Why Do the Seasons Lag? In the northern hemisphere, June

Convert the following Celsius temperatures to Fahrenheit: (a) the lowest temperature ever recorded in North America (February 3, 1947, Snag, Yukon); (b) the highest temperature ever recorded in the United States (July 10, 1913, Death Valley, California); (c) 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 body’s temperature can go as high as \(40^{\circ} \mathrm{C}\). 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 \(7\ \mathrm{C}^ {\circ}\) lower than the internal temperature. Express this temperature difference in kelvins and in Fahrenheit degrees. (d) Blood storage. Blood stored at \(4.0^{\circ} \mathrm{C}\) lasts safely for about 3 weeks, whereas blood stored at \(-160^{\circ} \mathrm{C}\) lasts for 5 years. Express both temperatures on the Fahrenheit and Kelvin scales. (e) Heat stroke. If the body’s temperature is above \(105^{\circ} \mathrm{F}\) 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 to in just 2 minutes. What was the temperature change in Celsius degrees? (b) The temperature in Browning, Montana, was on January 23, 1916. The next day the temperature plummeted to 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) \(\mathrm{F}^{\circ}\) and (b) \(\mathrm{C}^{\circ}\)?

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 K).

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 \(\mathrm CO_{2}\) solidifies?

A 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 to be and the pressure at the normal boiling point to be (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 were what pressure would the experimenter have measured at (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 \(\left(0^{\circ} \mathrm{R}\right)\). 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 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 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 \(20.0^{\circ} \mathrm{C}\). The coin is made of a metal alloy (mostly zinc) for which the coefficient of linear expansion is \(2.6\times10^{-5}\mathrm{\ K}^{-1}\). What would its diameter be on a hot day in Death Valley \(\left(48.0^{\circ} \mathrm{C}\right)\)? On a cold night in the mountains of Greenland \(\left(-53^{\circ} \mathrm{C}\right)\)?

Ensuring a Tight Fit. Aluminum rivets used in airplane construction are made slightly larger than the rivet holes and cooled by dry ice before being driven. If the diameter of a hole is 4.500 mm, what should be the diameter of a rivet at if its diameter is to equal that of the hole when the rivet is cooled to the temperature of dry ice? Assume that the expansion coefficient remains constant at the value given in Table 17.1.

The outer diameter of a glass jar and the inner diameter of its iron lid are both 725 mm at room temperature What will be the size of the difference in these diameters if the lid is briefly held under hot water until its temperature rises to without changing the temperature of the glass?

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 \(-15^{\circ} \mathrm{C}\). How much more interior space does the dome have in the summer, when the temperature is \(35^{\circ} \mathrm{C}\)? Text Transcription: -15^circ C 35^circ C

A copper cylinder is initially at \(20.0^{\circ} \mathrm{C}\). At what temperature will its volume be 0.150% larger than it is at \(20.0^{\circ} \mathrm{C}\)?

A steel tank is completely filled with of ethanol when both the tank and the ethanol are at a temperature of When the tank and its contents have cooled to what additional volume of ethanol can be put into the tank?

A glass flask whose volume is at is completely filled with mercury at this temperature. When flask and mercury are warmed to of mercury overflow. If the coefficient of volume expansion of mercury is compute the coefficient of volume expansion of the glass.

(a) If an area measured on the surface of a solid body is at some initial temperature and then changes by \(\Delta A\) when the temperature changes by \(\Delta T\), show that \(\Delta A=(2 \alpha) A_{0} \Delta T\) where \(\alpha\) is the coefficient of linear expansion. (b) A circular sheet of aluminum is 55.0 cm in diameter at \(15.0^{\circ} \mathrm{C}\). By how much does the area of one side of the sheet change when the temperature increases to \(27.5^{\circ} \mathrm{C}\)?

A machinist bores a hole of diameter 1.35 cm in a steel plate at a temperature of What is the cross- sectional area of the hole (a) at and (b) when the temperature of the plate is increased to Assume that the coefficient of linear expansion remains constant over this temperature range. (Hint: See Exercise 17.20.)

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 and Assume that the coefficients of expansion are constant over this temperature range. (a) If the piston just fits inside the chamber at will the engines be able to run at higher temperatures? Explain. (b) If the cylindrical pistons are 25.000 cm in diameter at what should be the minimum diameter of the cylinders at that temperature so the pistons will operate at

(a) A wire that is 1.50 m long at is found to increase in length by 1.90 cm when warmed to Compute its average coefficient of linear expansion for this temperature range. (b) The wire is stretched just taut (zero tension) at Find the stress in the wire if it is cooled to without being allowed to contract. Youngs modulus for the wire is

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

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 (a) How much space must be left between adjacent rails if they are just to touch on a summer day when their temperature is (b) If the rails are originally laid in contact, what is the stress in them on a summer day when their temperature is

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 to (b) How much time is required? Assume that all of the heaters power goes into heating the water.

An aluminum tea kettle with mass 1.50 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 to

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 what amount of heat is needed to warm to body temperature the 0.50 L of air exchanged with each breath? Assume that the specific heat of air is and that 1.0 L of air has mass (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 \(1.25 \times 10^{4} \ \mathrm{J}\) of heat energy to the sample and find that its temperature rises \(18.0 ^{\circ} \mathrm{C}\). What is the sample's 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 running, a 70-kg student generates thermal energy at a rate of 1200 W. For the runner to maintain a constant body temperature of \(37^{\circ} \mathrm{C}\), this energy must be removed by perspiration or other mechanisms. If these mechanisms failed and the heat 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 \(44^{\circ} \mathrm{C}\) or higher. The specific heat of a typical human body is \(3480\mathrm{\ J}/\mathrm{kg}\cdot\mathrm{K}\), slightly less than that of water. The difference is due to the presence of protein, fat, and minerals, which have lower specific heats.)

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 crate of fruit with mass 35.0 kg and specific heat slides down a ramp inclined at below the horizontal. The ramp is 8.00 m long. (a) If the crate was at rest at the top of the incline and has a speed of at the bottom, how much work was done on the crate by friction? (b) If an amount of heat equal to the magnitude of the work done by friction goes into the crate of fruit and the fruit reaches a uniform final temperature, what is its temperature change?

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 station's 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 \(1.20\mathrm{\ kg}/\mathrm{m}^3\) and its specific heat to be \(1020\mathrm{\ J}/\mathrm{kg}\cdot\mathrm{K}\).

A nail driven into a board increases in temperature. If we assume that 60% of the kinetic energy delivered by a 1.80-kg hammer with a speed of is transformed into heat that flows into the nail and does not flow out, what is the temperature increase of an 8.00-g aluminum nail after it is struck ten times?

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 to (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?

As a physicist, you put heat into a 500.0-g solid sample at the rate of while recording its temperature as a function of time. You plot your data and obtain the graph shown in Fig. E17.38. (a) What is the latent heat of fusion for this solid? (b) What are the specific heats of the liquid and solid states of the material?

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 After waiting and gently stirring for 5.00 minutes, you observe that the waters temperature has reached a constant value of (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) What 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 0C to lower the body temperature, which prevents damage to the brain. In one set of tests, patients were cooled until their internal temperature reached 32.0C. To treat a 70.0-kg patient, what is the minimum amount of ice (at 0C) you need in the bath so that its temperature remains at 0C? The specific heat of the human body is and recall that normal body temperature is 37.0C.

A copper pot with a mass of 0.500 kg contains 0.170 kg of water, and both are at a temperature of 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.

Bicycling on a Warm Day. If the air temperature is the same as the temperature of your skin (about \(30^{\circ} \mathrm{C}\)), 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 person's 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 person's body evaporate in an hour to get rid of this heat? The heat of vaporization of water at body temperature is \(2.42\times10^6\mathrm{\ J}/\mathrm{kg}\). (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 the preceeding problem increase in an hour if he were unable to get rid of the excess heat? (b) Is this emperature 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 )

In a container of negligible mass, 0.200 kg of ice at an initial temperature of \(-40.0^{\circ} \mathrm{C}\) is mixed with a mass m of water that has an initial temperature of \(80.0^{\circ} \mathrm{C}\). No heat is lost to the surroundings. If the final temperature of the system is \(20.0^{\circ} \mathrm{C}\), what is the mass m of the water that was initially at \(80.0^{\circ} \mathrm{C}\)?

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 . 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?

O Before going in for his annual physical, a 70.0-kg man whose body temperature is consumes an entire 0.355-L can of a soft drink (mostly water) at (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 (b) Is the change in his body temperature great enough to be measured by a medical thermometer?

In the situation described in Exercise 17.46, the mans metabolism will eventually return the temperature of his body (and of the soft drink that he consumed) to If his body releases energy at a rate of (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 0.350 kg of water at How much heat must be removed to cool the water to and freeze it? Express your answer in joules, calories, and Btu.

How much heat is required to convert 12.0 g of ice at \(-10.0^{\circ} \mathrm{C}\) to steam at \(100.0^{\circ} \mathrm{C}\)? Express your answer in joules, calories, and Btu.

