of $20 ° C$ . What will the pressure be if the temperature is lowered to $- 20 ° C ?$
|| A rigid, hollow sphere is submerged in boiling water in a room where the air pressure is 1.0 atm. The sphere has an open valve with its inlet just above the water level. After a long period of time has elapsed, the valve is closed. What will be the pressure inside the sphere if it is then placed in a mixture of ice and water?
|| A 24-cm-diameter vertical cylinder is sealed at the top by a frictionless 20 kg piston. The piston is 84 cm above the bottom when the gas temperature is $303 ° C$ . The air above the piston is at 1.00 atm pressure.
1. What is the gas pressure inside the cylinder?
2. What will the height of the piston be if the temperature is lowered to $15 ° C ?$
|| 0.10 mol of argon gas is admitted to an evacuated $50 {cm}^{3}$ container at $20 ° C$ . The gas then undergoes an isochoric heating to a temperature of $300 ° C$ .
1. What is the final pressure of the gas?
2. Show the process on a $p V$ diagram. Include a proper scale on both axes.
|| 0.10 mol of argon gas is admitted to an evacuated $50 {cm}^{3}$ container at $20 ° C$ . The gas then undergoes an isothermal expansion to a volume of $200 {cm}^{3}$ .
1. What is the final pressure of the gas?
2. Show the process on a $p V$ diagram. Include a proper scale on both axes.
|| 0.10 mol of argon gas is admitted to an evacuated $50 {cm}^{3}$ container at $20 ° C$ . The gas then undergoes an isobaric heating to a temperature of $300 ° C$ .
1. What is the final volume of the gas?
2. Show the process on a $p V$ diagram. Include a proper scale on both axes.
|| A gas with an initial temperature of $900 ° C$ undergoes the process shown in FIGURE EX18.35.
1. What type of process is this?
2. What is the final temperature in $° C$ ?
3. How many moles of gas are there?
FIGURE EX18.35

FIGURE EX18.35 Full Alternative Text
| 0.0040 mol of gas undergoes the process shown in FIGURE EX18.36.
1. What type of process is this?
2. What are the initial and final temperatures in $° C$ ?
FIGURE EX18.36

FIGURE EX18.36 Full Alternative Text
|| 0.0050 mol of gas undergoes the process $1 \to 2 \to 3$ shown in FIGURE EX18.37. What are (a) temperature $T_{1}$ , (b) pressure $p_{2}$ , and (c) volume $V_{3} ?$

FIGURE EX18.37

FIGURE EX18.37 Full Alternative Text
|| 0.020 mol of gas undergoes the process shown in FIGURE EX18.38.
1. What type of process is this?
2. What is the final temperature in $° C$ ?
3. What is the final volume $V_{2} ?$
FIGURE EX18.38

FIGURE EX18.38 Full Alternative Text
|| An ideal gas starts with pressure $p_{1}$ and volume $V_{1}$ . Draw a $p V$ diagram showing the process in which the gas undergoes an isothermal process during which the volume is halved, then an isochoric process during which the pressure is halved, followed by an isobaric process during which the volume is doubled. Label each of the three processes.
|| A sealed container holds 3.2 g of oxygen at 1 atm pressure and $20 ° C$ . The gas first undergoes an isobaric process that doubles the absolute temperature, then an isothermal process that halves the pressure. What is the final volume of the gas in L?

Problems

|| The molecular mass of water $(H_{2} O)$ is 18 u. How many protons are there in 1.0 L of liquid water?
| The 828-m-tall Burj Khalifa in Dubai is the world’s tallest building. It’s essentially a steel building wrapped in exterior paneling and glass. During construction, when the beams were exposed to the elements, the building was 36 cm taller on the hottest afternoon of the year than on the coldest morning. By how much did the temperature vary throughout the year?
||| A brass ring with inner diameter 2.00 cm and outer diameter 3.00 cm needs to fit over a 2.00-cm-diameter steel rod, but at $20 ° C$ the hole through the brass ring is $50 μ m$ too small in diameter. To what temperature, in $° C$ , must the rod and ring be heated so that the ring just barely slips over the rod?
|| The semiconductor industry manufactures integrated circuits in large vacuum chambers where the pressure is $1.0 \times 10^{- 10} mm$ of Hg.
1. What fraction is this of atmospheric pressure?
2. At $T = 20 ° C,$ how many molecules are in a cylindrical chamber 40 cm in diameter and 30 cm tall?
|| INT A 6.0-cm-diameter, 10-cm-long cylinder contains $100 mg$ of oxygen $(O_{2})$ at a pressure less than 1 atm. The cap on one end of the cylinder is held in place only by the pressure of the air. One day when the atmospheric pressure is 100 kPa, it takes a 184 N force to pull the cap off. What is the temperature of the gas?
|| A cylinder with a moveable piston contains $150 {cm}^{3}$ of carbon dioxide gas at the $CO_{2}$ triple-point temperature and pressure of $- 56 ° C$ and 5.2 atm. The gas undergoes a process described by a vertical line on the $CO_{2}$ phase diagram until the pressure is 1.0 atm. What is the final volume?
|| On average, each person in the industrialized world is responsible for the emission of 10,000 kg of carbon dioxide $(CO_{2})$ every year. This includes $CO_{2}$ that you generate directly, by burning fossil fuels to operate your car or your furnace, as well as $CO_{2}$ generated on your behalf by electric generating stations and manufacturing plants. $CO_{2}$ is a greenhouse gas that contributes to global warming. If you were to store your yearly $CO_{2}$ emissions in a cube at STP, how long would each edge of the cube be?
|| INT To determine the mass of neon contained in a rigid, 2.0 L cylinder, you vary the cylinder’s temperature while recording the reading of a pressure gauge. Your data are as follows:

