Solution: A long rod, insulated to prevent heat loss along

Explain why it would not make sense to use a full-size glass thermometer to measure the temperature of a thimbleful of hot water

If you heat the air inside a rigid, sealed container until its Kelvin temperature doubles, the air pressure in the container will also double. Is the same thing true if you double the Celsius temperature of the air in the container? Explain

If you heat the air inside a rigid, sealed container until its Kelvin temperature doubles, the air pressure in the container will also double. Is the same thing true if you double the Celsius temperature of the air in the container? Explain

Why do frozen water pipes burst? Would a mercury thermometer break if the temperature went below the freezing temperature of mercury? Why or why not?

Two bodies made of the same material have the same external dimensions and appearance, but one is solid and the other is hollow. When their temperature is increased, is the overall volume expansion the same or different? Why?

Why is it sometimes possible to loosen caps on screw-top bottles by dipping the capped bottle briefly into hot water?

The inside of an oven is at a temperature of 200C 1392F2. You can put your hand in the oven without injury as long as you dont touch anything. But since the air inside the oven is also at 200C, why isnt your hand burned just the same?

A newspaper article about the weather states that the temperature of a body measures how much heat the body contains. Is this description correct? Why or why not?

A student asserts that a suitable unit for specific heat is 1 m2>s 2 # C. Is she correct? Why or why not?

In some household air conditioners used in dry climates, air is cooled by blowing it through a water-soaked filter, evaporating some of the water. How does this cool the air? Would such a system work well in a high-humidity climate? Why or why not?

The units of specific heat c are \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\), but the units of heat of fusion \(L_{f}\) or heat of vaporization \(L_{\mathrm{V}}\) are simply J/kg. Why do the units of \(L_{f}\) and \(L_{\mathrm{V}}\) not include a factor of \((\mathrm{K})^{-1}\) to account for a temperature change?

Why is a hot, humid day in the tropics generally more uncomfortable for human beings than a hot, dry day in the desert?

A piece of aluminum foil used to wrap a potato for baking in a hot oven can usually be handled safely within a few seconds after the potato is removed from the oven. The same is not true of the potato, however! Give two reasons for this difference

Desert travelers sometimes keep water in a canvas bag. Some water seeps through the bag and evaporates. How does this cool the water inside the bag?

When you first step out of the shower, you feel cold. But as soon as you are dry you feel warmer, even though the room temperature does not change. Why?

The climate of regions adjacent to large bodies of water (like the Pacific and Atlantic coasts) usually features a narrower range of temperature than the climate of regions far from large bodies of water (like the prairies). Why?

When water is placed in ice-cube trays in a freezer, why doesn’t the water freeze all at once when the temperature has reached 0°C? In fact, the water freezes first in a layer adjacent to the sides of the tray. Why?

Before giving you an injection, a physician swabs your arm with isopropyl alcohol at room temperature. Why does this make your arm feel cold? (Hint: The reason is not the fear of the injection! The boiling point of isopropyl alcohol is 82.4C.)

A cold block of metal feels colder than a block of wood at the same temperature. Why? A hot block of metal feels hotter than a block of wood at the same temperature. Again, why? Is there any temperature at which the two blocks feel equally hot or cold? What temperature is this?

A person pours a cup of hot coffee, intending to drink it five minutes later. To keep the coffee as hot as possible, should she put cream in it now or wait until just before she drinks it? Explain.

When a freshly baked apple pie has just been removed from the oven, the crust and filling are both at the same temperature. Yet if you sample the pie, the filling will burn your tongue but the crust will not. Why is there a difference? (Hint: The filling is moist while the crust is dry.)

Old-time kitchen lore suggests that things cook better (evenly and without burning) in heavy cast-iron pots. What desirable characteristics do such pots have?

In coastal regions in the winter, the temperature over the land is generally colder than the temperature over the nearby ocean; in the summer, the reverse is usually true. Explain. (Hint: The specific heat of soil is only 0.20.8 times as great as that of water.)

It is well known that a potato bakes faster if a large nail is stuck through it. Why? Does an aluminum nail work better than a steel one? Why or why not? (Note: Dont try this in a microwave oven!) There is also a gadget on the market to hasten the roasting of meat; it consists of a hollow metal tube containing a wick and some water. This is claimed to work much better than a solid metal rod. How does it work?

Glider pilots in the Midwest know that thermal updrafts are likely to occur in the vicinity of freshly plowed fields. Why?

Some folks claim that ice cubes freeze faster if the trays are filled with hot water, because hot water cools off faster than cold water. What do you think?

Were lucky that the earth isnt in thermal equilibrium with the sun (which has a surface temperature of 5800 K). But why arent the two bodies in thermal equilibrium?

When energy shortages occur, magazine articles sometimes urge us to keep our homes at a constant temperature day and night to conserve fuel. They argue that when we turn down the heat at night, the walls, ceilings, and other areas cool off and must be reheated in the morning. So if we keep the temperature constant, these parts of the house will not cool off and will not have to be reheated. Does this argument make sense? Would we really save energy by following this advice?

Convert the following Celsius temperatures to Fahrenheit: (a) -62.8C, the lowest temperature ever recorded in North America (February 3, 1947, Snag, Yukon); (b) 56.7C, the highest temperature ever recorded in the United States (July 10, 1913, Death Valley, California); (c) 31.1C, the worlds highest average annual temperature (Lugh Ferrandi, Somalia)

Temperatures in Biomedicine. (a) Normal body temperature. The average normal body temperature measured in the mouth is 310 K. What would Celsius and Fahrenheit thermometers read for this temperature? (b) Elevated body temperature. During very vigorous exercise, the bodys temperature can go as high as 40C. What would Kelvin and Fahrenheit thermometers read for this temperature? (c) Temperature difference in the body. The surface temperature of the body is normally about 7 C lower than the internal temperature. Express this temperature difference in kelvins and in Fahrenheit degrees. (d) Blood storage. Blood stored at 4.0C lasts safely for about 3 weeks, whereas blood stored at -160C lasts for 5 years. Express both temperatures on the Fahrenheit and Kelvin scales. (e) Heat stroke. If the bodys temperature is above 105F for a prolonged period, heat stroke can result. Express this temperature on the Celsius and Kelvin scales.

(a) On January 22, 1943, the temperature in Spearfish, South Dakota, rose from -4.0F to 45.0F in just 2 minutes. What was the temperature change in Celsius degrees? (b) The temperature in Browning, Montana, was 44.0F on January 23, 1916. The next day the temperature plummeted to -56F. What was the temperature change in Celsius degrees?

(a) Calculate the one temperature at which Fahrenheit and Celsius thermometers agree with each other. (b) Calculate the one temperature at which Fahrenheit and Kelvin thermometers agree with each other.

You put a bottle of soft drink in a refrigerator and leave it until its temperature has dropped 10.0 K. What is its temperature change in (a) F and (b) C?

Convert the following Kelvin temperatures to the Celsius and Fahrenheit scales: (a) the midday temperature at the surface of the moon (400 K); (b) the temperature at the tops of the clouds in the atmosphere of Saturn (95 K); (c) the temperature at the center of the sun 11.55 * 107 K2.

