Exercises and Problems

Problems labeled INT integrate material from earlier chapters.

Exercises

Section 19.1 It’s All About Energy

Section 19.2 Work in Ideal-Gas Processes

|| How much work is done on the gas in the process shown in FIGURE EX19.1?

FIGURE EX19.1

FIGURE EX19.1 Full Alternative Text

Video Tutor Solution: Problem 19.1
|| –60 J of work are done on the gas in the process shown in FIGURE EX19.2. What is $p_{1}$ in kPa?

FIGURE EX19.2

FIGURE EX19.2 Full Alternative Text
|| 80 J of work are done on the gas in the process shown in FIGURE EX19.3. What is $V_{1}$ in ${cm}^{3} ?$

FIGURE EX19.3

FIGURE EX19.3 Full Alternative Text
|| 500 J of work must be done to compress a gas to half its initial volume at constant temperature. How much work must be done to compress the gas by a factor of 10, starting from its initial volume?
|| A $2000 {cm}^{3}$ container holds 0.10 mol of helium gas at $300 ° C$ . How much work must be done to compress the gas to $1000 {cm}^{3}$ at (a) constant pressure and (b) constant temperature?

Section 19.3 Heat

Section 19.4 The First Law of Thermodynamics

| Draw a first-law bar chart (see Figure 19.12) for the gas process in FIGURE EX19.6.

FIGURE EX19.6

FIGURE EX19.6 Full Alternative Text
| Draw a first-law bar chart (see Figure 19.12) for the gas process in FIGURE EX19.7.

FIGURE EX19.7

FIGURE EX19.7 Full Alternative Text
| Draw a first-law bar chart (see Figure 19.12) for the gas process in FIGURE EX19.8.

FIGURE EX19.8

FIGURE EX19.8 Full Alternative Text
| Draw a first-law bar chart (see Figure 19.12) for the gas process in FIGURE EX19.9.

FIGURE EX19.9

FIGURE EX19.9 Full Alternative Text
| A gas is compressed from $600 {cm}^{3}$ to $200 {cm}^{3}$ at a constant pressure of 400 kPa. At the same time, 400 J of heat energy is transferred out of the gas. What is the change in thermal energy of the gas during this process?
| 500 J of heat energy are transferred to a gas during a process in which the gas expands at constant pressure from $400 {cm}^{3}$ to $800 {cm}^{3}$ . The gas’s thermal energy increases by 300 J during this process. What is the gas pressure?

Section 19.5 Thermal Properties of Matter

|| A rapidly spinning paddle wheel raises the temperature of 200 mL of water from $21 ° C$ to $25 ° C$ . How much (a) heat is transferred and (b) work is done in this process?
|| How much heat energy must be added to a 6.0-cm-diameter copper sphere to raise its temperature from $- 50 ° C$ to $150 ° C ?$
|
1. 100 J of heat energy are transferred to 20 g of mercury. By how much does the temperature increase?
2. How much heat is needed to raise the temperature of 20 g of water by the same amount?
|| How much heat is needed to change 20 g of mercury at $20 ° C$ into mercury vapor at the boiling point?
|| What is the maximum mass of ethyl alcohol you could boil with 1000 J of heat, starting from $20 ° C ?$
| BIO One way you keep from overheating is by perspiring. Evaporation—a phase change—requires heat, and the heat energy is removed from your body. Evaporation is much like boiling, only water’s heat of vaporization at $35 ° C$ is a somewhat larger $24 \times 10^{5} J / kg$ because at lower temperatures more energy is required to break the molecular bonds. Very strenuous activity can cause an adult human to produce 30 g of perspiration per minute. If all the perspiration evaporates, rather than dripping off, at what rate (in J/s) is it possible to exhaust heat by perspiring?
|| INT The specific heat of rubber is $2000 J / kg K$ . Suppose that a rubber ball dropped from a height of 5.0 m bounces back to a height of 2.5 m. What is the temperature increase of the ball? Assume that no energy is transferred to the air or the ground.
|| INT Two cars collide head-on while each is traveling at 80 km/h. Suppose all their kinetic energy is transformed into the thermal energy of the wrecks. What is the temperature increase of each car? You can assume that each car’s specific heat is that of iron.
|| An experiment measures the temperature of a 500 g substance while steadily supplying heat to it. FIGURE EX19.20 shows the results of the experiment. What are the (a) specific heat of the solid phase, (b) specific heat of the liquid phase, (c) melting and boiling temperatures, and (d) heats of fusion and vaporization?