An open container holds 0.550 kg of ice at \(-15.0^{\circ} \mathrm{C}\). 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 \(0.0^{\circ} \mathrm{C} \text{? }\) (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 a temperature of 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

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 when it is cooled to skin temperature (b) By 25.0 g of water initially at when it is cooled to (c) What does this tell you about the relative severity of steam and hot water burns?

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 overnight and rise to 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 by evaporation of sweat during the day (12 hours) instead of letting it rise to (Note: The specific heat of a camel or other mammal is about the same as that of a typical human, The heat of vaporization of water at

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 137°C2 is \(2.42 \times 10^{6} J/kg\). The specific heat of a typical human body is \(3480 J / kg \cdot K\) (see Exercise 17.31). (b) What volume of water must the man drink to replenish the evaporated water? Compare to the volume of a soft-drink can \((355 cm^{3})\). Text Transcription: 2.42 x 10^6 J/kg 3480 J/kg cdot K

An asteroid with a diameter of 10 km and a mass of impacts the earth at a speed of 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 what mass of water will be boiled away by the collision? (For comparison, the mass of water contained in Lake Superior is about

A laboratory technician drops a 0.0850-kg sample of unknown solid material, at a temperature of \(100.0^{\circ} \mathrm{C}\) into a calorimeter. The calorimeter can, initially at \(19.0^{\circ} \mathrm{C}\), 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 \(26.1^{\circ} \mathrm{C}\). Compute the specific heat of the sample.

An insulated beaker with negligible mass contains 0.250 kg of water at a temperature of How many kilograms of ice at a temperature of must be dropped into the water to make the final temperature of the system

A glass vial containing a 16.0-g sample of an enzyme is cooled in an ice bath. The bath contains water and 0.120 kg of ice. The sample has specific heat the glass vial has mass 6.00 g and specific heat How much ice melts in cooling the enzyme sample from room temperature to the temperature of the ice bath?

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

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 a temperature of is dropped into the calorimeter can, what is the final temperature? Assume that no heat is lost to the surroundings.

A vessel whose walls are thermally insulated contains 2.40 kg of water and 0.450 kg of ice, all at a temperature of 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 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 0.200 m and the length of the copper section is 0.800 m. Each segment has cross- sectional area . The free end of the brass segment is in boiling water and the free end of the copper segment is in an ice and 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. 17.23a is made of copper, is 45.0 cm long, and has a cross-sectional area of Let and (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 and the Styrofoam has The interior surface temperature is and the exterior surface temperature is (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 \(0.040\mathrm{\ W}/\mathrm{m}\cdot\mathrm{K}\). (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 heat directly to the surface where it can radiate away. Nevertheless, this heat must still travel through the skin before it can radiate away. We shall assume that the blood is brought to the bottom layer of skin at a temperature of 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 we shall assume an average thickness of 0.75 mm. A 165-lb, 6-ft-tall person has a surface area of about 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 ice-water mixture at the other (Fig. E17.68). The rod consists of a 1.00-m section of copper (one end in boiling water) joined end to end to a length \(L_{2}\) of steel (one end in the ice-water mixture). Both sections of the rod have cross-sectional areas of \(4.00\mathrm{\ cm}^2\). The temperature of the copper-steel junction is \(65.0^{\circ} \mathrm{C}\) 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 \(L_{2}\) 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 The water inside the pot is at 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 from a furnace at to a container of boiling water under 1 atmosphere. What must the rods diameter be?

A picture window has dimensions of and is made of glass 5.20 mm thick. On a winter day, the outside temperature is while the inside temperature is a comfortable (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 )?

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

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 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 a temperature of 3000.0 K if heat conduction along the supports is neglected?

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 \(2.7\times10^{32}\mathrm{\ W}\) and has surface temperature 11,000 K; (b) Procyon B (visible only using a telescope), which radiates energy at a rate of \(2.1\times10^{23}\mathrm{\ W}\) 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.)

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

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

CP, CALC 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 (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 that is filled with olive oil to a height of 2.00 mm below the top of the cup. Initially, the cup and oil are at room temperature 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 a temperature of The distance between two points, as measured by this tape on a day when its temperature is is 25.970 m. What is the true distance between the points?

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

You pour of ethanol, at a temperature of into a graduated cylinder initially at filling it to the very top. The cylinder is made of glass with a specific heat of and a coefficient of volume expansion of its mass is 0.110 kg. The mass of the ethanol is 0.0873 kg. (a) What will be the final temperature of the ethanol, once thermal equilibrium is reached? (b) How much ethanol will overflow the cylinder before thermal equilibrium is reached?

A metal rod that is 30.0 cm long expands by 0.0650 cm when its temperature is raised from to 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 and Find the length of each portion of the composite rod.

On a cool 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 she checks the fuel level and finds only 103.4 L of gasoline in the 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 reached by the fuel and the tank on Saturday afternoon? The coefficient of volume expansion of gasoline is and the tank is made of aluminum. (b) In order 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 when its temperature changes by the stress is equal to where F is the tension on the rod, is the original length of the rod, A its cross- sectional area, its coefficient of linear expansion, and Y its Youngs modulus. (b) A heavy brass bar has projections at its ends, as in Fig. P17.85. Two fine steel wires, fastened between the pro jections, are just taut (zero tension) when the whole system is at What is the tensile stress in the steel wires when the temperature of the system is raised to Make any simplifying assumptions you think are justified, but state what they are.

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

Out of Tune. The B-string of a guitar is made of steel (density \(\left.7800 \mathrm{ \ kg} / \mathrm{m}^{3}\right)\)), is 63.5 cm long, and has diameter 0.406 mm. The fundamental frequency is f = 247.0 Hz. (a) Find the string tension. (b) If the tension F is changed by a small amount \(\Delta F\), the frequency f changes by a small amount \(\Delta f\). Show that \(\frac{\Delta f}{f}=\frac{\Delta F}{2 F}\) (c) The string is tuned to a fundamental frequency of 247.0 Hz when its temperature is \(18.5^{\circ} \mathrm{C}\). Strenuous playing can make the temperature of the string rise, changing its vibration frequency. Find \(\Delta f\) if the temperature of the string rises to \(29.5^{\circ} \mathrm{C}\). The steel string has a Young's modulus of \(2.00\times10^{11}\mathrm{\ Pa}\) and a coefficient of linear expansion of \(1.20\times10^{-5}\ \left(\mathrm{C}^{\circ}\right)^{-1}\). Assume that the temperature of the body of the guitar remains constant. Will the vibration frequency rise or fall?

A steel rod 0.450 m long and an aluminum rod 0.250 m long, both with the same diameter, are placed end to end between rigid supports with no initial stress in the rods. The temperature of the rods is now raised by What is the stress in each rod? (Hint: The length of the combined rod remains the same, but the lengths of the individual rods do not. See Problem 17.85.)

A steel ring with a 2.5000-in. inside diameter at \(20.0^{\circ} \mathrm{C}\) is to be warmed and slipped over a brass shaft with a 2.5020-in. outside diameter at \(20.0^{\circ} \mathrm{C}\). (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?

Bulk Stress Due to a Temperature Increase. (a) Prove that, if an object under pressure has its temperature raised but is not allowed to expand, the increase in pressure is where the bulk modulus B and the average coefficient of volume expansion are both assumed positive and constant. (b) What pressure is necessary to prevent a steel block from expanding when its temperature is increased from

A liquid is enclosed in a metal cylinder that is provided with a piston of the same metal. The system is originally at a pressure of 1.00 atm and at a temperature of The piston is forced down until the pressure on the liquid is increased by 50.0 atm, and then clamped in this position. Find the new temperature at which the pressure of the liquid is again 1.00 atm. Assume that the cylinder is sufficiently strong so that its volume is not altered by changes in pressure, but only by changes in temperature. Use the result derived in Problem 17.90. (Hint: See Section 11.4.) Compressibility of liquid: Coefficient of volume expansion of liquid: Coefficient of volume expansion of metal:

You cool a 100.0-g slug of red-hot iron by dropping it into an insulated cup of negligible mass containing 85.0 g of water at 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?

CP Spacecraft Reentry. A spacecraft made of aluminum circles the earth at a speed of (a) Find the ratio of its kinetic energy to the energy required to raise its temperature from to (The melting point of aluminum is Assume a constant specific heat of (b) Discuss the bearing of your answer on the problem of the reentry of a manned space vehicle into the earths atmosphere.

A capstan is a rotating drum or cylinder over which a rope or cord slides in order to provide a great amplification of the rope's tension while keeping both ends free (Fig. P17.94). Since the added tension in the rope is due to friction, the capstan generates thermal energy. (a) If the difference in tension between the two ends of the rope is 520.0 N and the capstan has a diameter of 10.0 cm and turns once in 0.900 s, find the rate at which thermal energy is generated. Why does the number of turns not matter? (b) If the capstan is made of iron and has mass 6.00 kg, at what rate does its temperature rise? Assume that the temperature in the capstan is uniform and that all the thermal energy generated flows into it.