The pressure of a gas at the triple point of water is 1.35 atm. If its volume remains unchanged, what will its pressure be at the temperature at which CO2 solidifies?

A constant-volume gas thermometer registers an absolute pressure corresponding to 325 mm of mercury when in contact with water at the triple point. What pressure does it read when in contact with water at the normal boiling point?

A Constant-Volume Gas Thermometer. An experimenter using a gas thermometer found the pressure at the triple point of water (0.01C) to be 4.80 * 104 Pa and the pressure at the normal boiling point (100C) to be 6.50 * 104 Pa. (a) Assuming that the pressure varies linearly with temperature, use these two data points to find the Celsius temperature at which the gas pressure would be zero (that is, find the Celsius temperature of absolute zero). (b) Does the gas in this thermometer obey Eq. (17.4) precisely? If that equation were precisely obeyed and the pressure at 100C were 6.50 * 104 Pa, what pressure would the experimenter have measured at 0.01C? (As we will learn in Section 18.1, Eq. (17.4) is accurate only for gases at very low density.)

Like the Kelvin scale, the Rankine scale is an absolute temperature scale: Absolute zero is zero degrees Rankine 10R2. However, the units of this scale are the same size as those of the Fahrenheit scale rather than the Celsius scale. What is the numerical value of the triple-point temperature of water on the Rankine scale?

The Humber Bridge in England has the worlds longest single span, 1410 m. Calculate the change in length of the steel deck of the span when the temperature increases from -5.0C to 18.0C

One of the tallest buildings in the world is the Taipei 101 in Taiwan, at a height of 1671 feet. Assume that this height was measured on a cool spring day when the temperature was 15.5C. You could use the building as a sort of giant thermometer on a hot summer day by carefully measuring its height. Suppose you do this and discover that the Taipei 101 is 0.471 foot taller than its official height. What is the temperature, assuming that the building is in thermal equilibrium with the air and that its entire frame is made of steel?

A U.S. penny has a diameter of 1.9000 cm at 20.0C. The coin is made of a metal alloy (mostly zinc) for which the coefficient of linear expansion is 2.6 * 10-5 K-1 . What would its diameter be on a hot day in Death Valley (48.0C)? On a cold night in the mountains of Greenland 1-53C2?

Ensuring a Tight Fit. Aluminum rivets used in airplane construction are made slightly larger than the rivet holes and cooled by “dry ice” (solid \(\mathrm {CO}_2\)) before being driven. If the diameter of a hole is 4.500 mm, what should be the diameter of a rivet at \(23.0^\circ \mathrm C\) if its diameter is to equal that of the hole when the rivet is cooled to\(-78.0^\circ \mathrm C\) , the temperature of dry ice? Assume that the expansion coefficient remains constant at the value given in Table 17.1.

A copper cylinder is initially at 20.0C. At what temperature will its volume be 0.150% larger than it is at 20.0C?

A geodesic dome constructed with an aluminum framework is a nearly perfect hemisphere; its diameter measures 55.0 m on a winter day at a temperature of -15C. How much more interior space does the dome have in the summer, when the temperature is 35C?

A glass flask whose volume is 1000.00 cm3 at 0.0C is completely filled with mercury at this temperature. When flask and mercury are warmed to 55.0C, 8.95 cm3 of mercury overflow. If the coefficient of volume expansion of mercury is 18.0 * 10-5 K-1 , compute the coefficient of volume expansion of the glass.

A steel tank is completely filled with 1.90 m3 of ethanol when both the tank and the ethanol are at 32.0C. When the tank and its contents have cooled to 18.0C, what additional volume of ethanol can be put into the tank?

A machinist bores a hole of diameter 1.35 cm in a steel plate that is at 25.0C. What is the cross-sectional area of the hole (a) at 25.0C and (b) when the temperature of the plate is increased to 175C? Assume that the coefficient of linear expansion remains constant over this temperature range

As a new mechanical engineer for Engines Inc., you have been assigned to design brass pistons to slide inside steel cylinders. The engines in which these pistons will be used will operate between \(20.0^\circ \mathrm {C}\) and \(150.0^\circ \mathrm {C}\). Assume that the coefficients of expansion are constant over this temperature range. (a) If the piston just fits inside the chamber at \(20.0^\circ \mathrm {C}\), will the engines be able to run at higher temperatures? Explain. (b) If the cylindrical pistons are 25.000 cm in diameter at \(20.0^\circ \mathrm {C}\), what should be the minimum diameter of the cylinders at that temperature so the pistons will operate at \(150.0^\circ \mathrm {C}\)?

Steel train rails are laid in 12.0-m-long segments placed end to end. The rails are laid on a winter day when their temperature is -9.0C. (a) How much space must be left between adjacent rails if they are just to touch on a summer day when their temperature is 33.0C? (b) If the rails are originally laid in contact, what is the stress in them on a summer day when their temperature is 33.0C?

A brass rod is 185 cm long and 1.60 cm in diameter. What force must be applied to each end of the rod to prevent it from contracting when it is cooled from 120.0C to 10.0C?

An aluminum tea kettle with mass 1.10 kg and containing 1.80 kg of water is placed on a stove. If no heat is lost to the surroundings, how much heat must be added to raise the temperature from \(20.0^{\circ} \mathrm{C} \text { to } 85.0^{\circ} \mathrm{C} \text { ? }\)

In an effort to stay awake for an all-night study session, a student makes a cup of coffee by first placing a 200-W electric immersion heater in 0.320 kg of water. (a) How much heat must be added to the water to raise its temperature from 20.0°C to 80.0°C? (b) How much time is required? Assume that all of the heater’s power goes into heating the water

While running, a 70-kg student generates thermal energy at a rate of 1200 W. For the runner to maintain a constant body temperature of 37C, this energy must be removed by perspiration or other mechanisms. If these mechanisms failed and the energy could not flow out of the students body, for what amount of time could a student run before irreversible body damage occurred? (Note: Protein structures in the body are irreversibly damaged if body temperature rises to 44C or higher. The specific heat of a typical human body is 3480 J>kg # K, slightly less than that of water. The difference is due to the presence of protein, fat, and minerals, which have lower specific heats.)

Heat Loss During Breathing. In very cold weather a significant mechanism for heat loss by the human body is energy expended in warming the air taken into the lungs with each breath. (a) On a cold winter day when the temperature is -20C, what amount of heat is needed to warm to body temperature 137C2 the 0.50 L of air exchanged with each breath? Assume that the specific heat of air is 1020 J>kg # K and that 1.0 L of air has mass 1.3 * 10-3 kg. (b) How much heat is lost per hour if the respiration rate is 20 breaths per minute?

You are given a sample of metal and asked to determine its specific heat. You weigh the sample and find that its weight is 28.4 N. You carefully add 1.25 * 104 J of heat energy to the sample and find that its temperature rises 18.0 C. What is the samples specific heat?