FIGURE EX19.20

FIGURE EX19.20 Full Alternative Text

Section 19.6 Calorimetry

| A 750 g aluminum pan is removed from the stove and plunged into a sink filled with 10.0 L of water at $20.0 ° C$ . The water temperature quickly rises to $24.0 ° C$ . What was the initial temperature of the pan in $° C$ and in $° F ?$
|| 30 g of copper pellets are removed from a $300 ° C$ oven and immediately dropped into 100 mL of water at $20 ° C$ in an insulated cup. What will the new water temperature be?
|| A 50.0 g thermometer is used to measure the temperature of 200 mL of water. The specific heat of the thermometer, which is mostly glass, is 750 J/kg K, and it reads $20.0 ° C$ while lying on the table. After being completely immersed in the water, the thermometer’s reading stabilizes at $71.2 ° C$ . What was the actual water temperature before it was measured?
|| A $65 {cm}^{3}$ block of iron is removed from an $800 ° C$ furnace and immediately dropped into 200 mL of $20 ° C$ water. What fraction of the water boils away?
|| 10 g of steam at the boiling point are combined with 50 g of ice at the freezing point. What is the final temperature of the system?

Section 19.7 The Specific Heats of Gases

| A monatomic gas follows the process $1 \to 2 \to 3$ shown in FIGURE EX19.26. How much heat is needed for (a) process $1 \to 2$ and (b) process $2 \to 3 ?$

FIGURE EX19.26

FIGURE EX19.26 Full Alternative Text
| 0.10 mol of nitrogen gas follow the two processes shown in FIGURE EX19.27. How much heat is required for each?

FIGURE EX19.27

FIGURE EX19.27 Full Alternative Text
|| A container holds 1.0 g of oxygen at a pressure of 8.0 atm.
1. How much heat is required to increase the temperature by $100 ° C$ at constant pressure?
2. How much will the temperature increase if this amount of heat energy is transferred to the gas at constant volume?
| A container holds 1.0 g of argon at a pressure of 8.0 atm.
1. How much heat is required to increase the temperature by $100 ° C$ at constant volume?
2. How much will the temperature increase if this amount of heat energy is transferred to the gas at constant pressure?
|| A gas cylinder holds 0.10 mol of $O_{2}$ at $150 ° C$ and a pressure of 3.0 atm. The gas expands adiabatically until the pressure is halved. What are the final (a) volume and (b) temperature?
| The volume of a gas is halved during an adiabatic compression that increases the pressure by a factor of 2.5.
1. What is the specific heat ratio $γ ?$
2. By what factor does the temperature increase?
|| A gas cylinder holds 0.10 mol of $O_{2}$ at $150 ° C$ and a pressure of 3.0 atm. The gas expands adiabatically until the volume is doubled. What are the final (a) pressure and (b) temperature?