CALC Debyes Law. At very low temperatures the molar heat capacity of rock salt varies with temperature according to Debyes law: where and (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 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 16.0 cm deep. The water is at a temperature of Suppose that the water were to cool down spontaneously to form ice at 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 90 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 of air in the room. The air has specific heat and density 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?

CALC The molar heat capacity of a certain substance varies with temperature according to the empirical equation \(C=29.5 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K}+\left(8.20 \times 10^{-3} \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K}^{2}\right) T\) How much heat is necessary to change the temperature of 3.00 mol of this substance from 27°C to 227°C? (Hint: Use Eq. (17.18) in the form dQ = nC dT and integrate.) Text Transcription: C = 29.5 J / mol cdot K + (8.20 x 10^-3 J / mp; cdot K^2) T

For your cabin in the wilderness, you decide to build a primitive refrigerator out of Styrofoam, planning to keep the interior cool with a block of ice that has an initial mass of 24.0 kg. The box has dimensions of 0.500 m x 0.800 m x 0.500 m. Water from melting ice collects in the bottom of the box. Suppose the ice block is at 0.00°C and the outside temperature is 21.0°C. If the top of the empty box is never opened and you want the interior of the box to remain at 5.00°C for exactly one week, until all the ice melts, what must be the thickness of the Styrofoam?

Hot Water Versus Steam Heating. In a household hot-water heating system, water is delivered to the radiators at \(70.0^{\circ}\mathrm{C}\ \left(158.0^{\circ}\mathrm{F}\right)\) and leaves at \(28.0^{\circ}\mathrm{C\ }\left(82.4^{\circ}\mathrm{F}\right)\). 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 \(35.0^{\circ}\mathrm{C\ }\left(95.0^{\circ}\mathrm{F}\right)\). How many kilograms of steam will supply the same heat as was supplied by 1.00 kg of hot water in the first system?

A copper calorimeter can with mass 0.446 kg contains 0.0950 kg of ice. The system is initially at (a) If 0.0350 kg of steam at 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 is added to the mixture in the bucket, and when thermal equilibrium has been reached, the total mass of ice in the bucket is 0.868 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 \(100.0^{\circ} \mathrm{C}\) and atmospheric pressure is added to 0.200 kg of water at \(50.0^{\circ} \mathrm{C}\). (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 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 fatinner fur boundary? (b) How thick should the air layer (contained within the fur) be so that the bear loses heat at a rate of 50.0 W?

A worker pours 1.250 kg of molten lead at a temperature of \(327.3^{\circ} \mathrm{C}\) into 0.5000 kg of water at a temperature of in an insulated bucket of negligible mass. Assuming no heat loss to the surroundings, calculate the mass of lead and water remaining in the bucket when the materials have reached thermal equilibrium.

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 above the temperature of the outer surface. The total area of the box is and the wall thickness is 3.90 cm. Find the thermal conductivity of the material in SI units

Effect of a Window in a Door. A carpenter builds a solid wood door with dimensions Its thermal conductivity is 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 and the outside air temperature is (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 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.

A wood ceiling with thermal resistance is covered with a layer of insulation with thermal resistance Prove that the effective thermal resistance of the combination is

Compute the ratio of the rate of heat loss through a single-pane window with area 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 The air films on the room and outdoor surfaces of either window have a combined thermal resistance of

Rods of copper, brass, and steel are welded together to form a Y-shaped figure. The cross-sectional area of each rod is \(2.00\mathrm{\ cm}^2\) The free end of the copper rod is maintained at \(100.0^{\circ} \mathrm{C}\), and the free ends of the brass and steel rods at \(0.0^{\circ} \mathrm{C}\). Assume 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. (a) What is the temperature of the junction point? (b) What is the heat current in each of the three rods?

CALC Time Needed for a Lake to Freeze Over. (a) When the air temperature is below the water at the surface of a lake freezes to form an ice sheet. Why doesnt 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 and the bottom surface is at 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?

A rod is initially at a uniform temperature of throughout. One end is kept at and the other is brought into contact with a steam bath at The surface of the rod is insulated so that heat can flow only lengthwise along the rod. The cross-sectional area of the rod is its length is 120 cm, its thermal conductivity is its density is and its specific heat is Consider a short cylindrical element of the rod 1.00 cm in length. (a) If the temperature gradient at the cooler end of this element is how many joules of heat energy flow across this end per second? (b) If the average temperature of the element is increasing at the rate of what is the temperature gradient at the other end of the element?

A rustic cabin has a floor area of Its walls, which are tall, are made of wood 1.80 cm thick and are further insulated with 1.50 cm of a synthetic material. When the outside temperature is it is found necessary to heat the room at a rate of 1.25 kW to maintain its temperature at Calculate the thermal conductivity of the insulating material. Neglect the heat lost through the ceiling and floor. Assume the inner and outer surfaces of the wall have the same termperature as the air inside and outside the cabin.

The rate at which radiant energy from the sun reaches the earth's upper atmosphere is about \(1.50\mathrm{\ kW}/\mathrm{m}^2\). The distance from the earth to the sun is \(1.50\times10^{11}\mathrm{\ m}\), and the radius of the sun is \(6.96\times10^{8}\mathrm{\ m}\). (a) What is the rate of radiation of energy per unit area from the sun's 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 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.

BIO 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 (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.

Jogging in the Heat of the Day. You have probably seen people jogging in extremely hot weather and wondered Why? As we shall see, there are good reasons not to do this! When jogging strenuously, an average runner of mass 68 kg and surface area \(1.85\mathrm{\ m}^2\) 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 skin's temperature can be elevated to around \(33^{\circ} \mathrm{C}\) instead of the usual \(30^{\circ} \mathrm{C}\). (We shall neglect 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 \(40.0^{\circ}\mathrm{C}\ \left(104^{\circ}\mathrm{F}\right)\)? (Remember that he radiates out, but the environment radiates back in.) (c) What is the total amount of excess heat this runner's body must get rid of per second? (d) How much water must the jogger's body evaporate every minute due to his activity? The heat of vaporization of water at body temperature is \(2.42\times10^6\mathrm{\ J}/\mathrm{kg}\) (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.

BIO Overheating While Jogging. (a) If the jogger in the preceding problem were not able to get rid of the excess heat, by how much would his body temperature increase above the normal 37C in a half hour of jogging? The specific heat for a human is about (b) How high a fever (in F) would this temperature increase be equivalent to? Is the increase large enough to be of concern? (Recall that normal body temperature is 98.6F.)

An engineer is developing an electric water heater to provide a continuous supply of hot water. One trial design is shown in Fig. P17.119. Water is flowing at the rate of the inlet thermometer registers the voltmeter reads 120 V, and the ammeter reads 15.0 A corresponding to a power input of . (a) When a steady state is finally reached, what is the reading of the outlet thermometer? (b) Why is it unnecessary to take into account the heat capacity mc of the apparatus itself?

Food Intake of a Hamster. The energy output of an animal engaged in an activity is called the basal metabolic rate (BMR) and is a measure of the conversion of food energy into other forms of energy. A simple calorimeter to measure the BMR consists of an insulated box with a thermometer to measure the temperature of the air. The air has density and specific heat A 50.0-g hamster is placed in a calorimeter that contains of air at room temperature. (a) When the hamster is running in a wheel, the temperature of the air in the calorimeter rises per hour. How much heat does the running hamster generate in an hour? Assume that all this heat goes into the air in the calorimeter. You can ignore the heat that goes into the walls of the box and into the thermometer, and assume that no heat is lost to the surroundings. (b) Assuming that the hamster converts seed into heat with an efficiency of 10% and that hamster seed has a food energy value of how many grams of seed must the hamster eat per hour to supply this energy?

The icecaps of Greenland and Antarctica contain about 1.75% of the total water (by mass) on the earths surface; the oceans contain about 97.5%, and the other 0.75% is mainly groundwater. Suppose the icecaps, currently at an average temperature of about somehow slid into the ocean and melted. What would be the resulting temperature decrease of the ocean? Assume that the average temperature of ocean water is currently

Why Do the Seasons Lag? In the northern hemisphere, June 21 (the summer solstice) is both the longest day of the year and the day on which the sun's rays strike the earth most vertically, hence delivering the greatest amount of heat to the surface. Yet the hottest summer weather usually occurs about a month or so later. Let us see why this is the case. Because of the large specific heat of water, the oceans are slower to warm up than the land (and also slower to cool off in winter). In addition to perusing pertinent information in the tables included in this book, it is useful to know that approximately two-thirds of the earth's surface is ocean composed of salt water having a specific heat of \(3890\mathrm{\ J}/\mathrm{kg}\cdot\mathrm{K}\) and that the oceans, on the average, are 4000 m deep. Typically, an average of \(1050\mathrm{\ W}/\mathrm{m}^2\) of solar energy falls on the earth's surface, and the oceans absorb essentially all of the light that strikes them. However, most of that light is absorbed in the upper 100 m of the surface. Depths below that do not change temperature seasonally. Assume that the sunlight falls on the surface for only 12 hours per day and that the ocean retains all the heat it absorbs. What will be the rise in temperature of the upper 100 m of the oceans during the month following the summer solstice? Does this seem to be large enough to be perceptible?