On-Demand Water Heaters. Conventional hot-water heaters consist of a tank of water maintained at a fixed temperature. The hot water is to be used when needed. The drawbacks are that energy is wasted because the tank loses heat when it is not in use and that you can run out of hot water if you use too much. Some utility companies are encouraging the use of on-demand water heaters (also known as flash heaters), which consist of heating units to heat the water as you use it. No water tank is involved, so no heat is wasted. A typical household shower flow rate is 2.5 gal>min (9.46 L>min) with the tap water being heated from 50F (10C) to 120F (49C) by the on-demand heater. What rate of heat input (either electrical or from gas) is required to operate such a unit, assuming that all the heat goes into the water?

While painting the top of an antenna 225 m in height, a worker accidentally lets a 1.00-L water bottle fall from his lunchbox. The bottle lands in some bushes at ground level and does not break. If a quantity of heat equal to the magnitude of the change in mechanical energy of the water goes into the water, what is its increase in temperature?

(a) On January 22, 1943, the temperature in Spearfish, South Dakota, rose from -4.0°F to 45.0°F in just 2 minutes. What was the temperature change in Celsius degrees? (b) The temperature in Browning, Montana, was 44.0°F on January 23, 1916. The next day the temperature plummeted to -56°F. What was the temperature change in Celsius degrees?

A 25,000-kg subway train initially traveling at 15.5 m>s slows to a stop in a station and then stays there long enough for its brakes to cool. The stations dimensions are 65.0 m long by 20.0 m wide by 12.0 m high. Assuming all the work done by the brakes in stopping the train is transferred as heat uniformly to all the air in the station, by how much does the air temperature in the station rise? Take the density of the air to be 1.20 kg>m3 and its specific heat to be 1020 J>kg # K.

A technician measures the specific heat of an unidentified liquid by immersing an electrical resistor in it. Electrical energy is converted to heat transferred to the liquid for 120 s at a constant rate of 65.0 W. The mass of the liquid is 0.780 kg, and its temperature increases from 18.55C to 22.54C. (a) Find the average specific heat of the liquid in this temperature range. Assume that negligible heat is transferred to the container that holds the liquid and that no heat is lost to the surroundings. (b) Suppose that in this experiment heat transfer from the liquid to the container or surroundings cannot be ignored. Is the result calculated in part (a) an overestimate or an underestimate of the average specific heat? Explain

A 15.0-g bullet traveling horizontally at 865 m>s passes through a tank containing 13.5 kg of water and emerges with a speed of 534 m>s. What is the maximum temperature increase that the water could have as a result of this event?

You have 750 g of water at \(10.0^\circ \mathrm C\) in a large insulated beaker. How much boiling water at \(100.0^\circ \mathrm C\) must you add to this beaker so that the final temperature of the mixture will be \(75^\circ \mathrm C\)?

A 500.0-g chunk of an unknown metal, which has been in boiling water for several minutes, is quickly dropped into an insulating Styrofoam beaker containing 1.00 kg of water at room temperature 120.0C2. After waiting and gently stirring for 5.00 minutes, you observe that the waters temperature has reached a constant value of 22.0C. (a) Assuming that the Styrofoam absorbs a negligibly small amount of heat and that no heat was lost to the surroundings, what is the specific heat of the metal? (b) Which is more useful for storing thermal energy: this metal or an equal weight of water? Explain. (c) If the heat absorbed by the Styrofoam actually is not negligible, how would the specific heat you calculated in part (a) be in error? Would it be too large, too small, or still correct? Explain.

Treatment for a Stroke. One suggested treatment for a person who has suffered a stroke is immersion in an ice-water bath at 0°C to lower the body temperature, which prevents damage to the brain. In one set of tests, patients were cooled until their internal temperature reached 32.0°C. To treat a 70.0-kg patient, what is the minimum amount of ice (at 0°C) you need in the bath so that its temperature remains at 0°C? The specific heat of the human body is \(3480 \mathrm{\ J} / \mathrm{kg} \cdot \mathrm{C}^{\circ}\), and recall that normal body temperature is 37.0°C

A blacksmith cools a 1.20-kg chunk of iron, initially at 650.0C, by trickling 15.0C water over it. All of the water boils away, and the iron ends up at 120.0C. How much water did the blacksmith trickle over the iron?

A copper calorimeter can with mass 0.100 kg contains 0.160 kg of water and 0.0180 kg of ice in thermal equilibrium at atmospheric pressure. If 0.750 kg of lead at 255°C is dropped into the calorimeter can, what is the final temperature? Assume that no heat is lost to the surroundings.

A copper pot with a mass of 0.500 kg contains 0.170 kg of water, and both are at 20.0C. A 0.250-kg block of iron at 85.0C is dropped into the pot. Find the final temperature of the system, assuming no heat loss to the surroundings

In a container of negligible mass, 0.200 kg of ice at an initial temperature of -40.0C is mixed with a mass m of water that has an initial temperature of 80.0C. No heat is lost to the surroundings. If the final temperature of the system is 28.0C, what is the mass m of the water that was initially at 80.0C?

A 6.00-kg piece of solid copper metal at an initial temperature T is placed with 2.00 kg of ice that is initially at -20.0C. The ice is in an insulated container of negligible mass and no heat is exchanged with the surroundings. After thermal equilibrium is reached, there is 1.20 kg of ice and 0.80 kg of liquid water. What was the initial temperature of the piece of copper?

Before going in for his annual physical, a 70.0-kg man whose body temperature is 37.0C consumes an entire 0.355-L can of a soft drink (mostly water) at 12.0C. (a) What will his body temperature be after equilibrium is attained? Ignore any heating by the mans metabolism. The specific heat of the mans body is 3480 J>kg # K. (b) Is the change in his body temperature great enough to be measured by a medical thermometer?

Basal Metabolic Rate. In the situation described in Exercise 17.42, the mans metabolism will eventually return the temperature of his body (and of the soft drink that he consumed) to 37.0C. If his body releases energy at a rate of 7.00 * 103 kJ>day (the basal metabolic rate, or BMR), how long does this take? Assume that all of the released energy goes into raising the temperature

An ice-cube tray of negligible mass contains 0.290 kg of water at 18.0C. How much heat must be removed to cool the water to 0.00C and freeze it? Express your answer in joules, calories, and Btu

How much heat is required to convert 18.0 g of ice at -10.0C to steam at 100.0C? Express your answer in joules, calories, and Btu

An open container holds 0.550 kg of ice at -15.0C. The mass of the container can be ignored. Heat is supplied to the container at the constant rate of 800.0 J>min for 500.0 min. (a) After how many minutes does the ice start to melt? (b) After how many minutes, from the time when the heating is first started, does the temperature begin to rise above 0.0C? (c) Plot a curve showing the temperature as a function of the elapsed time

What must the initial speed of a lead bullet be at 25.0°C so that the heat developed when it is brought to rest will be just sufficient to melt it? Assume that all the initial mechanical energy of the bullet is converted to heat and that no heat flows from the bullet to its surroundings. (Typical rifles have muzzle speeds that exceed the speed of sound in air, which is 347 m/s at 25.0°C.)