Section 19.8 Heat-Transfer Mechanisms

|| A $10 m \times 14 m$ house is built on a 12-cm-thick concrete slab. What is the heat-loss rate through the slab if the ground temperature is $5 ° C$ while the interior of the house is $22 ° C ?$
| The ends of a 20-cm-long, 2.0-cm-diameter rod are maintained at $0 ° C$ and $100 ° C$ by immersion in an ice-water bath and boiling water. Heat is conducted through the rod at $4.5 \times 10^{4} J$ per hour. Of what material is the rod made?
|| You are boiling pasta and absentmindedly grab a copper stirring spoon rather than your wooden spoon. The copper spoon has a $20 mm \times 1.5 mm$ rectangular cross section, and the distance from the boiling water to your $35 ° C$ hand is 18 cm. How long does it take the spoon to transfer 25 J of energy to your hand?
|| What maximum power can be radiated by a 10-cm-diameter solid lead sphere? Assume an emissivity of 1.
||| BIO Radiation from the head is a major source of heat loss from the human body. Model a head as a 20-cm-diameter, 20-cm-tall cylinder with a flat top. If the body’s surface temperature is $35 ° C,$ what is the net rate of heat loss on a chilly $5 ° C$ day? All skin, regardless of color, is effectively black in the infrared where the radiation occurs, so use an emissivity of 0.95.

Problems

|| A 5.0-m-diameter garden pond is 30 cm deep. Solar energy is incident on the pond at an average rate of $400 W / m^{2}$ . If the water absorbs all the solar energy and does not exchange energy with its surroundings, how many hours will it take to warm from $15 ° C$ to $25 ° C ?$
|| A 5.0 g ice cube at $- 20 ° C$ is in a rigid, sealed container from which all the air has been evacuated. How much heat is required to change this ice cube into steam at $200 ° C ?$ Steam has $c_{V} = 1500 J / kg K$ and $c_{P} = 1960 J / kg K$ .
|| The burner on an electric stove has a power output of 2.0 kW. A 750 g stainless steel teakettle is filled with $20 ° C$ water and placed on the already hot burner. If it takes 3.0 min for the water to reach a boil, what volume of water, in ${cm}^{3}$ , was in the kettle? Stainless steel is mostly iron, so you can assume its specific heat is that of iron.
|| BIO When air is inhaled, it quickly becomes saturated with water vapor as it passes through the moist airways. Consequently, an adult human exhales about 25 mg of evaporated water with each breath. Evaporation—a phase change—requires heat, and the heat energy is removed from your body. Evaporation is much like boiling, only water’s heat of vaporization at $35 ° C$ is a somewhat larger $24 \times 10^{5} J / kg$ because at lower temperatures more energy is required to break the molecular bonds. At 12 breaths/min, on a dry day when the inhaled air has almost no water content, what is the body’s rate of energy loss (in J/s) due to exhaled water? (For comparison, the energy loss from radiation, usually the largest loss on a cool day, is about 100 J/s.)
|| A 150 L $(\approx 40 gal)$ electric hot-water tank has a 5.0 kW heater. How many minutes will it take to raise the water temperature from $65 ° F$ to $140 ° F ?$
|| 512 g of an unknown metal at a temperature of $15 ° C$ is dropped into a 100 g aluminum container holding 325 g of water at $98 ° C$ . A short time later, the container of water and metal stabilizes at a new temperature of $78 ° C$ . Identify the metal.
||| CALC The specific heat of most solids is nearly constant over a wide temperature range. Not so for diamond. Between 200 K and 600 K, the specific heat of diamond is reasonably well described by $c = 2.8 T - 350 J / kg K$ , where $T$ is in K. For gemstone diamonds, 1 carat $= 200 mg$ . How much heat energy is needed to raise the temperature of a 3.5 carat diamond from $- 50 ° C$ to $250 ° C ?$
||| The beaker in FIGURE P19.45, with a thin metal bottom, is filled with 20 g of water at $20 ° C$ . It is brought into good thermal contact with a $4000 {cm}^{3}$ container holding 0.40 mol of a monatomic gas at 10 atm pressure. Both containers are well insulated from their surroundings. What is the gas pressure after a long time has elapsed? You can assume that the containers themselves are nearly massless and do not affect the outcome.