CALC Suppose that both ends of the rod in Fig. 17.23a are kept at a temperature of and that the initial temperature distribution along the rod is given by where x is measured from the left end of the rod. Let the rod be copper, with length and cross-sectional area (a) Show the initial temperature distribution in a diagram. (b) What is the final temperature distribution after a very long time has elapsed? (c) Sketch curves that you think would represent the temperature distribution at intermediate times. (d) What is the initial temperature gradient at the ends of the rod? (e) What is the initial heat current from the ends of the rod into the bodies making contact with its ends? (f) What is the initial heat current at the center of the rod? Explain. What is the heat current at this point at any later time? (g) What is the value of the thermal diffusivity for copper, and in what unit is it expressed? (Here k is the thermal conductivity, is the density, and c is the specific heat.) (h) What is the initial time rate of change of temperature at the center of the rod? (i) How much time would be required for the center of the rod to reach its final temperature if the temperature continued to decrease at this rate? (This time is called the relaxation time of the rod.) (j) From the graphs in part (c), would you expect the magnitude of the rate of temperature change at the midpoint to remain constant, increase, or decrease as a function of time? (k) What is the initial rate of change of temperature at a point in the rod 2.5 cm from its left end?

(a) A spherical shell has inner and outer radii a and b, respectively, and the temperatures at the inner and outer surfaces are \(T_{2}\) and \(T_{1}\) The thermal conductivity of the material of which the shell is made is k. Derive an equation for the total heat current through the shell. (b) Derive an equation for the temperature variation within the shell in part (a); that is, calculate T as a function of r, the distance from the center of the shell. (c) A hollow cylinder has length L, inner radius a, and outer radius b, and the temperatures at the inner and outer surfaces are \(T_{2}\) and \(T_{1}\). (The cylinder could represent an insulated hot-water pipe, for example.) The thermal conductivity of the material of which the cylinder is made is k. Derive an equation for the total heat current through the walls of the cylinder. (d) For the cylinder of part (c), derive an equation for the temperature variation inside the cylinder walls. (e) For the spherical shell of part (a) and the hollow cylinder of part (c), show that the equation for the total heat current in each case reduces to Eq. (17.21) for linear heat flow when the shell or cylinder is very thin.

A steam pipe with a radius of 2.00 cm, carrying steam at 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 This in turn is surrounded by a cylindrical jacket made of a brand of Styrofoam with thermal conductivity and having inner and outer radii 4.00 cm and 6.00 cm (Fig. P17.125). The outer surface of the Styrofoam is in contact with air at Assume that this outer surface has a temperature of (a) What is the temperature at a radius of 4.00 cm, where the two insulating layers meet? (b) What is the total rate of transfer of heat out of a 2.00-m length of pipe? (Hint: Use the expression derived in part (c) of Challenge Problem 17.124.)

Temperature Change in a Clock. A pendulum clock is designed to tick off one second on each side-to- side swing of the pendulum (two ticks per complete period). (a) Will a pendulum clock gain time in hot weather and lose it in cold, or the reverse? Explain your reasoning. (b) A particular pendulum clock keeps correct time at The pendulum shaft is steel, and its mass can be ignored compared with that of the bob. What is the fractional change in the length of the shaft when it is cooled to (c) How many seconds per day will the clock gain or lose at (d) How closely must the temperature be controlled if the clock is not to gain or lose more than 1.00 s a day? Does the answer depend on the period of the pendulum?

BIO A Walk in the Sun. Consider a poor lost soul walking at on a hot day in the desert, wearing only a bathing suit. This persons 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 where is the exposed skin area is the air temperature is and the skin temperature is (iii) the skin absorbs radiant energy from the sun at a rate of (iv) the skin absorbs radiant energy from the environment, which has temperature (a) Calculate the net rate (in watts) at which the persons skin is heated by all four of these mechanisms. Assume that the emissivity of the skin is and that the skin temperature is initially Which mechanism is the most important? (b) At what rate must perspiration evaporate from this persons skin to maintain a constant skin temperature? (The heat of vaporization of water at is (c) Suppose instead the person is protected by light-colored clothing so that the exposed skin area is only What rate of perspiration is required now? Discuss the usefulness of the traditional clothing worn by desert peoples.

Problem 1DQ Explain why it would not make sense to use a full-size glass thermometer to measure the temperature of a thimbleful of hot water.

Problem 1E Convert the following Celsius temperatures to Fahrenheit: (a) - 62.8o C, the lowest temperature ever recorded in North America (February 3, 1947, Snag, Yukon); (b) 56.7o C, the highest temperature ever recorded in the United States (July 10, 1913, Death Valley, California); (c) 31.1o C, the world’s highest average annual temperature (Lugh Ferrandi, Somalia).

Problem 2DQ 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.

Problem 2E BIO 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 body’s temperature can go as high as 40o C. 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 Co lower than the internal temperature. Express this temperature difference in kelvins and in Fahrenheit degrees. (d) Blood storage. Blood stored at 4.0o C lasts safely for about 3 weeks, whereas blood stored at – 160o C lasts for 5 years. Express both temperatures on the Fahrenheit and Kelvin scales. (e) Heat stroke. If the body’s temperature is above 105o F for a prolonged period, heat stroke can result. Express this temperature on the Celsius and Kelvin scales.

Problem 3DQ Many automobile engines have cast-iron cylinders and aluminum pistons. What kinds of problems could occur if the engine gets too hot? (The coefficient of volume expansion of cast iron is approximately the same as that of steel.)

Problem 3E (a) On January 22, 1943, the temperature in Spearfish, South Dakota, rose from - 4.0o F to 45.0o F in just 2 minutes. What was the temperature change in Celsius degrees? (b) The temperature in Browning, Montana, was 44.0o F on January 23, 1916. The next day the temperature plummeted to - 56o F. What was the temperature change in Celsius degrees?

Problem 4DQ 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?

Problem 4E (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.

Problem 5DQ 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?

Problem 5E 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) Fo and (b) Co?

Problem 6DQ The inside of an oven is at a temperature of 200o C (392o F). You can put your hand in the oven without injury as long as you don’t touch anything. But since the air inside the oven is also at 200o C, why isn’t your hand burned just the same?

Problem 6E 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 (1.55 X 107 K).

Problem 7DQ 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?

Problem 7E 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?

Problem 8DQ To raise the temperature of an object, must you add heat to it? If you add heat to an object, must you raise its temperature? Explain.

Problem 8E 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?

Problem 9DQ A student asserts that a suitable unit for specific heat is 1 m2/s2 ? Co. Is she correct? Why or why not?

A Constant-Volume Gas Thermometer. An experimenter using a gas thermometer found the pressure at the triple point of water \(\left(0.01^{\circ} \mathrm{C}\right)\) to be \(4.80 \times 10^{4}\) Pa and the pressure at the normal boiling point \(\left(100^{\circ} \mathrm{C}\right)\) to be \(6.50 \times 10^{4}\) 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 \(100^{\circ} \mathrm{C}\) were \(6.50 \times 10^{4}\) Pa, what pressure would the experimenter have measured at \(0.01^{\circ} \mathrm{C}\) (As we will learn in Section 18.1, Eq. (17.4) is accurate only for gases at very low density.) Equation Transcription: Text Transcription: (0.01degC) 4.80x10^4 (100degC) 6.50x10^4 100degC 6.50x10^4 0.01degC

Problem 10DQ 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?

Problem 10E Like the Kelvin scale, the Rankine scale is an absolute temperature scale: Absolute zero is zero degrees Rankine (0oR). 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?

Problem 11DQ The units of specific heat c are J/kg ? K, but the units of heat of fusion Lf or heat of vaporization Lv are simply J/kg. Why do the units of Lf and Lv not include a factor of (K)-1 to account for a temperature change?

Problem 11E 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 - 5.0o C to 18.0o C.

Problem 12DQ Why is a hot, humid day in the tropics generally more uncomfortable for human beings than a hot, dry day in the desert?

Problem 12E 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 15.5o C. 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 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.

Problem 13E A U.S. penny has a diameter of 1.9000 cm at 20.0o C. The coin is made of a metal alloy (mostly zinc) for which the coefficient of linear expansion is 2.6 X 10-5 K-1. What would its diameter be on a hot day in Death Valley (48.0°C)? On a cold night in the mountains of Greenland (– 53o C)?

Problem 14DQ 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?

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. Equation Transcription: Text Transcription: CO2 23.0degC -78.0degC

Problem 15DQ 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?

Problem 15E The outer diameter of a glass jar and the inner diameter of its iron lid are both 725 mm at r??m temperature (20.0°C). What will be the size of the difference in these diameters if the lid is briefly held under hot water until its temperature rises to 50.0°C, without changing the temperature of the glass?

Problem 16DQ 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?

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 - 15\(^\circ\)C. How much more interior space does the dome have in the summer, when the temperature is 35\(^\circ\)C?

Problem 17DQ 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 0o C? In fact, the water freezes first in a layer adjacent to the sides of the tray. Why?