Steam Burns Versus Water Burns. What is the amount of heat input to your skin when it receives the heat released (a) by 25.0 g of steam initially at 100.0C, when it is cooled to skin temperature 134.0C2? (b) By 25.0 g of water initially at 100.0C, when it is cooled to 34.0C? (c) What does this tell you about the relative severity of burns from steam versus burns from hot water?

The Ship of the Desert. Camels require very little water because they are able to tolerate relatively large changes in their body temperature. While humans keep their body temperatures constant to within one or two Celsius degrees, a dehydrated camel permits its body temperature to drop to 34.0C overnight and rise to 40.0C during the day. To see how effective this mechanism is for saving water, calculate how many liters of water a 400-kg camel would have to drink if it attempted to keep its body temperature at a constant 34.0C by evaporation of sweat during the day (12 hours) instead of letting it rise to 40.0C. (Note: The specific heat of a camel or other mammal is about the same as that of a typical human, 3480 J>kg # K. The heat of vaporization of water at 34C is 2.42 * 106 J>kg.)

Evaporation of sweat is an important mechanism for temperature control in some warm-blooded animals. (a) What mass of water must evaporate from the skin of a 70.0-kg man to cool his body 1.00 C? The heat of vaporization of water at body temperature 137C2 is 2.42 * 106 J>kg. The specific heat of a typical human body is 3480 J>kg # K (see Exercise 17.25). (b) What volume of water must the man drink to replenish the evaporated water? Compare to the volume of a soft-drink can 1355 cm3 2.

An asteroid with a diameter of 10 km and a mass of 2.60 * 1015 kg impacts the earth at a speed of 32.0 km>s, landing in the Pacific Ocean. If 1.00% of the asteroids kinetic energy goes to boiling the ocean water (assume an initial water temperature of 10.0C), what mass of water will be boiled away by the collision? (For comparison, the mass of water contained in Lake Superior is about 2 * 1015 kg.)

A laboratory technician drops a 0.0850-kg sample of unknown solid material, at 100.0C, into a calorimeter. The calorimeter can, initially at 19.0C, is made of 0.150 kg of copper and contains 0.200 kg of water. The final temperature of the calorimeter can and contents is 26.1C. Compute the specific heat of the sample.

. An insulated beaker with negligible mass contains 0.250 kg of water at 75.0C. How many kilograms of ice at -20.0C must be dropped into the water to make the final temperature of the system 40.0C?

A 4.00-kg silver ingot is taken from a furnace, where its temperature is 750.0C, and placed on a large block of ice at 0.0C. Assuming that all the heat given up by the silver is used to melt the ice, how much ice is melted?

A vessel whose walls are thermally insulated contains 2.40 kg of water and 0.450 kg of ice, all at 0.0C. The outlet of a tube leading from a boiler in which water is boiling at atmospheric pressure is inserted into the water. How many grams of steam must condense inside the vessel (also at atmospheric pressure) to raise the temperature of the system to 28.0C? You can ignore the heat transferred to the container.

Two rods, one made of brass and the other made of copper, are joined end to end. The length of the brass section is 0.300 m and the length of the copper section is 0.800 m. Each segment has a cross-sectional area \(0.00500 \mathrm{\ m}^{2}\). The free end of the brass segment is in boiling water and the free end of the copper segment is in an ice–water mixture, in both cases under normal atmospheric pressure. The sides of the rods are insulated so there is no heat loss to the surroundings. (a) What is the temperature of the point where the brass and copper segments are joined? (b) What mass of ice is melted in 5.00 min by the heat conducted by the composite rod?

Suppose that the rod in Fig. 17.24a is made of copper, is 45.0 cm long, and has a cross-sectional area of 1.25 cm2 . Let TH = 100.0C and TC = 0.0C. (a) What is the final steady-state temperature gradient along the rod? (b) What is the heat current in the rod in the final steady state? (c) What is the final steady-state temperature at a point in the rod 12.0 cm from its left end?

One end of an insulated metal rod is maintained at \(100.0^{\circ} \mathrm{C}\), and the other end is maintained at \(0.00^{\circ} \mathrm{C}\) by an ice– water mixture. The rod is 60.0 cm long and has a cross-sectional area of \(1.25\mathrm{\ cm}^2\). The heat conducted by the rod melts 8.50 g of ice in 10.0 min. Find the thermal conductivity k of the metal.

A carpenter builds an exterior house wall with a layer of wood 3.0 cm thick on the outside and a layer of Styrofoam insulation 2.2 cm thick on the inside wall surface. The wood has k = 0.080 W>m # K, and the Styrofoam has k = 0.027 W/m # K. The interior surface temperature is 19.0C, and the exterior surface temperature is -10.0C. (a) What is the temperature at the plane where the wood meets the Styrofoam? (b) What is the rate of heat flow per square meter through this wall?

An electric kitchen range has a total wall area of \(1.40 \ \mathrm m^2\) and is insulated with a layer of fiberglass 4.00 cm thick. The inside surface of the fiberglass has a temperature of \(175^\circ \ \mathrm C\), and its outside surface is at \(35.0^\circ \ \mathrm C\). The fiberglass has a thermal conductivity of \(0.040 \ \mathrm {W/m} \cdot \mathrm K\). (a) What is the heat current through the insulation, assuming it may be treated as a flat slab with an area of \(1.40 \ \mathrm m^2\) ? (b) What electric-power input to the heating element is required to maintain this temperature?

Conduction Through the Skin. The blood plays an important role in removing heat from the body by bringing this energy directly to the surface where it can radiate away. Nevertheless, this heat must still travel through the skin before it can radiate away. Assume that the blood is brought to the bottom layer of skin at 37.0C and that the outer surface of the skin is at 30.0C. Skin varies in thickness from 0.50 mm to a few millimeters on the palms and soles, so assume an average thickness of 0.75 mm. A 165-lb, 6-ft-tall person has a surface area of about 2.0 m2 and loses heat at a net rate of 75 W while resting. On the basis of our assumptions, what is the thermal conductivity of this persons skin?

A long rod, insulated to prevent heat loss along its sides, is in perfect thermal contact with boiling water (at atmospheric pressure) at one end and with an icewater mixture at the other (Fig. E17.62). The rod consists of a 1.00-m section of copper (one end in boiling water) joined end to end to a length L2 of steel (one end in the icewater mixture). Both sections of the rod have crosssectional areas of 4.00 cm2 . The temperature of the coppersteel junction is 65.0C after a steady state has been set up. (a) How much heat per second flows from the boiling water to the ice water mixture? (b) What is the length L2 of the steel section?

A pot with a steel bottom 8.50 mm thick rests on a hot stove. The area of the bottom of the pot is \(0.150 \ \mathrm m^2\). The water inside the pot is at \(100.0^\circ \mathrm C\), and 0.390 kg are evaporated every 3.00 min. Find the temperature of the lower surface of the pot, which is in contact with the stove.

You are asked to design a cylindrical steel rod 50.0 cm long, with a circular cross section, that will conduct 190.0 J>s from a furnace at 400.0C to a container of boiling water under 1 atmosphere. What must the rods diameter be?