FIGURE P19.45

FIGURE P19.45 Full Alternative Text
|| Your 300 mL cup of coffee is too hot to drink when served at $90 ° C$ . What is the mass of an ice cube, taken from a $- 20 ° C$ freezer, that will cool your coffee to a pleasant $60 ° C ?$
|| BIO Suppose you take and hold a deep breath on a chilly day, inhaling 3.0 L of air at $0 ° C$ and 1 atm.
1. How much heat must your body supply to warm the air to your internal body temperature of $37 ° C ?$
2. By how much does the air’s volume increase as it warms?
| A typical nuclear reactor generates 1000 MW (1000 MJ/s) of electric energy. In doing so, it produces 2000 MW of “waste heat” that must be removed from the reactor to keep it from melting down. Many reactors are sited next to large bodies of water so that they can use the water for cooling. Consider a reactor where the intake water is at $18 ° C$ . State regulations limit the temperature of the output water to $30 ° C$ so as not to harm aquatic organisms. How many liters of cooling water have to be pumped through the reactor each minute?
|| A 6.0-cm-diameter cylinder of nitrogen gas has a 4.0-cm-thick movable copper piston. The cylinder is oriented vertically, as shown in FIGURE P19.49, and the air above the piston is evacuated. When the gas temperature is $20 ° C,$ the piston floats 20 cm above the bottom of the cylinder.
1. What is the gas pressure?
2. How many gas molecules are in the cylinder?
  
  Then 2.0 J of heat energy are transferred to the gas.
3. What is the new equilibrium temperature of the gas?
4. What is the final height of the piston?
5. How much work is done on the gas as the piston rises?
FIGURE P19.49

FIGURE P19.49 Full Alternative Text
|| 2.0 mol of gas are at $30 ° C$ and a pressure of 1.5 atm. How much work must be done on the gas to compress it to one third of its initial volume at (a) constant temperature and (b) constant pressure? (c) Show both processes on a single pV diagram.
|| 0.25 mol of a gas are compressed at a constant pressure of 250 kPa from $6000 {cm}^{3}$ to $2000 {cm}^{3}$ , then expanded at a constant temperature back to $6000 {cm}^{3}$ . What is the net work done on the gas?
|| CALC An ideal-gas process is described by $p = c V^{1 / 2}$ , where $c$ is a constant.
1. Find an expression for the work done on the gas in this process as the volume changes from $V_{1}$ to $V_{2}$ .
2. 0.033 mol of gas at an initial temperature of $150 ° C$ is compressed, using this process, from $300 {cm}^{3}$ to $200 {cm}^{3}$ . How much work is done on the gas?
3. What is the final temperature of the gas in $° C ?$
|| A 10-cm-diameter cylinder contains argon gas at 10 atm pressure and a temperature of $50 ° C$ . A piston can slide in and out of the cylinder. The cylinder’s initial length is 20 cm. 2500 J of heat are transferred to the gas, causing the gas to expand at constant pressure. What are (a) the final temperature and (b) the final length of the cylinder?
|| An 8.0-cm-diameter, well-insulated vertical cylinder containing nitrogen gas is sealed at the top by a 5.1 kg frictionless piston. The air pressure above the piston is 100 kPa.
1. What is the gas pressure inside the cylinder?
2. Initially, the piston height above the bottom of the cylinder is 26 cm. What will be the piston height if an additional 3.5 kg are placed on top of the piston?
||| A cube 20 cm on each side contains 3.0 g of helium at $20 ° C$ . 1000 J of heat energy are transferred to this gas. What are (a) the final pressure if the process is at constant volume and (b) the final volume if the process is at constant pressure? (c) Show and label both processes on a single pV diagram.
|| $n$ moles of an ideal gas at temperature $T_{1}$ and volume $V_{1}$ expand isothermally until the volume has doubled. In terms of $n$ , $T_{1}$ , and $V_{1}$ , what are (a) the final temperature, (b) the work done on the gas, and (c) the heat energy transferred to the gas?
|| 5.0 g of nitrogen gas at $20 ° C$ and an initial pressure of 3.0 atm undergo an isobaric expansion until the volume has tripled.
1. What are the gas volume and temperature after the expansion?
2. How much heat energy is transferred to the gas to cause this expansion?
  