Problem 17E A copper cylinder is initially at 20.0o C. At what temperature will its volume be 0.150% larger than it is at 20.0o C?

Problem 18DQ 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 82.4o C.)

Problem 18E A steel tank is completely filled with 2.80 m3 of ethanol when both the tank and the ethanol are at a temperature of 32.0°C. When the tank and its contents have c??led to 18.0°C, what additional volume of ethanol can be put into the tank?

Problem 19DQ 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?

Problem 19E A glass flask whose volume is 1000.00 cm3 at 0.0o C is completely filled with mercury at this temperature. When flask and mercury are warmed to 55.0o C, 8.95 cm3 of mercury overflow. If the coefficient of volume expansion of mercury is 18.0 X 10-5 K-1, compute the coefficient of volume expansion of the glass.

Problem 20DQ 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.

Problem 20E (a) If an area measured on the surface of a solid body is A0 at some initial temperature and then changes by ?A when the temperature changes by ?T, show that ?A = (2?)A0?T where ? is the coefficient of linear expansion. (b) A circular sheet of aluminum is 55.0 cm in diameter at 15.0°C. By how much does the area of one side of the sheet change when the temperature increases to 27.5°C?

Problem 21DQ 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.)

Problem 21E A machinist bores a hole of diameter 1.35 cm in a steel plate that is at 25.0o C. What is the cross-sectional area of the hole (a) at 25.0o C and (b) when the temperature of the plate is increased to 175o C? Assume that the coefficient of linear expansion remains constant over this temperature range.

Problem 22DQ 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?

Problem 22E 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 20.0o C and 150.0o C. Assume that the coefficients of expansion are constant over this temperature range. (a) If the piston just fits inside the chamber at 20.0o C, will the engines be able to run at higher temperatures? Explain. (b) If the cylindrical pistons are 25.000 cm in diameter at 20.0o C, what should be the minimum diameter of the cylinders at that temperature so the pistons will operate at 150.0o C?

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 0.2–0.8 times as great as that of water.)

Problem 23E (a) A wire that is 1.50 m long at 20.0°C is found to increase in length by 1.90 cm when warmed to 420.0°C. Compute its average coefficient of linear expansion for this temperature range. (b) The wire is stretched just taut (zero tension) at 420.0°C. Find the stress in the wire if it is c??led to 20.01°C without being allowed to contract. Young’s modulus for the wire is 2.0 × 1011 Pa.

Problem 24DQ 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: Don’t 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?

Problem 24E 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 120.0o C to 10.0o C?

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

Problem 25E 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 ?2.0°C. (a) How much space must be left between adjacent rails if they are just to touch on a summer day when their temperature is 33.0°C? (b) If the rails are originally laid in contact, what is the stress in them on a summer day when their temperature is 33.0°C?

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?

Problem 26E 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 20.0o C to 80.0o C? (b) How much time is required? Assume that all of the heater’s power goes into heating the water.

Problem 27DQ We’re lucky that the earth isn’t in thermal equilibrium with the sun (which has a surface temperature of 5800 K). But why aren’t the two bodies in thermal equilibrium?

Problem 27E An aluminum tea kettle with mass 1.50 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°C to 85.0°C?

Problem 28DQ 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?

BIO 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 – 20\(^\circ\)C, what amount of heat is needed to warm to body temperature (37\(^\circ\)C) 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 \(1.3 \times 10^{-3}\) kg. (b) How much heat is lost per hour if the respiration rate is 20 breaths per minute?

Problem 29E 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 X 104 J of heat energy to the sample and find that its temperature rises 18.0 Co. What is the sample’s specific heat?

Problem 30E 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 50°F (10°C) to 120°F (49°C) 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?

Problem 31E BIO 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 37o C, this energy must be removed by perspiration or other mechanisms. If these mechanisms failed and the energy could not flow out of the student’s 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 44o C 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.)

Problem 32E CP 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 crate of fruit with mass 35.0 kg and specific heat \(3650 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) slides down a ramp inclined at \(36.9^{\circ}\) below the horizontal. The ramp is 8.00 m long. (a) If the crate was at rest at the top of the incline and has a speed of 2.50 m/s at the bottom, how much work was done on the crate by friction? (b) If an amount of heat equal to the magnitude of the work done by friction goes into the crate of fruit and the fruit reaches a uniform final temperature, what is its temperature change?

Problem 34E CP 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 station’s 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/m 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 \(18.55^{\circ} \mathrm{C} \text { to } 22.54^{\circ} \mathrm{C}\). (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.

Problem 37E CP 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?

As a physicist, you put heat into a \(500.0-\mathrm{g}\) solid sample at the rate of \(10.0 \mathrm{~kJ} / \mathrm{min}\), while recording its temperature as a function of time. You plot your data and obtain the graph shown in Fig. E17.38. (a) What is the latent heat of fusion for this solid? (b) What are the specific heats of the liquid and solid states of the material? Equation Transcription: Text Transcription: 500.0-g 10.0 kJ/min T(C) t(min)

Problem 39E 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 (20.0o C). After waiting and gently stirring for 5.00 minutes, you observe that the water’s temperature has reached a constant value of 22.0o C. (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.

Problem 40E BIO 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 pre-vents 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 J/kg ? Co, and recall that normal body temperature is 37.0°C.

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

Bicycling on a Warm Day. If the air temperature is the same as the temperature of your skin (about 30°C), 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 person’s 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 person’s body evaporate in an hour to get rid of this heat? The heat of vaporization of water at body temperature is \(2.42\times10^6\mathrm{\ J}/\mathrm{kg}\). (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.)

Problem 43E Overheating. (a) By how much would the body temperature of the bicyclist in the proceeding problem 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 98.6°F and the specific heat of the body is 3480 J/kg · C°.)

Problem 44E In a container of negligible mass, 0.200 kg of ice at an initial temperature of - 40.0o C is mixed with a mass m of water that has an initial temperature of 80.0o C. No heat is lost to the surroundings. If the final temperature of the system is 28.0o C, what is the mass m of the water that was initially at 80.0o C?

Problem 45E 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 - 20.0o C. 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?

Problem 46E BIO Before going in for his annual physical, a 70.0-kg man whose body temperature is 37.0o C consumes an entire 0.355-L can of a soft drink (mostly water) at 12.0o C. (a) What will his body temperature be after equilibrium is attained? Ignore any heating by the man’s metabolism. The specific heat of the man’s body is 3480 J/kg ? K. (b) Is the change in his body temperature great enough to be measured by a medical thermometer?

Problem 47E BIO Basal Metabolic Rate. In the situation described in Exercise 17.42, the man’s metabolism will eventually return the temperature of his body (and of the soft drink that he consumed) to 37.0o C. If his body releases energy at a rate of 7.00 X 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. 17.42 . BIO Before going in for his annual physical, a 70.0-kg man whose body temperature is 37.0o C consumes an entire 0.355-L can of a soft drink (mostly water) at 12.0o C. (a) What will his body temperature be after equilibrium is attained? Ignore any heating by the man’s metabolism. The specific heat of the man’s body is 3480 J/kg ? K. (b) Is the change in his body temperature great enough to be measured by a medical thermometer?

Problem 48E An ice-cube tray of negligible mass contains 0.350 kg of water at 18 °C. How much heat must be removed to c??l the water to 0.00°C and freeze it? Express your answer in joules, calories, and Btu.

How much heat is required to convert 12.0 g of ice at ?10.0\(^\circ\)C to steam at 100.0\(^\circ\)C? Express your answer in joules, calories, and Btu.

An open container holds 0.550 kg of ice at - 15.0\(^\circ\)C. 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 0.0\(^\circ\)C? (c) Plot a curve showing the temperature as a function of the elapsed time.

Problem 51E CP What must the initial speed of a lead bullet be at 25.0o 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 25.0o C.)

BIO 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 100.0\(^\circ\)C, when it is cooled to skin temperature (34.0\(^\circ\)C)? (b) By 25.0 g of water initially at 100.0\(^\circ\)C, when it is cooled to 34.0\(^\circ\)C? (c) What does this tell you about the relative severity of burns from steam versus burns from hot water?

Problem 53E BIO “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 34.0o C overnight and rise to 40.0o C 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 34.0o C by evaporation of sweat during the day (12 hours) instead of letting it rise to 40.0o C. (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 34o C is 2.42 X 106 J/kg.)

Problem 54E BIO 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 Co? The heat of vaporization of water at body temperature (37o C) is 2.42 X 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 (355 m3). 17.25 . BIO 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 37o C, this energy must be removed by perspiration or other mechanisms. If these mechanisms failed and the energy could not flow out of the student’s 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 44o C 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.)

Problem 55E CP An asteroid with a diameter of 10 km and a mass of 2.60 X 1015 kg impacts the earth at a speed of 32.0 km/s, landing in the Pacific Ocean. If 1.00% of the asteroid’s kinetic energy goes to boiling the ocean water (assume an initial water temperature of 10.0o C), what mass of water will be boiled away by the collision? (For comparison, the mass of water contained in Lake Superior is about 2 X 1015 kg.)