A picture window has dimensions of 1.40 m * 2.50 m and is made of glass 5.20 mm thick. On a winter day, the temperature of the outside surface of the glass is -20.0C, while the temperature of the inside surface is a comfortable 19.5C. (a) At what rate is heat being lost through the window by conduction? (b) At what rate would heat be lost through the window if you covered it with a 0.750-mm-thick layer of paper (thermal conductivity 0.0500 W/m # K)?

What is the rate of energy radiation per unit area of a blackbody at (a) 273 K and (b) 2730 K?

A spherical pot contains 0.75 L of hot coffee (essentially water) at an initial temperature of 95C. The pot has an emissivity of 0.60, and the surroundings are at 20.0C. Calculate the coffees rate of heat loss by radiation

The emissivity of tungsten is 0.350. A tungsten sphere with radius 1.50 cm is suspended within a large evacuated enclosure whose walls are at 290.0 K. What power input is required to maintain the sphere at 3000.0 K if heat conduction along the supports is ignored?

Size of a Light-Bulb Filament. The operating temperature of a tungsten filament in an incandescent light bulb is 2450 K, and its emissivity is 0.350. Find the surface area of the filament of a 150-W bulb if all the electrical energy consumed by the bulb is radiated by the filament as electromagnetic waves. (Only a fraction of the radiation appears as visible light.)

The Sizes of Stars. The hot glowing surfaces of stars emit energy in the form of electromagnetic radiation. It is a good approximation to assume e = 1 for these surfaces. Find the radii of the following stars (assumed to be spherical): (a) Rigel, the bright blue star in the constellation Orion, which radiates energy at a rate of 2.7 * 1032 W and has surface temperature 11,000 K; (b) Procyon B (visible only using a telescope), which radiates energy at a rate of 2.1 * 1023 W and has surface temperature 10,000 K. (c) Compare your answers to the radius of the earth, the radius of the sun, and the distance between the earth and the sun. (Rigel is an example of a supergiant star, and Procyon B is an example of a white dwarf star.)

A Foucault pendulum consists of a brass sphere with a diameter of 35.0 cm suspended from a steel cable 10.5 m long (both measurements made at 20.0°C). Due to a design oversight, the swinging sphere clears the floor by a distance of only 2.00 mm when the temperature is 20.0°C. At what temperature will the sphere begin to brush the floor?

Suppose that a steel hoop could be constructed to fit just around the earths equator at 20.0C. What would be the thickness of space between the hoop and the earth if the temperature of the hoop were increased by 0.500 C?

You propose a new temperature scale with temperatures given in M. You define 0.0M to be the normal melting point of mercury and 100.0M to be the normal boiling point of mercury. (a) What is the normal boiling point of water in M? (b) A temperature change of 10.0 M corresponds to how many C?

A 250-kg weight is hanging from the ceiling by a thin copper wire. In its fundamental mode, this wire vibrates at the frequency of concert A (440 Hz). You then increase the temperature of the wire by 40 C. (a) By how much will the fundamental frequency change? Will it increase or decrease? (b) By what percentage will the speed of a wave on the wire change? (c) By what percentage will the wavelength of the fundamental standing wave change? Will it increase or decrease?

You are making pesto for your pasta and have a cylindrical measuring cup 10.0 cm high made of ordinary glass 3b = 2.7 * 10-5 1C2-1 4 that is filled with olive oil 3b = 6.8 * 10-4 1C2-1 4 to a height of 3.00 mm below the top of the cup. Initially, the cup and oil are at room temperature 122.0C2. You get a phone call and forget about the olive oil, which you inadvertently leave on the hot stove. The cup and oil heat up slowly and have a common temperature. At what temperature will the olive oil start to spill out of the cup?

A surveyors 30.0-m steel tape is correct at 20.0C. The distance between two points, as measured by this tape on a day when its temperature is 5.00C, is 25.970 m. What is the true distance between the points?

A metal rod that is 30.0 cm long expands by 0.0650 cm when its temperature is raised from 0.0C to 100.0C. A rod of a different metal and of the same length expands by 0.0350 cm for the same rise in temperature. A third rod, also 30.0 cm long, is made up of pieces of each of the above metals placed end to end and expands 0.0580 cm between 0.0C and 100.0C. Find the length of each portion of the composite rod.

On a cool 14.0C2 Saturday morning, a pilot fills the fuel tanks of her Pitts S-2C (a two-seat aerobatic airplane) to their full capacity of 106.0 L. Before flying on Sunday morning, when the temperature is again 4.0C, she checks the fuel level and finds only 103.4 L of gasoline in the aluminum tanks. She realizes that it was hot on Saturday afternoon and that thermal expansion of the gasoline caused the missing fuel to empty out of the tanks vent. (a) What was the maximum temperature 1in C2 of the fuel and the tank on Saturday afternoon? The coefficient of volume expansion of gasoline is 9.5 * 10-4 K-1 . (b) To have the maximum amount of fuel available for flight, when should the pilot have filled the fuel tanks?

(a) Equation (17.12) gives the stress required to keep the length of a rod constant as its temperature changes. Show that if the length is permitted to change by an amount L when its temperature changes by T, the stress is equal to F A = Y a L L0 - a Tb where F is the tension on the rod, L0 is the original length of the rod, A its cross-sectional area, a its coefficient of linear expansion, and Y its Youngs modulus. (b) A heavy brass bar has projections at its ends (Fig. P17.79). Two fine steel wires, fastened between the projections, are just taut (zero tension) when the whole system is at 20C. What is the tensile stress in the steel wires when the temperature of the system is raised to 140C? Make any simplifying assumptions you think are justified, but state them

A metal wire, with density r and Youngs modulus Y, is stretched between rigid supports. At temperature T, the speed of a transverse wave is found to be v1. When the temperature is increased to T + T, the speed decreases to v2 6 v1. Determine the coefficient of linear expansion of the wire.

A steel ring with a 2.5000-in. inside diameter at 20.0C is to be warmed and slipped over a brass shaft with a 2.5020-in. outside diameter at 20.0C. (a) To what temperature should the ring be warmed? (b) If the ring and the shaft together are cooled by some means such as liquid air, at what temperature will the ring just slip off the shaft?

Doughnuts: Breakfast of Champions! A typical doughnut contains 2.0 g of protein, 17.0 g of carbohydrates, and 7.0 g of fat. Average food energy values are 4.0 kcal>g for protein and carbohydrates and 9.0 kcal>g for fat. (a) During heavy exercise, an average person uses energy at a rate of 510 kcal>h. How long would you have to exercise to work off one doughnut? (b) If the energy in the doughnut could somehow be converted into the kinetic energy of your body as a whole, how fast could you move after eating the doughnut? Take your mass to be 60 kg, and express your answer in m>s and in km>h.

Shivering. Shivering is your bodys way of generating heat to restore its internal temperature to the normal 37C, and it produces approximately 290 W of heat power per square meter of body area. A 68-kg, 1.78-m-tall woman has approximately 1.8 m2 of surface area. How long would this woman have to shiver to raise her body temperature by 1.0 C, assuming that the body loses none of this heat? The bodys specific heat capacity is about 3500 J>kg # K

You cool a 100.0-g slug of red-hot iron (temperature 745C) by dropping it into an insulated cup of negligible mass containing 85.0 g of water at 20.0C. Assuming no heat exchange with the surroundings, (a) what is the final temperature of the water and (b) what is the final mass of the iron and the remaining water?