  The gas pressure is then decreased at constant volume until the original temperature is reached.
3. What is the gas pressure after the decrease?
4. What amount of heat energy is transferred from the gas as its pressure decreases?
5. Show the total process on a pV diagram. Provide an appropriate scale on both axes.
|| 0.10 mol of nitrogen gas follow the two processes shown in FIGURE P19.58. How much heat is required for each?

FIGURE P19.58

FIGURE P19.58 Full Alternative Text
|| INT A 1.0-cm-diameter, 50-cm-long aluminum rod is heated to 250°C and then dropped into a tray that contains 250 mL of 30°C water. By how much does the length of the rod shrink?
||| Your laboratory assignment for the week is to measure the specific heat ratio $γ$ of carbon dioxide. The gas is contained in a cylinder with a movable piston and a thermometer. When the piston is withdrawn as far as possible, the cylinder’s length is 20 cm. You decide to push the piston in very rapidly by various amounts and, for each push, to measure the temperature of the carbon dioxide. Before each push, you withdraw the piston all the way and wait several minutes for the gas to come to the room temperature of $21 ° C$ . Your data are as follows:

Push (cm) Temperature $(° C)$

5 35

10 68

13 110

15 150

Use the best-fit line of an appropriate graph to determine $γ$ for carbon dioxide.
|| Two cylinders each contain 0.10 mol of a diatomic gas at 300 K and a pressure of 3.0 atm. Cylinder A expands isothermally and cylinder B expands adiabatically until the pressure of each is 1.0 atm.
1. What are the final temperature and volume of each?
2. Show both processes on a single pV diagram. Use an appropriate scale on both axes.
|| FIGURE P19.62 shows a thermodynamic process followed by 120 mg of helium.
1. Determine the pressure (in atm), temperature (in $° C$ ), and volume (in ${cm}^{3}$ ) of the gas at points 1, 2, and 3. Put your results in a table for easy reading.
2. How much work is done on the gas during each of the three segments?
3. How much heat energy is transferred to or from the gas during each of the three segments?
FIGURE P19.62

FIGURE P19.62 Full Alternative Text
| Two containers of a diatomic gas have the same initial conditions. One container, heated at constant pressure, has a temperature increase of $20 ° C$ . The other container receives the same quantity of heat energy, but at constant volume. What is its temperature increase?
|| 14 g of nitrogen gas at STP are adiabatically compressed to a pressure of 20 atm. What are (a) the final temperature, (b) the work done on the gas, (c) the heat transfer to the gas, and (d) the compression ratio $V_{max} / V_{min} ?$ (e) Show the process on a pV diagram, using proper scales on both axes.
|| 14 g of nitrogen gas at STP are pressurized in an isochoric process to a pressure of 20 atm. What are (a) the final temperature, (b) the work done on the gas, (c) the heat transfer to the gas, and (d) the pressure ratio $p_{max} / p_{min} ?$ (e) Show the process on a pV diagram, using proper scales on both axes.
|| When strong winds rapidly carry air down from mountains to a lower elevation, the air has no time to exchange heat with its surroundings. The air is compressed as the pressure rises, and its temperature can increase dramatically. These warm winds are called Chinook winds in the Rocky Mountains and Santa Ana winds in California. Suppose the air temperature high in the mountains behind Los Angeles is $0 ° C$ at an elevation where the air pressure is 60 kPa. What will the air temperature be, in $° C$ and $° F,$ when the Santa Ana winds have carried this air down to an elevation near sea level where the air pressure is 100 kPa?
|| Liquid helium, with a boiling point of 4.2 K, is used in ultralow-temperature experiments and also for cooling the superconducting magnets used in MRI imaging in medicine. Storing liquid helium so far below room temperature is a challenge because even a small “heat leak” will boil the helium away. A standard helium dewar, shown in FIGURE P19.67, has an inner stainless-steel cylinder filled with liquid helium surrounded by an outer cylindrical shell filled with liquid nitrogen at –196°C. The space between is a vacuum. The small structural supports have very low thermal conductivity, so you can assume that radiation is the only heat transfer between the helium and its surroundings. Suppose the helium cylinder is 16 cm in diameter and 30 cm tall and that all walls have an emissivity of 0.25. The density of liquid helium is $125 {kg/m}^{3}$ and its heat of vaporization is $2.1 \times 10^{4} J / kg$ .
1. What is the mass of helium in the filled cylinder?
2. In how many hours will half the helium boil away?
FIGURE P19.67

