Chapter 17

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

On a cool \(\left(4,0^{\circ} \mathrm{C}\right)\) Saturfay moming, a pilot fills the fuel tanks of her Pitts \(S-2 C\) (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 \(\mathrm{L}\) 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 (in "C) reached by the fuel and the tank on Saturday aftemoon? The coefficient of volume expansion of gasoline is \(9.5 \times 10^{-4} \mathrm{K}^{-1}\) , 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?

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 $$\Delta p=B \beta \Delta T$$ where the bulk modulus \(B\) and the average coefficient of volume expansion \(\beta\) are both assumed positive and constant. (b) What pressure is necessary to prevent a steel block from expanding when its temperature is increased from \(20.0^{\circ} \mathrm{C}\) to \(35.0^{\circ} \mathrm{C} ?\)

Spacecraft Reentry. A spacecraft made of aluminum circles the earth at a speed of 7700 \(\mathrm{m} / \mathrm{s}\) . (a) Find the ratio of its kinetic energy to the energy required to raise its temperature from \(0^{\circ} \mathrm{C}\) to \(600^{\circ} \mathrm{C}\) . (The melting point of aluminum is \(660^{\circ} \mathrm{C}\) . Assume a constant specific heat of \(910 \mathrm{J} / \mathrm{kg} \cdot \mathrm{K} .\) (b) Discuss the bearing of your answer on the problem of the reentry of a manned space vehicle into the earth's atmosphere.

A capstan is a rotating drum or cylinder over which a rope or cond slides in order to provide a great amplification of the rope's tension while keeping both ends free (Fig. 17.33). 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 \(\mathrm{N}\) and the capstan has a diameter of 10.0 \(\mathrm{cm}\) and turns once in 0.900 \(\mathrm{s}\) , 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 \(\mathrm{kg}\) , 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.

Hot Air in a Physics Lecture. (a) Atypical student listening attentively to a physics lecture has a heat output of 100 \(\mathrm{W}\) . 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 \(\mathrm{m}^{3}\) of air in the room. The air has specific heat capacity 1020 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) and density 1.20 \(\mathrm{kg} / \mathrm{m}^{3}\) . 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 \(\mathrm{W}\) . What is the temperature rise during 50 \(\mathrm{min}\) in this case?

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 3.00 mol 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. \()\)

A copper calorimeter can with mass 0.446 kg contains 0.0950 \(\mathrm{kg}\) of ice. The system is initially at \(0.0^{\circ} \mathrm{C}\) (a) If 0.0350 \(\mathrm{kg}\) of steam at \(100.0^{\circ} \mathrm{C}\) 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?

A Styrofoam bucket of negligible mass contains 1.75 \(\mathrm{kg}\) of water and 0.450 \(\mathrm{kg}\) of ice. More ice, from a refrigerator at \(-15.0^{\circ} \mathrm{C},\) is added to the mixture in the bucket, and when thermal equilibrium has been reached, the total mass of ice in the bucket is 0.778 \(\mathrm{kg}\) . Assuming no hear exchange with the surroundings, what mass of ice was added?

In a container of negligible mass, 0.0400 \(\mathrm{kg}\) of steam at \(100^{\circ} \mathrm{C}\) and atmospheric pressure is added to 0.200 \(\mathrm{kg}\) 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?

A tube leads from a \(0.150-\mathrm{kg}\) calorimeter to a flask in which water is boiling under atmospheric pressure. The calorimeter has specific heat capacity 420 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) , and it originally contains 0.340 \(\mathrm{kg}\) of water at \(15.0^{\circ} \mathrm{C}\) . Steam is allowed to condense in the calorimeter at atmospheric pressure until the temperature of the calorimeter and contents reaches \(71.0^{\circ} \mathrm{C}\) , at which point the total mass of the calorimeter and its contents is found to be 0.525 \(\mathrm{kg}\) . Compute the heat of vaporization of water from these data.

A worker pours 1.250 kg of molten lead at a temperature of \(365.0^{\circ} \mathrm{C}\) into 0.5000 \(\mathrm{kg}\) of water at a temperature of \(75.00^{\circ} \mathrm{C}\) 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.