C Debyes T3 Law. At very low temperatures the molar heat capacity of rock salt varies with temperature according to Debyes T3 law: C = k T3 u3 where k = 1940 J>mol # K and u = 281 K. (a) How much heat is required to raise the temperature of 1.50 mol of rock salt from 10.0 K to 40.0 K? (Hint: Use Eq. (17.18) in the form dQ = nC dT and integrate.) (b) What is the average molar heat capacity in this range? (c) What is the true molar heat capacity at 40.0 K?

A person of mass 70.0 kg is sitting in the bathtub. The bathtub is 190.0 cm by 80.0 cm; before the person got in, the water was 24.0 cm deep. The water is at \(37.0^{\circ} \mathrm{C} \text {. }\) Suppose that the water were to cool down spontaneously to form ice at \(0.0^{\circ} \mathrm{C}\), and that all the energy released was used to launch the hapless bather vertically into the air. How high would the bather go? (As you will see in Chapter 20, this event is allowed by energy conservation but is prohibited by the second law of thermodynamics.)

. Hot Air in a Physics Lecture. (a) A typical student listening attentively to a physics lecture has a heat output of 100 W. How much heat energy does a class of 140 physics students release into a lecture hall over the course of a 50-min lecture? (b) Assume that all the heat energy in part (a) is transferred to the 3200 m3 of air in the room. The air has specific heat 1020 J>kg # K and density 1.20 kg>m3 . If none of the heat escapes and the air conditioning system is off, how much will the temperature of the air in the room rise during the 50-min lecture? (c) If the class is taking an exam, the heat output per student rises to 280 W. What is the temperature rise during 50 min in this case?

The molar heat capacity of a certain substance varies with temperature according to the empirical equation C = 29.5 J>mol # K + 18.20 * 10-3 J>mol # K2 2T How much heat is necessary to change the temperature of 3.00 mol of this substance from 27C to 227C? (Hint: Use Eq. (17.18) in the form dQ = nC dT and integrate.)

Bicycling on a Warm Day. If the air temperature is the same as the temperature of your skin (about 30C), your body cannot get rid of heat by transferring it to the air. In that case, it gets rid of the heat by evaporating water (sweat). During bicycling, a typical 70-kg persons body produces energy at a rate of about 500 W due to metabolism, 80% of which is converted to heat. (a) How many kilograms of water must the persons body evaporate in an hour to get rid of this heat? The heat of vaporization of water at body temperature is 2.42 * 106 J>kg. (b) The evaporated water must, of course, be replenished, or the person will dehydrate. How many 750-mL bottles of water must the bicyclist drink per hour to replenish the lost water? (Recall that the mass of a liter of water is 1.0 kg.)

Overheating. (a) By how much would the body temperature of the bicyclist in Problem 17.89 increase in an hour if he were unable to get rid of the excess heat? (b) Is this temperature increase large enough to be serious? To find out, how high a fever would it be equivalent to, in F? (Recall that the normal internal body temperature is 98.6F and the specific heat of the body is 3480 J>kg # C.)

A Thermodynamic Process in an Insect. The African bombardier beetle (Stenaptinus insignis) can emit a jet of defensive spray from the movable tip of its abdomen (Fig. P17.91). The beetles body has reservoirs containing two chemicals; when the beetle is disturbed, these chemicals combine in a reaction chamber, producing a compound that is warmed from 20C to 100C by the heat of reaction. The high pressure produced allows the compound to be sprayed out at speeds up to 19 m>s 168 km>h2, scaring away predators of all kinds. (The beetle shown in Fig. P17.91 is 2 cm long.) Calculate the heat of reaction of the two chemicals (in J>kg). Assume that the specific heat of the chemicals and of the spray is the same as that of water, 4.19 * 103 J>kg # K, and that the initial temperature of the chemicals is 20C.

Hot Water Versus Steam Heating. In a household hot-water heating system, water is delivered to the radiators at 70.0C 1158.0F2 and leaves at 28.0C 182.4F2. The system is to be replaced by a steam system in which steam at atmospheric pressure condenses in the radiators and the condensed steam leaves the radiators at 35.0C 195.0F2. How many kilograms of steam will supply the same heat as was supplied by 1.00 kg of hot water in the first system?

You have 1.50 kg of water at 28.0C in an insulated container of negligible mass. You add 0.600 kg of ice that is initially at -22.0C. Assume that no heat exchanges with the surroundings. (a) After thermal equilibrium has been reached, has all of the ice melted? (b) If all of the ice has melted, what is the final temperature of the water in the container? If some ice remains, what is the final temperature of the water in the container, and how much ice remains?

A thirsty nurse cools a 2.00-L bottle of a soft drink (mostly water) by pouring it into a large aluminum mug of mass 0.257 kg and adding 0.120 kg of ice initially at -15.0C. If the soft drink and mug are initially at 20.0C, what is the final temperature of the system, assuming that no heat is lost?

A copper calorimeter can with mass 0.446 kg contains 0.0950 kg of ice. The system is initially at 0.0C. (a) If 0.0350 kg of steam at 100.0C and 1.00 atm pressure is added to the can, what is the final temperature of the calorimeter can and its contents? (b) At the final temperature, how many kilograms are there of ice, how many of liquid water, and how many of steam?

A Styrofoam bucket of negligible mass contains 1.75 kg of water and 0.450 kg of ice. More ice, from a refrigerator at -15.0C, is added to the mixture in the bucket, and when thermal equilibrium has been reached, the total mass of ice in the bucket is 0.884 kg. Assuming no heat exchange with the surroundings, what mass of ice was added?

In a container of negligible mass, 0.0400 kg of steam at 100C and atmospheric pressure is added to 0.200 kg of water at 50.0C. (a) If no heat is lost to the surroundings, what is the final temperature of the system? (b) At the final temperature, how many kilograms are there of steam and how many of liquid water?

Mammal Insulation. Animals in cold climates often depend on two layers of insulation: a layer of body fat (of thermal conductivity 0.20 W>m # K) surrounded by a layer of air trapped inside fur or down. We can model a black bear (Ursus americanus) as a sphere 1.5 m in diameter having a layer of fat 4.0 cm thick. (Actually, the thickness varies with the season, but we are interested in hibernation, when the fat layer is thickest.) In studies of bear hibernation, it was found that the outer surface layer of the fur is at 2.7C during hibernation, while the inner surface of the fat layer is at 31.0C. (a) What is the temperature at the fat inner fur boundary so that the bear loses heat at a rate of 50.0 W? (b) How thick should the air layer (contained within the fur) be?