FIGURE P19.67 Full Alternative Text
|| You would like to put a solar hot water system on your roof, but you’re not sure it’s feasible. A reference book on solar energy shows that the ground-level solar intensity in your city is $800 {W/m}^{2}$ for at least 5 hours a day throughout most of the year. Assuming that a completely black collector plate loses energy only by radiation, and that the air temperature is $20 ° C,$ what is the equilibrium temperature of a collector plate directly facing the sun? Note that while a plate has two sides, only the side facing the sun will radiate because the opposite side will be well insulated.
|| Some gases in the earth’s atmosphere—notably carbon dioxide and water vapor—keep the earth warmer than it would be otherwise by absorbing some of the infrared radiation that the earth is trying to radiate into space. This is the greenhouse effect, and it’s a natural feature of the earth’s climate. You can make a good estimate of the size of the greenhouse effect. Model the earth as a sphere of uniform temperature $T$ and emissivity $e = 1$ radiating energy into outer space at $T_{space} \approx 0 K$ . The earth’s temperature is not really uniform, but the uniform-temperature assumption will allow you to find the earth’s average temperature. Assume that nothing absorbs or interferes with this radiation. At the same time, the earth is constantly absorbing energy from the sun. The solar intensity $I$ just above the earth’s atmosphere is $1360 {W/m}^{2}$ . Seen from the sun, the absorbing area of the earth is a circular disk. Measurements show that 70% of the solar energy striking the earth is absorbed, 30% is reflected back into space. Averaged over many days, as the earth rotates, the energy absorbed from the sun balances the energy radiated into space. Now calculate $T_{earth}$ , what the earth’s average temperature in °C would be in the complete absence of infrared-absorbing greenhouse gases. The actual average temperature is a much more pleasant 15°C, which shows that the greenhouse effect is essential to life on earth.
|| A cubical box 20 cm on a side is constructed from 1.2-cm-thick concrete panels. A 100 W lightbulb is sealed inside the box. What is the air temperature inside the box when the light is on if the surrounding air temperature is $20 ° C ?$
|| A cylindrical copper rod and an iron rod with exactly the same dimensions are welded together end to end. The outside end of the copper rod is held at $100 ° C$ , and the outside end of the iron rod is held at $0 ° C$ . What is the temperature at the midpoint where the rods are joined together?
|| Most stars are main-sequence stars, a group of stars for which size, mass, surface temperature, and radiated power are closely related. The sun, for instance, is a yellow main-sequence star with a surface temperature of 5800 K. For a main-sequence star whose mass $M$ is more than twice that of the sun, the total radiated power, relative to the sun, is approximately $P / P_{sun} = 1.5 (M / M_{sun})^{3.5}$ . The star Regulus A is a bluish main-sequence star with mass $3.8 M_{sun}$ and radius $3.1 R_{sun}$ . What is the surface temperature of Regulus A?
|| One cylinder in the diesel engine of a truck has an initial volume of $600 {cm}^{3}$ . Air is admitted to the cylinder at $30 ° C$ and a pressure of 1.0 atm. The piston rod then does 400 J of work to rapidly compress the air. What are its final temperature and volume?
|| A flow-through electric water heater has a 20 kW electric heater inside an insulated 2.0-cm-diameter pipe so that water flowing through the pipe will have good thermal contact with the heater. Assume that all the heat energy is transferred to the water. Suppose the inlet water temperature is 12°C and the flow rate is 8.0 L/min (about that of a standard shower head). What is the outlet temperature?

Push (cm)	Temperature $(° C)$
5	35
10	68
13	110
15	150

In Problems 75, 76, and 77 you are given the equation used to solve a problem. For each of these, you are to

Write a realistic problem for which this is the correct equation.
Finish the solution of the problem.