Problem 56E A laboratory technician drops a 0.0850-kg sample of unknown solid material, at 100.0o C, into a calorimeter. The calorimeter can, initially at 19.0o C, 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 26.1o C. Compute the specific heat of the sample.

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

Problem 58E A glass vial containing a 16.0-g sample of an enzyme is c??led in an ice bath. The bath contains water and 0.120 kg of ice. The sample has specific heat 2250 J/kg · K; the glass vial has mass 6.00 g and specific heat 2800 J/kg · K. How much ice melts in c??ling the enzyme sample from r??m temperature (19.5°C) to the temperature of the ice bath?

Problem 59E A 4.00-kg silver ingot is taken from a furnace, where its temperature is 750.0o C, and placed on a large block of ice at 0.0o C. Assuming that all the heat given up by the silver is used to melt the ice, how much ice is melted?

Problem 60E 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 a temperature of 255°C is dropped into the calorimeter can, what is the final temperature? Assume that no heat is lost to the surroundings.

Problem 61E A vessel whose walls are thermally insulated contains 2.40 kg of water and 0.450 kg of ice, all at 0.0o C. 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 28.0o C? You can ignore the heat transferred to the container.

Problem 62E 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 cross-sectional area 0.00500 m2. 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. 17.23a is made of copper, is 45.0 cm long, and has a cross-sectional area of 1.25 \(\mathrm{cm}^{2}\). Let \(T_{H}=100.0^{\circ} \mathrm{C}\) and \(T_{C}=0.0^{\circ} \mathrm{C}\). (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? Equation Transcription: Text Transcription: cm^2 TH=100.0degC TC=0.0degC

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.

Problem 65E 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 19.0o C, and the exterior surface temperature is - 10.0o C. (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?

Problem 66E An electric kitchen range has a total wall area of 1.40 m2 and is insulated with a layer of fiberglass 4.00 cm thick. The inside surface of the fiberglass has a temperature of 175o C, and its outside surface is at 35.0o C. The fiberglass has a thermal conductivity of 0.040 W/.m ? 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 m2? (b) What electric-power input to the heating element is required to maintain this temperature?

BIO 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 37.0\(^\circ\)C and that the outer surface of the skin is at 30.0\(^\circ\)C. 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 \(2.0 m^{2}\) 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 person’s 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 ice–water mixture at the other (Fig. E17.68). The rod consists of a \(\text { 1.00-m }\) section of copper (one end in boiling water) joined end to end to a length \(L_{2}\) of steel (one end in the ice–water mixture). Both sections of the rod have cross sectional areas of 4.00 \(\mathrm{cm}^{2}\). The temperature of the copper–steel junction is \(65.0^{\circ} \mathrm{C}\) 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 \(L_{2}\) of the steel section? Equation Transcription: Text Transcription: 1.00-m L2 cm^2 65.0degC L2

Problem 69E 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 m2. The water inside the pot is at 100.0o 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 150.0 J/s from a furnace at 400.0°C to a container of boiling water under 1 atmosphere. What must the rod’s diameter be?

Problem 71E A picture window has dimensions of 1.40 m X 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 - 20.0o C, while the temperature of the inside surface is a comfortable 19.5o C. (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)?

Problem 72E What is the rate of energy radiation per unit area of a blackbody at (a) 273 K and (b) 2730 K?

Problem 73E 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.)

Problem 74E 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?

Problem 75E 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 X 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 X 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.)

Problem 76P Suppose that a steel hoop could be constructed to fit just around the earth’s equator at 20.0o C. What would be the thickness of space between the hoop and the earth if the temperature of the hoop were increased by 0.500 Co?

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

Problem 79P You are making pesto for your pasta and have a cylindrical measuring cup 10.0 cm high made of ordinary glass [? = 2.7 × 10–5 (C°)–1] that is tilled with olive oil [? = 6.8 × 10–4 (C°)–1] to a height of 2.00 mm below the top of the cup. Initially, the cup and oil are at room temperature (22.0°C). 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?

Problem 80P A surveyor’s 30.0-m steel tape is correct at 20.0o C. The distance between two points, as measured by this tape on a day when its temperature is 5.00o C, is 25.970 m. What is the true distance between the points?

Problem 81P CP 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 20.0o C). Due to a design over-sight, the swinging sphere clears the floor by a distance of only 2.00 mm when the temperature is 20.0o C. At what temperature will the sphere begin to brush the floor?

Problem 82P You pour 108 cm3 of ethanol, at a temperature of 10.0°C, into a graduated cylinder initially at 20.0°C, filling it to very top. The cylinder is made of glass with a specific heat of 840 J/kg · K. and a coefficient of volume expansion of 1.2 × 10–5 K–1; its mass is 0.110 kg. The mass of the ethanol is 0.0873 kg. (a) What will be the final temperature of the ethanol, once thermal equilibrium is reached? (b) How much ethanol will overflow the cylinder before thermal equilibrium is reached?

Problem 83P A metal rod that is 30.0 cm long expands by 0.0650 cm when its temperature is raised from 0.0o C to 100.0o C. 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 0.0o C and 100.0o C. Find the length of each portion of the composite rod.

On a cool \(\left(4.0^{\circ} \mathrm{C}\right)\) 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 \(4.0^{\circ} \mathrm{C}\), she checks the fuel level and finds only 103.4 L of gasoline in the 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 tank’s vent. (a) What was the maximum temperature \(\left(\text { in }^{\circ} \mathrm{C}\right)\) reached by the fuel and the tank on Saturday afternoon? The coefficient of volume expansion of gasoline is \(9.5\times10^{-4}\mathrm{\ K}^{-1}\), and the tank is made of aluminum. (b) In order 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 \(\Delta L\) when its temperature changes by \(\Delta T\), the stress is equal to \(\frac{F}{A}=Y\left(\frac{\Delta L}{L_{0}}-\alpha \Delta T\right)\) where is the tension on the rod, is the original length of the rod, its cross-sectional area, \(\alpha\) its coefficient of linear expansion, and its Young's modulus. (b) A heavy brass bar has projections at its ends, as in Fig. P17.85. Two fine steel wires, fastened between the pro- jections, are just taut (zero tension) when the whole system is at \(20^{\circ} \mathrm{C}\). What is the tensile stress in the steel wires when the temperature of the system is raised to \(140^{\circ} \mathrm{C}\)? Make any simplifying assumptions you think are justified, but state what they are. Equation Transcription: Text Transcription: Delta L Delta T F/A=Y(changeL/L0-alpha delta T) L0 Alpha 20degC 140degC

Problem 86P CP A metal wire, with density ? and Young’s 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 < v1. Determine the coefficient of linear expansion of the wire.

Out of Tune. The B-string of a guitar is made of steel \((density\ 7800\mathrm{\ kg}/\mathrm{m}^3)\), is \(63.5\mathrm{\ cm}\) long, and has diameter \(0.406\mathrm{\ mm}\). The fundamental frequency is \(f=247.0\mathrm{\ Hz}\). (a) Find the string tension. (b) If the tension F is changed by a small amount \(\Delta F\), the frequency f changes by a small amount \(\Delta f\). Show that \(\quad \quad \quad \quad \quad \frac{\Delta f}{f}=\frac{\Delta F}{2F}\) (c) The string is tuned to a fundamental frequency of \(247.0\mathrm{\ Hz}\) when its temperature is \(18.5^{\circ} \mathrm{C}\). Strenuous playing can make the temperature of the string rise, changing its vibration frequency. Find \(\Delta f\) if the temperature of the string rises to \(29.5^{\circ} \mathrm{C}\). The steel string has a Young's modulus of \(2.00 \times 10^{11}\mathrm{\ Pa}\) and a coefficient of linear expansion of \(1.20 \times 10^{-5}\ \left(\mathrm{C}^{\circ}\right)^{-1}\). Assume that the temperature of the body of the guitar remains constant. Will the vibration frequency rise or fall?

A steel rod 0.450 m long and an aluminum rod 0.250 m long, both with the same diameter, are placed end to end between rigid supports with no initial stress in the rods. The temperature of the rods is now raised by 60.0 \(\mathrm{C}^{\circ}\). What is the stress in each rod? (Hint: The length of the combined rod remains the same, but the lengths of the individual rods do not. See Problem 17.85.) Equation Transcription: Text Transcription: C^o

Problem 89P A steel ring with a 2.5000-in. inside diameter at 20.0o C is to be warmed and slipped over a brass shaft with a 2.5020-in. outside diameter at 20.0o C. (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?

Problem 90P Bulk Stress Due to a Temperature Increase. (a) Prove that, if an object under pressure has its temperature raised but is not allowed to expand, the increase in pressure is ?p = B??T where the bulk modulus B and the average coefficient of volume expansion ? are both assumed positive and constant. (b) What pressure is necessary to prevent a steel block from expanding when its temperature is increased from 20.0°C to 35.0°C?