Effect of a Window in a Door. A carpenter builds a solid wood door with dimensions 2.00 m * 0.95 m * 5.0 cm. Its thermal conductivity is k = 0.120 W>m # K. The air films on the inner and outer surfaces of the door have the same combined thermal resistance as an additional 1.8-cm thickness of solid wood. The inside air temperature is 20.0C, and the outside air temperature is -8.0C. (a) What is the rate of heat flow through the door? (b) By what factor is the heat flow increased if a window 0.500 m on a side is inserted in the door? The glass is 0.450 cm thick, and the glass has a thermal conductivity of 0.80 W>m # K. The air films on the two sides of the glass have a total thermal resistance that is the same as an additional 12.0 cm of glass

One experimental method of measuring an insulating materials thermal conductivity is to construct a box of the material and measure the power input to an electric heater inside the box that maintains the interior at a measured temperature above the outside surface. Suppose that in such an apparatus a power input of 180 W is required to keep the interior surface of the box 65.0 C 1about 120 F2 above the temperature of the outer surface. The total area of the box is 2.18 m2 , and the wall thickness is 3.90 cm. Find the thermal conductivity of the material in SI units.

Compute the ratio of the rate of heat loss through a single-pane window with area 0.15 m2 to that for a double-pane window with the same area. The glass of a single pane is 4.2 mm thick, and the air space between the two panes of the double-pane window is 7.0 mm thick. The glass has thermal conductivity 0.80 W/m # K. The air films on the room and outdoor surfaces of either window have a combined thermal resistance of 0.15 m2 # K>W.

Rods of copper, brass, and steel—each with a cross-sectional area of \(2.00\mathrm{\ cm}^2\) —are welded together to form a Y-shaped figure. The free end of the copper rod is maintained at 100.0°C, and the free ends of the brass and steel rods at 0.0°C. Assume that there is no heat loss from the surfaces of the rods. The lengths of the rods are: copper, 13.0 cm; brass, 18.0 cm; steel, 24.0 cm. What is (a) the temperature of the junction point; (b) the heat current in each of the three rods?

A brass rod 12.0 cm long, a copper rod 18.0 cm long, and an aluminum rod 24.0 cm long—each with cross-sectional area \(2.30\mathrm{\ cm}^3-\text{ are }\) welded together end to end to form a rod 54.0 cm long, with copper as the middle section. The free end of the brass section is maintained at 100.0°C, and the free end of the aluminum section is maintained at 0.0°C. Assume that there is no heat loss from the curved surfaces and that the steady-state heat current has been established. What is (a) the temperature \(T_{1}\) at the junction of the brass and copper sections; (b) the temperature \(T_{2}\) at the junction of the copper and aluminum sections; (c) the heat current in the aluminum section?

Basal Metabolic Rate. The basal metabolic rate is the rate at which energy is produced in the body when a person is at rest. A 75-kg (165-lb) person of height 1.83 m (6 ft) has a body surface area of approximately 2.0 m2 . (a) What is the net amount of heat this person could radiate per second into a room at 18C (about 65F) if his skins surface temperature is 30C? (At such temperatures, nearly all the heat is infrared radiation, for which the bodys emissivity is 1.0, regardless of the amount of pigment.) (b) Normally, 80% of the energy produced by metabolism goes into heat, while the rest goes into things like pumping blood and repairing cells. Also normally, a person at rest can get rid of this excess heat just through radiation. Use your answer to part (a) to find this persons basal metabolic rate.

Time Needed for a Lake to Freeze Over. (a) When the air temperature is below 0°C, the water at the surface of a lake freezes to form an ice sheet. Why doesn’t freezing occur throughout the entire volume of the lake? (b) Show that the thickness of the ice sheet formed on the surface of a lake is proportional to the square root of the time if the heat of fusion of the water freezing on the underside of the ice sheet is conducted through the sheet. (c) Assuming that the upper surface of the ice sheet is at -10°C and the bottom surface is at 0°C, calculate the time it will take to form an ice sheet 25 cm thick. (d) If the lake in part (c) is uniformly 40 m deep, how long would it take to freeze all the water in the lake? Is this likely to occur?

The rate at which radiant energy from the sun reaches the earths upper atmosphere is about 1.50 kW/m2 . The distance from the earth to the sun is 1.50 * 1011 m, and the radius of the sun is 6.96 * 108 m. (a) What is the rate of radiation of energy per unit area from the suns surface? (b) If the sun radiates as an ideal blackbody, what is the temperature of its surface?

A Thermos for Liquid Helium. A physicist uses a cylindrical metal can 0.250 m high and 0.090 m in diameter to store liquid helium at 4.22 K; at that temperature the heat of vaporization of helium is 2.09 * 104 J>kg. Completely surrounding the metal can are walls maintained at the temperature of liquid nitrogen, 77.3 K, with vacuum between the can and the surrounding walls. How much helium is lost per hour? The emissivity of the metal can is 0.200. The only heat transfer between the metal can and the surrounding walls is by radiation.

A metal sphere with radius 3.20 cm is suspended in a large metal box with interior walls that are maintained at 30.0C. A small electric heater is embedded in the sphere. Heat energy must be supplied to the sphere at the rate of 0.660 J>s to maintain the sphere at a constant temperature of 41.0C. (a) What is the emissivity of the metal sphere? (b) What power input to the sphere is required to maintain it at 82.0C? What is the ratio of the power required for 82.0C to the power required for 41.0C? How does this ratio compare with 24 ? Explain.

Jogging in the Heat of the Day. You have probably seen people jogging in extremely hot weather. There are good reasons not to do this! When jogging strenuously, an average runner of mass 68 kg and surface area 1.85 m2 produces energy at a rate of up to 1300 W, 80% of which is converted to heat. The jogger radiates heat but actually absorbs more from the hot air than he radiates away. At such high levels of activity, the skins temperature can be elevated to around 33C instead of the usual 30C. (Ignore conduction, which would bring even more heat into his body.) The only way for the body to get rid of this extra heat is by evaporating water (sweating). (a) How much heat per second is produced just by the act of jogging? (b) How much net heat per second does the runner gain just from radiation if the air temperature is 40.0C (104F)? (Remember: He radiates out, but the environment radiates back in.) (c) What is the total amount of excess heat this runners body must get rid of per second? (d) How much water must his body evaporate every minute due to his activity? The heat of vaporization of water at body temperature is 2.42 * 106 J>kg. (e) How many 750-mL bottles of water must he drink after (or preferably before!) jogging for a half hour? Recall that a liter of water has a mass of 1.0 kg.

The Humber Bridge in England has the world’s longest single span, 1410 m. Calculate the change in length of the steel deck of the span when the temperature increases from -5.0°C to 18.0°C.

As a physicist, you put heat into a 500.0-g solid sample at the rate of 10.0 kJ/min while recording its temperature as a function of time. You plot your data as shown in Fig. P17.111. (a) What is the latent heat of fusion for this solid? (b) What are the specific heats of the liquid and solid states of this material?