$50 J = - n (8.31 J/mol K) (350K)ln (\frac{1}{3})$
$\begin{array}{l} (200 \times 10^{- 6} m^{3}) (13, 600 {kg/m}^{3}) \\ \times (140 J/kg K) (90 °C - 15 °C) \\ + (0.50 kg) (449 J/kg K) (90 °C - T_{i}) = 0 \end{array}$
$(10 atm) V_{2}^{1.40} = (1.0 atm) V_{1}^{1.40}$

Challenge Problems

||| 10 g of aluminum at $200 ° C$ and 20 g of copper are dropped into $50 {cm}^{3}$ of ethyl alcohol at $15 ° C$ . The temperature quickly comes to $25 ° C$ . What was the initial temperature of the copper?
||| A lava flow is threatening to engulf a small town. A 400-m-wide, 35-cm-thick tongue of $1200 ° C$ lava is advancing at the rate of 1.0 m per minute. The mayor devises a plan to stop the lava in its tracks by flying in large quantities of $20 ° C$ water and dousing it. The lava has density $2500 {kg/m}^{3}$ , specific heat $1100 J / kg K$ , melting temperature $800 ° C$ , and heat of fusion $4.0 \times 10^{5} J/kg$ . How many liters of water per minute, at a minimum, will be needed to save the town?
||| FIGURE CP19.80 shows a thermodynamic process followed by 0.015 mol of hydrogen. How much heat energy is transferred to the gas?

FIGURE CP19.80

FIGURE CP19.80 Full Alternative Text
||| CALC A satellite to reflect radar is a 2.0-m-diameter, 2.0-mm-thick spherical copper shell. While orbiting the earth, the satellite absorbs sunlight and is warmed to $50 ° C$ . When it passes into the earth’s shadow, the satellite radiates energy to deep space. The temperature of deep space is actually 3 K, as a result of the Big Bang 14 billion years ago, but it is so much colder than the satellite that you can assume a deep-space temperature of 0 K. If the satellite’s emissivity is 0.75, to what temperature, in $° C,$ will it drop during the 45 minutes it takes to move through the earth’s shadow?
||| CALC 0.020 mol of a diatomic gas, with initial temperature $20 ° C,$ are compressed from $1500 {cm}^{3}$ to $500 {cm}^{3}$ in a process in which $p V^{2} = constant$ . How much heat energy is added during this process?
||| CALC A monatomic gas fills the left end of the cylinder in FIGURE CP19.83. At 300 K, the gas cylinder length is 10.0 cm and the spring is compressed by 2.0 cm. How much heat energy must be added to the gas to expand the cylinder length to 16.0 cm?

FIGURE CP19.83

FIGURE CP19.83 Full Alternative Text
||| CALC For an object that cools by convection, the temperature $T$ often can be described by the equation

$\frac{d T}{d t} = - b (T - T_{env})$

where $T_{env}$ is the temperature of the surrounding environment and $b$ is an experimentally determined constant. The left side, $d T / d t$ , is the rate at which the temperature changes. The equation says that the rate of change is proportional to how much the object’s temperature differs from its environment: a rapid change when the temperature difference is large, slowing to a much smaller change when the temperature difference is small. This agrees with our everyday experience. The minus sign indicates that the object’s temperature falls, a negative rate of change, when it is hotter than its environment. This model of convection is often called Newton’s law of cooling.

To solve the equation, change the variable to $U = T - T_{env}$ , for which $d U = d T$ . Then integrate from $U_{i} = T_{0} - T_{env}$ at $t_{i} = 0$ to $U_{f} = T - T_{env}$ at a later time $t_{f} = t$ , where $T_{0}$ is the object’s initial temperature at $t = 0$ . The result is an expression for the temperature difference $T - T_{env}$ as a function of time. Suppose a mug of hot coffee in a 20°C room cools from 80°C to 60°C in 5.0 min. How long will it take to cool an additional 20°C, from 60°C to 40°C?