Problem 91P A liquid is enclosed in a metal cylinder that is provided with a piston of the same metal. The system is originally at a pressure of 1.00 atm (1.013 × 105 Pa) and at a temperature of 30.0°C. The piston is forced down until the pressure on the liquid is increased by 50.0 atm, and then clamped in this position. Find the new temperature at which the pressure of the liquid is again 1.00 atm. Assume that the cylinder is sufficiently strong so that its volume is not altered by changes in pressure, but only by changes in temperature. Use the result derived in Problem (Hint: See Section 11.4.) Compressibility of liquid: k = 8.50 × 10–10 Pa–1 Coefficient of volume expansion of liquid: ? = 4.80 × 10–4 K–1 Coefficient of volume expansion of metal: ? = 3.90 × 10–5 K–1 Problem: Bulk Stress Due to a Temperature Increase. (a) Prove that, if an object under pressure has its temperature raised but is not allowed to expand, the increase in pressure is ?p = B??T where the bulk modulus B and the average coefficient of volume expansion ? are both assumed positive and constant. (b) What pressure is necessary to prevent a steel block from expanding when its temperature is increased from 20.0°C to 35.0°C?

Problem 92P You cool a 100.0-g slug of red-hot iron (temperature 745o C) by dropping it into an insulated cup of negligible mass containing 85.0 g of water at 20.0o C. 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?

Problem 93P Spacecraft Reentry. A spacecraft made of aluminum circles the earth at a speed of 7700 m/s. (a) Find the ratio of its kinetic energy to the energy required to raise its temperature from 0°C to 600°C. (The melting point of aluminum is 660°C. Assume a constant specific heat of 910 J/kg · K.) (b) Discuss the bearing of your answer on the problem of the re-entry of a manned space vehicle into the earth ’s atmosphere.

CP A capstan is a rotating drum or cylinder over which a rope or cord slides in order to provide a great amplification of the rope’s tension while keeping both ends free (Fig. P17.94). Since the added tension in the rope is due to friction, the capstan generates thermal energy. (a) If the difference in tension between the two ends of the rope is 520.0 N and the capstan has a diameter of 10.0 cm and turns once in 0.900 s, find the rate at which thermal energy is generated. Why does the number of turns not matter? (b) If the capstan is made of iron and has mass 6.00 kg, at what rate does its temperature rise? Assume that the temperature in the capstan is uniform and that all the thermal energy generated flows into it. Equation Transcription: Text Transcription: T’<T0 T0

CALC Debye's \(T^{3}\) Law. At very low temperatures the molar heat capacity of rock salt varies with temperature according to Debye's \(T^{3}\) law: \(C=k \frac{T^{3}}{\theta^{3}}\) where \(k=1940 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K}\) and \(\theta\) = 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. in the form \(d q=n c D T\) 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? Equation Transcription: Text Transcription: T^3 T^3 C=k T^3/theta^3 k=1940 J/mol.K Theta dQ=nc DT

Problem 96P CP 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 37.0o C. Suppose that the water were to cool down spontaneously to form ice at 0.0o 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.)

Problem 97P 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 90 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?

CALC The molar heat capacity of a certain substance varies with temperature according to the empirical equation \(C=29.5 \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K}+\left(8.20 \times 10^{-3} \mathrm{~J} / \mathrm{mol} \cdot \mathrm{K}^{2}\right) \mathrm{T}\) How much heat is necessary to change the temperature of of this substance from \(27^{\circ} \mathrm{C}\) to \(227^{\circ} \mathrm{C}\)? (Hint: Use Eq. (17.18) in the form \(d Q=n C d T\) and integrate.) Equation Transcription: Text Transcription: C=29.5 J/mol.K+(8.20x10^-3 J/mol.K^2)T dQ=nC dT 27degC 227degC

Problem 99P For your cabin in the wilderness, you decide to build a primitive refrigerator out of Styrofoam, planning to keep the interior cool with a block of ice that has an initial mass of 24.0 kg. The boxhas dimensions of 0.500 m × 0.800 m × 0.500 m Water from melting ice collects in the bottom of the box. Suppose the ice block is at 0.00 °C and the outside temperature is 21.0°C, if the top of the empty boxis never opened and you want the interior of the boxto remain at 5.00°C for exactly one week, until all the ice melts. What must be the thickness of the Styrofoam?

Hot Water Versus Steam Heating. In a household hot-water heating system, water is delivered to the radiators at 70.0o C (158.0o F) and leaves at 28.0o C (82.4o F). 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 35.0o C (95.0o F). How many kilograms of steam will supply the same heat as was supplied by 1.00 kg of hot water in the first system?

Problem 101P A copper calorimeter can with mass 0.446 kg contains 0.0950 kg of ice. The system is initially at 0.0o C. (a) If 0.0350 kg of steam at 100.0o C 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?

Problem 103P In a container of negligible mass, 0.0400 kg of steam at 100o C and atmospheric pressure is added to 0.200 kg of water at 50.0o C. (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?

Problem 104P BIO 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 2.7°C during hibernation, while the inner surface of the fat layer is at 31.0°C. (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?

Problem 105P A worker pours 1.250 kg of molten lead at a temperature of 327.3°C into 0.5000 kg of water at a temperature of 75.00°C in an insulated bucket of negligible mass. Assuming no heat loss to the surroundings, calculate the mass of lead and water remaining in the bucket when the materials have reached thermal equilibrium.

Problem 108P A wood ceiling with thermal resistance R1 is covered with a layer of insulation with thermal resistance R2. Prove that the effective thermal resistance of the combination is R = R1+ R2.

Compute the ratio of the rate of heat loss through a single-pane window with area \(0.15 \mathrm{~m}^{2}\) 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 \(0.80 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The air films on the room and outdoor surfaces of either window have a combined thermal resistance of \(0.15 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\).

Problem 110P Rods of copper, brass, and steel—each with cross-sectional area of 2.00 cm2 —are welded together to form a Y-shaped figure. The free end of the copper rod is maintained at 100.0o C, and the free ends of the brass and steel rods at 0.0o 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?

Problem 111P CALC Time Needed for a Lake to Freeze Over. (a) When the air temperature is below 0o 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 thick-ness 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 – 10o C and the bottom surface is at 0o 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?

Problem 112P A rod is initially at a uniform temperature of 0°C throughout. One end is kept at 0°C, and the other is brought into contact with a steam bath at 100°C. The surface of the rod is insulated so that heat can flow only lengthwise along the rod. The cross-sectional area of the rod is 2.50 cm2, its length is 120 cm, its thermal conductivity is 380 W/m · K, its density is 1.00 × 104 kg/m3, and its specific heat is 520 J/kg · K. Consider a short cylindrical element of the rod 1.00 cm in length. (a) If the temperature gradient at the cooler end of this element is 140 C°/m, how many joules of heat energy flow across this end per second? (b) If the average temperature of the element is increasing at the rate of what is the temperature gradient at the other end of the element?

Problem 113P A rustic cabin has a floor area of 3.50 m × 3.00 m. (is walls, which are 2.50 m tall, are made of wood (thermal conductivity 0.0600 W/m · K) 1.80 cm thick and are further insulated with 1.50 cm of a synthetic material. When the outside temperature is 2.00 °C. it is found necessary to heat the room at a rate of 1.25 kW to maintain its temperature at 19.0°C. Calculate the thermal conductivity of the insulating material. Neglect the heat lost through the ceiling and floor. Assume the inner and outer surfaces of the wall have the same temperature as the air inside and outside the cabin.

Problem 114P The rate at which radiant energy from the sun reaches the earth’s upper atmosphere is about 1.50 kW/m2. The distance from the earth to the sun is 1.50 X 1011 m, and the radius of the sun is 6.96 X 108 m. (a) What is the rate of radiation of energy per unit area from the sun’s surface? (b) If the sun radiates as an ideal blackbody, what is the temperature of its surface?

Problem 115P 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 X 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.

Problem 116P BIO 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 18°C (about 65°F) if his skin’s surface temperature is 30°C? (At such temperatures, nearly all the heat is infrared radiation, for which the body’s 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 person’s basal metabolic rate.

Problem 117P BIO 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 skin’s temperature can be elevated to around 33°C instead of the usual 30°C. (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 40.0°C (104°F)? (Remember: He radiates out, but the environment radiates back in.) (c) What is the total amount of excess heat this runner’s 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 X 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.

Problem 118P Overheating While Jogging. (a) if The jogger in The preceding problem were not able to get rid of the excess heat, by how much would his body temperature increase above the normal 37°C in a half hour of jogging? The specific heat for a human is about 3500 J/kg · K. (b) How high a fever (in °F) would this temperature increase be equivalent to? Is the increase large enough to be of concern? (Recall that normal body temperature is 98.6°F.) PRECEDING PROBLEM: BIO 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 skin’s temperature can be elevated to around 33°C instead of the usual 30°C. (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 40.0°C (104°F)? (Remember: He radiates out, but the environment radiates back in.) (c) What is the total amount of excess heat this runner’s 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 X 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.