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 \(\mathrm{W}\) is required to keep the interior surface of the box 65.0 \(\mathrm{C}^{\circ}\) (about 120 \(\mathrm{F}^{\circ}\) ) above the temperature of the outer surface. The total area of the box is 2.18 \(\mathrm{m}^{2}\) , and the wall thickness is 3.90 \(\mathrm{cm}\) . 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 \(2.00 \mathrm{m} \times 0.95 \mathrm{m} \times 5.0 \mathrm{cm} .\) Its thermal conductivity is \(k=0.120 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}\) . The air films on the inner and outer surfaces of the door have the same combined thermal resistance as an additional \(1.8-\mathrm{cm}\) thickness of solid wood. The inside air temperature is \(20.0^{\circ} \mathrm{C}\) , and the outside air temperature is \(-8.0^{\circ} \mathrm{C} .\) (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 \(\mathrm{m}\) on a side is inserted in the door? The glass is 0.450 \(\mathrm{cm}\) thick, and the glass has a thermal conductivity of 0.80 \(\mathrm{W} / \mathrm{m} \cdot \mathrm{K}\) . The air films on the two sides of the glass have a total thermal resistance that is the same as an additional 12.0 \(\mathrm{cm}\) of glass.

A wood ceiling with thermal resistance \(R_{1}\) is covered with a layer of insulation with thermal resistance \(R_{2} .\) Prove that the effective thermal resistance of the combination is \(R=R_{1}+R_{2}\) .

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, \(13.0 \mathrm{cm} ;\) brass, \(18.0 \mathrm{cm} ;\) steel, \(24.0 \mathrm{cm} .\) (a) What is the temperature of the junction point? (b) What is the heat current in each of the three rods?

Arod is initially at a uniform temperature of \(0^{\circ} \mathrm{C}\) throughout. One end is kept at \(0^{\circ} \mathrm{C}\) , and the other is brought into contact with a steam bath at \(100^{\circ} \mathrm{C}\) . 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 \(2.50 \mathrm{cm}^{2},\) its length is 120 \(\mathrm{cm}\) , its thermal conductivity is \(380 \mathrm{W} / \mathrm{m} \cdot \mathrm{K},\) its density is \(1.00 \times 10^{4} \mathrm{kg} / \mathrm{m}^{3},\) and its specific heat is 520 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) . Consider a short cylindrical element of the rod 1.00 \(\mathrm{cm}\) in length. (a) If the temperature gradient at the cooler end of this element is 140 \(\mathrm{C}^{\circ} / \mathrm{m}\) , 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 0.250 \(\mathrm{C} \%\) /s, what is the temperature gradient at the other end of the element?

A rustic cabin has a floor area of \(3.50 \mathrm{m} \times 3.00 \mathrm{m} .\) Its walls, which are 2.50 \(\mathrm{m}\) tall, are made of wood (thermal conductivity 0.0600 \(\mathrm{W} / \mathrm{m} \cdot \mathrm{K} ) 1.80 \mathrm{cm}\) thick and are further insulated with 1.50 \(\mathrm{cm}\) of a synthetic material. When the outside temperature is \(2.00^{\circ} \mathrm{C},\) it is found necessary to heat the room at a rate of 1.25 \(\mathrm{kW}\) to maintain its temperature at \(19.0^{\circ} \mathrm{C}\) . 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.

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 \(1.50 \times 10^{11} \mathrm{m},\) and the radius of the sun is \(6.96 \times 10^{8} \mathrm{m}\) (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 black-body, what is the temperature of its surface?

A Thermos for Liquid Helium. A physicist uses a cylindrical metal can 0.250 \(\mathrm{m}\) high and 0.090 \(\mathrm{m}\) in diameter to store liquid helium at 4.22 \(\mathrm{K}\) ; at that temperature the heat of vaporization of helium is \(2.09 \times 10^{4} \mathrm{J} / \mathrm{kg}\) . Completely surrounding the metal can are walls maintained at the temperature of liquid nitrogen, 77.3 \(\mathrm{K}\) , 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.

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 \(\mathrm{kg} / \mathrm{m}^{3}\) and specific heat 1020 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) . A 50.0 -g hamster is placed in a calorimeter that contains 0.0500 \(\mathrm{m}^{3}\) 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 \(\mathrm{C}^{\circ}\) 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 \(\mathrm{J} / \mathrm{g}\) , 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 earth's surface; the occens contain about \(97.5 \%,\) and the other 0.75\(\%\) is mainly groundwater. Suppose the icecaps, currently at an average temperature of about \(-30^{\circ} \mathrm{C},\) 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 \(5.00^{\circ} \mathrm{C}\) .

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