Temperature $(° C)$ Pressure (atm)

100 6.52

150 7.80

200 8.83

250 9.59

Use the best-fit line of an appropriate graph to determine the mass of the neon.
|| INT The 3.0-m-long pipe in FIGURE P18.49 is closed at the top end. It is slowly pushed straight down into the water until the top end of the pipe is level with the water’s surface. What is the length $L$ of the trapped volume of air?

FIGURE P18.49

FIGURE P18.49 Full Alternative Text
|| A diving bell is a 3.0-m-tall cylinder closed at the upper end but open at the lower end. The temperature of the air in the bell is $20 ° C$ . The bell is lowered into the ocean until its lower end is 100 m deep. The temperature at that depth is $10 ° C$ .
1. How high does the water rise in the bell after enough time has passed for the air inside to reach thermal equilibrium?
2. A compressed-air hose from the surface is used to expel all the water from the bell. What minimum air pressure is needed to do this?
|| An electric generating plant boils water to produce high-pressure steam. The steam spins a turbine that is connected to the generator.
1. How many liters of water must be boiled to fill a $5.0 m^{3}$ boiler with 50 atm of steam at $400 ° C ?$
2. The steam has dropped to 2.0 atm pressure at $150 ° C$ as it exits the turbine. How much volume does it now occupy?
|| On a cool morning, when the temperature is $15 ° C,$ you measure the pressure in your car tires to be 30 psi. After driving 20 mi on the freeway, the temperature of your tires is $45 ° C$ . What pressure will your tire gauge now show?
|| A 10-cm-diameter, 40-cm-tall gas cylinder, sealed at the top by a frictionless 50 kg piston, is surrounded by a bath of $20 ° C$ water. Then 50 kg of sand is slowly poured onto the top of the piston, where it stays. Afterward, what is the height of the piston?
|| $10, 000 c m^{3}$ of $200 ° C$ steam at a pressure of 20 atm is cooled until it condenses. What is the volume of the liquid water? Give your answer in ${cm}^{3}$ .
|| The interior of a Boeing 737-800 can be modeled as a 32-m-long, 3.7-m-diameter cylinder. The air inside, at cruising altitude, is $20 ° C$ at a pressure of 82 kPa. What volume of outside air, at $- 40 ° C$ and a pressure of 23 kPa, must be drawn in, heated, and compressed to fill the plane?
|| INT The U-shaped tube in FIGURE P18.56 has a total length of 1.0 m. It is open at one end, closed at the other, and is initially filled with air at $20 ° C$ and 1.0 atm pressure. Mercury is poured slowly into the open end without letting any air escape, thus compressing the air. This is continued until the open side of the tube is completely filled with mercury. What is the length $L$ of the column of mercury?

FIGURE P18.56

FIGURE P18.56 Full Alternative Text
|| The 50 kg circular piston shown in FIGURE P18.57 floats on 0.12 mol of compressed air.
1. What is the piston height $h$ if the temperature is $30 ° C ?$
2. How far does the piston move if the temperature is increased by $100 ° C ?$
FIGURE P18.57

FIGURE P18.57 Full Alternative Text
||| Containers A and B in FIGURE P18.58 hold the same gas. The volume of B is four times the volume of A. The two containers are connected by a thin tube (negligible volume) and a valve that is closed. The gas in A is at 300 K and pressure of $1.0 \times 10^{5} Pa$ . The gas in B is at 400 K and pressure of $5.0 \times 10^{5} Pa$ . Heaters will maintain the temperatures of A and B even after the valve is opened. After the valve is opened, gas will flow one way or the other until A and B have equal pressure. What is this final pressure?

FIGURE P18.58

FIGURE P18.58 Full Alternative Text
|| INT A diver 50 m deep in $10 ° C$ fresh water exhales a 1.0-cm-diameter bubble. What is the bubble’s diameter just as it reaches the surface of the lake, where the water temperature is $20 ° C ?$

Hint: Assume that the air bubble is always in thermal equilibrium with the surrounding water.
| 8.0 g of helium gas follows the process $1 \to 2 \to 3$ shown in FIGURE P18.60. Find the values of $V_{1}$ , $V_{3}$ , $p_{2}$ , and $T_{3}$ .