An engineer is developing an electric water heater to provide a continuous supply of hot water. One trial design is shown in Fig. P17.119. Water is flowing at the rate of 0.500 kg/min, the inlet thermometer registers \(18.0^{\circ} \mathrm{C}\), the voltmeter reads 120 V, and the ammeter reads 15.0 A [corresponding to a power input of (120 V) x (15.0 A) = 1800 W]. (a) When a steady state is finally reached, what is the reading of the outlet thermometer? (b) Why is it unnecessary to take into account the heat capacity mc of the apparatus itself? Equation Transcription: Text Transcription: 18.0degC kg/min

Food Intake of a Hamster. The energy output of an animal engaged in an activity is called the basal metabolic rate (BMR) and is a measure of the conversion of food energy into other forms of energy. A simple calorimeter to measure the BMR consists of an insulated box with a thermometer to measure the temperature of the air. The air has density \(1.20 kg/m^{3}\) and specific heat 1020 J/kg \(\cdot\) K. A 50.0 g hamster is placed in a calorimeter that contains \(0.0500 m^{3}\) of air at room temperature. (a) When the hamster is running in a wheel, the temperature of the air in the calorimeter rises 1.60\(^\circ\)C per hour. How much heat does the running hamster generate in an hour? Assume that all this heat goes into the air in the calorimeter. You can ignore the heat that goes into the walls of the box and into the thermometer, and assume that no heat is lost to the surroundings. (b) Assuming that the hamster converts seed into heat with an efficiency of 10% and that hamster seed has a food energy value of 24 J/g, how many grams of seed must the hamster eat per hour to supply this energy?

The ice caps of Greenland and Antarctica contain about 1.75% of the total water (by mass) on the earth’s surface; the oceans contain about 97.5%, and the other 0.75% is mainly groundwater. Suppose the ice caps, currently at an average temperature of about – 30\(^\circ\)C, somehow slid into the ocean and melted. What would be the resulting temperature decrease of the ocean? Assume that the average temperature of ocean water is currently 5.00\(^\circ\)C.

Why Do the Seasons Lag? In the northern hemisphere, June 21 (the summer solstice) is both the longest day of the year and the day on which the sun’s rays strike the earth most vertically, hence delivering the greatest amount of heat to the surface. Yet the hottest summer weather usually occurs about a month or so later. Let us see why this is the case. Because of the large specific heat of water, the oceans are slower to warm up than the land (and also slower to cool off in winter). In addition to pursuing pertinent information in the tables included in this book, it is useful to know that approximately two-thirds of the earth’s surface is ocean composed of salt water having a specific heat of 3890 J/kg \(\cdot\) K and the oceans, on the average, are 4000 m deep. Typically, an average of \(1050 W/m^{2}\) of solar energy falls on the earth’s surface, and the oceans absorb essentially all of the light that strikes them. However most of that light is absorbed in the upper 100 m of the Depths below that do not change temperature seasonally. Assume that the sunlight falls on the surface for only 12 hours per day that the ocean retains all the heat it absorbs. What will be the rise in temperature of the upper 100 m of the oceans during the month following the summer solstice? Does this seem to be large enough to be perceptible?

CALC Suppose that both ends of the rod in Fig. 17.23a are kept at a temperature of \(0^{\circ} \mathrm{C}\), and that the initial temperature distribution along the rod is given by \(T=\left(100^{\circ} \mathrm{C}\right) \sin \pi x / L\) where x is measured from the left end of the rod. Let the rod be copper, with length \(L=0.100 m\) and cross-sectional area \(1.00 \ \mathrm{cm}^{2}\). (a) Show the initial temperature distribution in a diagram. (b) What is the final temperature distribution after a very long time has elapsed? (c) Sketch curves that you think would represent the temperature distribution at intermediate times. (d) What is the initial temperature gradient at the ends of the rod? (e) What is the initial heat current from the ends of the rod into the bodies making contact with its ends? (f) What is the initial heat current at the center of the rod? Explain. What is the heat current at this point at any later time? (g) What is the value of the thermal diffusivity \(k / \rho c\) for copper, and in what unit is it expressed? (Here k is the thermal conductivity, \(\rho=8.9 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\) is the density, and c is the specific heat.) (h) What is the initial time rate of change of temperature at the center of the rod? (i) How much time would be required for the center of the rod to reach its final temperature if the temperature continued to decrease at this rate? (This time is called the relaxation time of the rod.) (j) From the graphs in part (c), would you expect the magnitude of the rate of temperature change at the midpoint to remain constant, increase, or decrease as a function of time? (k) What is the initial rate of change of temperature at a point in the rod 2.5 cm from its left end? Equation Transcription: 0°C T=100°C sin ??x/L L=0.100 m 1.00 cm2 k/?c ?=8.9103 kg/m3 Text Transcription: 0^circ C T=100^circ C sin ??x/L L=0.100 m 1.00 cm^2 k/rho c rho=8.9103 kg/m^3

(a) A spherical shell has inner and outer radii a and b, respectively, and the temperatures at the inner and outer surfaces are \(T_2\) and \(T_1\). The thermal conductivity of the material of which the shell is made is k. Derive an equation for the total heat current through the shell. (b) Derive an equation for the temperature variation within the shell in part (a); that is, calculate T as a function of r, the distance from the center of the shell. (c) A hollow cylinder has length L, inner radius a, and outer radius b, and the temperatures at the inner and outer surfaces are \(T_2\) and \(T_1\). (The cylinder could represent an insulated hot-water pipe, for example.) The thermal conductivity of the material of which the cylinder is made is k. Derive an equation for the total heat current through the walls of the cylinder. (d) For the cylinder of part (c), derive an equation for the temperature variation inside the cylinder walls. (e) For the spherical shell of part (a) and the hollow cylinder of part (c), show that the equation for the total heat current in each case reduces to Eq. (17.21) for linear heat flow when the shell or cylinder is very thin.

Problem 126CP Temperature Change in a Clock. A pendulum clock is designed to tick off one second on each side-to side swing of the pendulum (two ticks per complete period). (a) Will a pendulum clock gain time in hot weather and lose it in cold, or the reserve? Explain your reasoning. (b) A particular pendulum clock keeps correct time at 20.0°C. The pendulum shaft is steel, and its mass can be ignored compared with that of the bob. What is the fractional change in the length of the shaft when it is cooled to 10.0°C (c) How many seconds per day will the clock gain or lose at 10.0°C? (d) How closely must the temperature be controlled if the clock is not to gain or lose more than 1.00 s a day? Does the answer depend on the period of the pendulum?

BI0 A Walk in the Sun. Consider a poor lost soul walking at 5 \(\mathrm{km} / h\) 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 , 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 \(k^{\prime} A_{s k i n}\left(T_{\text {air }}-T_{s k i n}\right)\), where \(k^{\prime}\) is 54 \(J / h \cdot C^{\circ} \cdot m^{2}\), the exposed skin area \(A_{s k i n}\) is 1.5 \(m^{2}\), the air temperature \(T_{\text {air }}\) is \(47^{\circ} \mathrm{C}\), and the skin temperature \(T_{s k i n}\) is \(36^{\circ} \mathrm{C}\); (iii) the skin absorbs radiant energy from the sun at a rate of 1400 \(W / m^{2}\); (iv) the skin absorbs radiant energy from the environment, which has temperature \(47^{\circ} \mathrm{C}\). (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 \(36^{\circ} \mathrm{C}\). 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 \(36^{\circ} \mathrm{C}\) 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 \(m^{2}\). What rate of perspiration is required now? Discuss the usefulness of the traditional clothing worn by desert peoples. Equation Transcription: Text Transcription: km/h k’A_skin(T_air-T_skin) k’ J/h.Cdeg.m^2 A_skin m^2 T_air 47degC T_skin 36degC W/m^2 47degC e=1 L/h 36degC 2.42x10^6 J/kg e almost 0 m^2

CP CALC 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 Co. (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?

Problem 102P A Styrofoam bucket of negligible mass contains 1.75 kg of water and 0.450 kg of ice. More ice, from a refrigerator at - 15.0o C, 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?

Problem 106P One experimental method of measuring an insulating material’s 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 Co (about 120 Fo) above the temperature of the outer sur-face. 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.

Problem 107P Effect of a Window in a Door. A carpenter builds a solid wood door with dimensions 2.00 m X 0.95 m X 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 20.0o C, and the outside air temperature is - 8.0o C. (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.

A steam pipe with a radius of , carrying steam at \(140^{\circ} \mathrm{C}\), is surrounded by a cylindrical jacket with inner and outer radii and and made of a type of cork with thermal conductivity \(4.00 \times 10^{-2} \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). This in turn is surrounded by a cylindrical jacket made of a brand of Styrofoam with thermal conductivity \(1.00 \times 10^{-2} \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and having inner and outer radii and (Fig. P17.125). The outer surface of the Styrofoam is in contact with air at \(15^{\circ} \mathrm{C}\). Assume that this outer surface has a temperature of \(15^{\circ} \mathrm{C}\). (a) What is the temperature at a radius of 4.00 cm, where the two insulating layers meet? (b) What is the total rate of transfer of heat out of a 2.00-m length of pipe? (Hint: Use the expression derived in part (c) of Challenge Problem 17.124.) Equation Transcription: Text Transcription: 140degC 4.00x10-2 W/m cdot K 1.00x10-2 W/m cdot K 15odegC 15odegC