At a chemical plant where you are an engineer, a tank contains an unknown liquid. You must determine the liquids specific heat capacity. You put 0.500 kg of the liquid into an insulated metal cup of mass 0.200 kg. Initially the liquid and cup are at 20.0C. You add 0.500 kg of water that has a temperature of 80.0o C. After thermal equilibrium has been reached, the final temperature of the two liquids and the cup is 58.1C. You then empty the cup and repeat the experiment with the same initial temperatures, but this time with 1.00 kg of the unknown liquid. The final temperature is 49.3C. Assume that the specific heat capacities are constant over the temperature range of the experiment and that no heat is lost to the surroundings. Calculate the specific heat capacity of the liquid and of the metal from which the cup is made.

During your mechanical engineering internship, you are given two uniform metal bars A and B, which are made from different metals, to determine their thermal conductivities. Measuring the bars, you determine that both have length 40.0 cm and uniform cross-sectional area 2.50 cm2 . You place one end of bar A in thermal contact with a very large vat of boiling water at 100.0C and the other end in thermal contact with an icewater mixture at 0.0C. To prevent heat loss along the bars sides, you wrap insulation around the bar. You weigh the amount of ice initially and find it to be 300 g. After 45.0 min has elapsed, you weigh the ice again and find that 191 g of ice remains. The icewater mixture is in an insulated container, so the only heat entering or leaving it is the heat conducted by the metal bar. You are confident that your data will allow you to calculate the thermal conductivity kA of bar A. But this measurement was tediousyou dont want to repeat it for bar B. Instead, you glue the bars together end to end, with adhesive that has very large thermal conductivity, to make a composite bar 80.0 m long. You place the free end of A in thermal contact with the boiling water and the free end of B in thermal contact with the icewater mixture. As in the first measurement, the composite bar is thermally insulated. You go to lunch; when you return, you notice that ice remains in the icewater mixture. Measuring the temperature at the junction of the two bars, you find that it is 62.4o C. After 10 minutes you repeat that measurement and get the same temperature, with ice remaining in the icewater mixture. From your data, calculate the thermal conductivities of bar A and of bar B.

A Walk in the Sun. Consider a poor lost soul walking at 5 km/h on a hot day in the desert, wearing only a bathing suit. This person’s skin temperature tends to rise due to four mechanisms: (i) energy is generated by metabolic reactions in the body at a rate of 280 W, and almost all of this energy is converted to heat that flows to the skin; (ii) heat is delivered to the skin by convection from the outside air at a rate equal to \(k^{\prime}A_{\text{skin }}\left(T_{\text{air }}-T_{\text{skin }}\right)\text{, where }k^{\prime}\text{ is }54\mathrm{\ J}/\mathrm{h}\cdot\mathrm{C}^{\circ}\cdot\mathrm{m}^2\), the exposed skin area \(A_{\text {skin }} \text { is } 1.5 \mathrm{\ m}^{2}\), the air temperature \(T_{\text {air }} \text { is } 47^{\circ} \mathrm{C}\), and the skin temperature \(T_{\text {skin }} \text { is } 36^{\circ} \mathrm{C}\); (iii) the skin absorbs radiant energy from the sun at a rate of \(1400\mathrm{\ W}/\mathrm{m}^2\); (iv) the skin absorbs radiant energy from the environment, which has temperature 47°C. (a) Calculate the net rate (in watts) at which the person’s skin is heated by all four of these mechanisms. Assume that the emissivity of the skin is \(e=1\) and that the skin temperature is initially 36°C. Which mechanism is the most important? (b) At what rate (in L/h) must perspiration evaporate from this person’s skin to maintain a constant skin temperature? (The heat of vaporization of water at \(36^{\circ}\mathrm{C}\text{ is }2.42\times 10^6\mathrm{\ J}/\mathrm{kg}.\)) (c) Suppose instead the person is protected by light-colored clothing \((e \approx 0)\) so that the exposed skin area is only 0.45 m2 . What rate of perspiration is required now? Discuss the usefulness of the traditional clothing worn by desert peoples.

A hollow cylinder has length L, inner radius a, and outer radius b, and the temperatures at the inner and outer surfaces are T2 and T1. (The cylinder could represent an insulated hot-water pipe.) The thermal conductivity of the material of which the cylinder is made is k. Derive an equation for (a) the total heat current through the walls of the cylinder; (b) the temperature variation inside the cylinder walls. (c) Show that the equation for the total heat current reduces to Eq. (17.21) for linear heat flow when the cylinder wall is very thin. (d) A steam pipe with a radius of 2.00 cm, carrying steam at 140C, is surrounded by a cylindrical jacket with inner and outer radii 2.00 cm and 4.00 cm and made of a type of cork with thermal conductivity 4.00 * 10-2 W>m # K. This in turn is surrounded by a cylindrical jacket made of a brand of Styrofoam with thermal conductivity 2.70 * 10-2 W>m # K and having inner and outer radii 4.00 cm and 6.00 cm (Fig. P17.115). The outer surface of the Styrofoam has a temperature of 15C. What is the temperature at a radius of 4.00 cm, where the two insulating layers meet? (e) What is the total rate of transfer of heat out of a 2.00-m length of pipe?

You place 35 g of this cryoprotectant at 22C in contact with a cold plate that is maintained at the boiling temperature of liquid nitrogen (77 K). The cryoprotectant is thermally insulated from everything but the cold plate. Use the values in the table to determine how much heat will be transferred from the cryoprotectant as it reaches thermal equilibrium with the cold plate. (a) 1.5 * 104 J; (b) 2.9 * 104 J; (c) 3.4 * 104 J; (d) 4.4 * 104 J

Careful measurements show that the specific heat of the solid phase depends on temperature (Fig. P17.117). How will the actual time needed for this cryoprotectant to come to equilibrium with the cold plate compare with the time predicted by using the values in the table? Assume that all values other than the specific heat (solid) are correct. The actual time (a) will be shorter; (b) will be longer; (c) will be the same; (d) depends on the density of the cryoprotectant.

In another experiment, you place a layer of this cryoprotectant between one \(10 \ \mathrm{cm} \times 10 \ \mathrm{cm}\) cold plate maintained at \(-40^\circ \mathrm C\) and a second cold plate of the same size maintained at liquid nitrogen’s boiling temperature (77 K). Then you measure the rate of heat transfer. Another lab wants to repeat the experiment but uses cold plates that are \(20 \ \mathrm{cm} \times 20 \ \mathrm{cm}\), with one at \(-40^\circ \mathrm C\) and the other at 77 K. How thick does the layer of cryoprotectant have to be so that the rate of heat transfer by conduction is the same as that when you use the smaller plates? (a) One-quarter the thickness; (b) half the thickness; (c) twice the thickness; (d) four times the thickness.

To measure the specific heat in the liquid phase of a newly developed cryoprotectant, you place a sample of the new cryoprotectant in contact with a cold plate until the solutions temperature drops from room temperature to its freezing point. Then you measure the heat transferred to the cold plate. If the system isnt sufficiently isolated from its room-temperature surroundings, what will be the effect on the measurement of the specific heat? (a) The measured specific heat will be greater than the actual specific heat; (b) the measured specific heat will be less than the actual specific heat; (c) there will be no effect because the thermal conductivity of the cryoprotectant is so low; (d) there will be no effect on the specific heat, but the temperature of the freezing point will change.