FIGURE P18.60

FIGURE P18.60 Full Alternative Text
|| FIGURE P18.61 shows two different processes by which 1.0 g of nitrogen gas moves from state 1 to state 2. The temperature of state 1 is $25 ° C$ . What are (a) pressure $p_{1}$ and (b) temperatures (in $° C$ ) $T_{2}$ , $T_{3}$ , and $T_{4} ?$

FIGURE P18.61

FIGURE P18.61 Full Alternative Text
|| CALC FIGURE P18.62 shows two different processes by which 80 mol of gas move from state 1 to state 2. The dashed line is an isotherm.
1. What is the temperature of the isothermal process?
2. What maximum temperature is reached along the straight-line process?
FIGURE P18.62

FIGURE P18.62 Full Alternative Text
|| 4.0 g of oxygen gas, starting at $820 ° C,$ follow the process $1 \to 2$ shown in FIGURE P18.63. What is temperature $T_{2}$ $(in ° C) ?$

FIGURE P18.63

FIGURE P18.63 Full Alternative Text
|| 10 g of dry ice (solid $CO_{2}$ ) is placed in a $10, 000 c m^{3}$ container, then all the air is quickly pumped out and the container sealed. The container is warmed to $0 ° C,$ a temperature at which $CO_{2}$ is a gas.
1. What is the gas pressure? Give your answer in atm.
  
  The gas then undergoes an isothermal compression until the pressure is 3.0 atm, immediately followed by an isobaric compression until the volume is $1000 {cm}^{3}$ .
2. What is the final temperature of the gas (in $° C$ )?
3. Show the process on a $p V$ diagram.
|| A container of gas at 2.0 atm pressure and $127 ° C$ is compressed at constant temperature until the volume is halved. It is then further compressed at constant pressure until the volume is halved again.
1. What are the final pressure and temperature of the gas?
2. Show this process on a $p V$ diagram.
|| Five grams of nitrogen gas at an initial pressure of 3.0 atm and at $20 ° C$ undergo an isobaric expansion until the volume has tripled.
1. What is the gas volume after the expansion?
2. What is the gas temperature after the expansion (in $° C$ )?
The gas pressure is then decreased at constant volume until the original temperature is reached.
1. What is the gas pressure after the decrease?
Finally, the gas is isothermally compressed until it returns to its initial volume.
1. What is the final gas pressure?
2. Show the full three-step process on a $p V$ diagram. Use appropriate scales on both axes.

Temperature $(° C)$	Pressure (atm)
100	6.52
150	7.80
200	8.83
250	9.59

In Problems 67, 68, 69, and 70 you are given the equation(s) used to solve a problem. For each of these, you are to

Write a realistic problem for which this is the correct equation(s).
Draw a $p V$ diagram.
Finish the solution of the problem.

$p_{2} = \frac{300 {cm}^{3}}{100 {cm}^{3}} \times 1 \times 2 atm$
$(T_{2} + 273) K = \frac{200 kPa}{500 kPa} \times 1 \times (400 + 273) K$
$V_{2} = \frac{(400 + 273) K}{(50 + 273) K} \times 1 \times 200 {cm}^{3}$
$\begin{array}{l} (2.0 \times 101, 300 Pa) (100 \times 10^{- 6} m^{3}) = n (8.31 J/mol K) T_{1} \\ n = \frac{0.12 g}{20 g/mol} \\ T_{2} = \frac{200 {cm}^{3}}{100 {cm}^{3}} \times 1 \times T_{1} \end{array}$

Challenge Problems

||| A $15 ° C$ , 2.0-cm-diameter aluminum bar just barely slips between two rigid steel walls 10.0 cm apart. If the bar is warmed to $25 ° C$ , how much force does it exert on each wall?
||| An inflated bicycle inner tube is 2.2 cm in diameter and 200 cm in circumference. A small leak causes the gauge pressure to decrease from 110 psi to 80 psi on a day when the temperature is $20 ° C$ . What mass of air is lost? Assume the air is pure nitrogen.
||| The cylinder in FIGURE CP18.73 has a moveable piston attached to a spring. The cylinder’s cross-section area is $10 {cm}^{2}$ , it contains 0.0040 mol of gas, and the spring constant is $1500 N / m$ . At $20 ° C$ the spring is neither compressed nor stretched. How far is the spring compressed if the gas temperature is raised to $100 ° C ?$

FIGURE CP18.73

FIGURE CP18.73 Full Alternative Text
||| The closed cylinder of FIGURE CP18.74 has a tight-fitting but frictionless piston of mass $M$ . The piston is in equilibrium when the left chamber has pressure $p_{0}$ and length $L_{0}$ while the spring on the right is compressed by $Δ L$ .
1. What is $Δ L$ in terms of $p_{0}$ , $L_{0}$ , $A$ , $M$ , and $k$ ?
2. Suppose the piston is moved a small distance $x$ to the right. Find an expression for the net force $(F_{x})_{net}$ on the piston. Assume all motions are slow enough for the gas to remain at the same temperature as its surroundings.
3. If released, the piston will oscillate around the equilibrium position. Assuming $x ≪ L_{0}$ find an expression for the oscillation period $T$ .
  
  Hint: Use the binomial approximation.
FIGURE CP18.74

FIGURE CP18.74 Full Alternative Text