Chapter 14

\(\bullet\) (a) While vacationing in Europe, you feel sick and are told that you have a temperature of \(40.2^{\circ} \mathrm{C}\) . Should you be concerned? What is your temperature in \(^{\circ} \mathrm{F} ?\) (b) The morning weather report in Sydney predicts a high temperature of \(12^{\circ} \mathrm{C}\) . Will you need to bring a jacket? What is this temperature in \(^{\circ} \mathrm{F} ?(\mathrm{c})\) A friend has suggested that you go swimming in a pool having water of temperature 350 \(\mathrm{K}\) . Is this safe to do? What would this temperature be on the Fahrenheit and Celsius scales?

Temperatures in biomedicine. (a) Normal body temperature. The average normal body temperature measured in the mouth is 310 \(\mathrm{K}\) . 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.

\(\bullet\) (a) On January \(22,1943,\) the temperature in Spearfish, South Dakota, rose from \(-4.0^{\circ} \mathrm{F}\) to \(45.0^{\circ} \mathrm{F}\) in just 2 minutes. What was the temperature change in Celsius degrees and in kelvins? (b) The temperature in Browning, Montana, was \(44.0^{\circ} \mathrm{Fon}\) January \(23,1916,\) and the next day it plummeted to \(-56.0^{\circ} \mathrm{F} .\) What was the temperature change in Celsius degrees and in kelvins?

\(\cdot\) Inside the earth and the sun. (a) Geophysicists have esti- mated that the temperature at the center of the earth's core is \(5000^{\circ} \mathrm{C}\) (or more), while the temperature of the sun's core is about 15 million \(\mathrm{K}\) . Express both of these temperatures in Fahrenheit degrees.

(a) At what temperature do the Fahrenheit and Celsius scales coincide? (b) Is there any temperature at which the Kelvin and Celsius scales coincide?

Convert the following Kelvin temperatures to the Celsius and Fahrenheit scales: (a) the midday temperature at the surface of the moon \((400 \mathrm{K}) ;\) (b) the temperature at the tops of the clouds in the atmosphere of Saturn \((95 \mathrm{K}) ;(\mathrm{c})\) the temperature at the center of the sun \(\left(1.55 \times 10^{7} \mathrm{K}\right)\) .

\(\cdot\) The Eiffel Tower in Paris is 984 ft tall and is made mostly of steel. If this is its height in winter when its temperature is \(-8.00^{\circ} \mathrm{C},\) how much additional vertical distance must you cover if you decide to climb it during a summer heat wave when its temperature is \(40.0^{\circ} \mathrm{C}\) ? (b) Express the coefficient of linear expansion of steel in terms of Fahrenheit degrees.

\(\cdot\) A metal rod is 40.125 \(\mathrm{cm}\) long at \(20.0^{\circ} \mathrm{C}\) and 40.148 \(\mathrm{cm}\) long at \(45.0^{\circ} \mathrm{C} .\) Calculate the average coefficient of linear expansion of the rod's material for this temperature range.

\(\bullet\) The markings on an aluminum ruler and a brass ruler begin at the left end; when the rulers are at \(0.00^{\circ} \mathrm{C}\) , they are perfectly aligned. How far apart will the 20.0 \(\mathrm{cm}\) marks be on the two rulers at \(100.0^{\circ} \mathrm{C}\) if the left-hand ends are kept precisely aligned?

\(\cdot\) In an effort to stay awake for an all-night study session, a student makes a cup of coffee by first placing a 200.0 \(\mathrm{W}\) electric immersion heater in 0.320 \(\mathrm{kg}\) of water. (a) How much heat must be added to the water to raise its temperature from \(20.0^{\circ} \mathrm{C}\) to \(80.0^{\circ} \mathrm{C} ?\) (b) How much time is required if all of the heater's power goes into heating the water?

\(\cdot\) 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 \(-20^{\circ} \mathrm{C},\) what is the amount of heat needed to warm to internal body temperature \(\left(37^{\circ} \mathrm{C}\right)\) the 0.50 \(\mathrm{L}\) of air exchanged with each breath? Assume that the specific heat capacity of 1.3 \(\mathrm{g}\) is 1020 \(\mathrm{J} /(\mathrm{kg} \cdot \mathrm{K})\) and that 1.0 \(\mathrm{L}\) of air has a mass of 1.3 \(\mathrm{g}\) . (b) How much heat is lost per hour if the respiration rate is 20 breaths per minute?

\(\cdot\) A nail driven into a board increases in temperature. If 60\(\%\) of the kinetic energy delivered by a 1.80 kg hammer with a speed of 7.80 \(\mathrm{m} / \mathrm{s}\) is transformed into heat that flows into the nail and does not flow out, what is the increase in temperature of an 8.00 g aluminum nail after it is struck 10 times?

\(\cdot\) 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 \(\mathrm{N}\) . You carefully add \(1.25 \times 10^{4} \mathrm{J}\) of heat energy to the sample and find that its temperature rises 18.0 \(\mathrm{C}^{\circ} .\) What is the sample's specific heat?

\(\bullet\) A \(25,000\) -kg subway train initially traveling at 15.5 \(\mathrm{m} / \mathrm{s}\) 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 \(\mathrm{m}\) long by 20.0 \(\mathrm{m}\) wide by 12.0 \(\mathrm{m}\) 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 \(\mathrm{J} /(\mathrm{kg} \cdot \mathrm{K})\)

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

Maintaining body temperature. While running, a 70 \(\mathrm{kg}\) student generates thermal energy at a rate of 1200 \(\mathrm{W}\) . 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 student's body, for what amount of time could a student run before irreversible body damage occurred? (Protein structures in the body are damaged irreversibly if the body temperature rises to \(44^{\circ} \mathrm{C}\) or above. The specific heat capacity 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 heat capacities.)

A technician measures the specific heat capacity of an unidentified liquid by immersing an electrical resistor in it. Electrical energy is converted to heat, which is then transferred to the liquid for 120 s at a constant rate of 65.0 W. The mass of the liquid is \(0.780 \mathrm{kg},\) and its temperature increases from \(18.55^{\circ} \mathrm{C}\) to \(22.54^{\circ} \mathrm{C}\) . (a) Find the average specific heat capacity 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 its surroundings cannot be ignored. Is the result calculated in part (a) an overestimate or an underestimate of the average specific heat capacity? Explain.

\(\bullet\) Much of the energy of falling water in a waterfall is converted into heat. If all the mechanical energy is converted into heat that stays in the water, how much of a rise in temperature occurs in a 100 m waterfall?

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

An asteroid with a diameter of 10 \(\mathrm{km}\) and a mass of \(2.60 \times 10^{15} \mathrm{kg}\) impacts the earth at a speed of 32.0 \(\mathrm{km} / \mathrm{s}\) 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 \(10.0^{\circ} \mathrm{C} ),\) what mass of water will be boiled away by the collision? (For comparison, the mass of water contained in Lake Superior is about \(2 \times 10^{15} \mathrm{kg.} )\)

\(\cdot\) Evaporative cooling. The evaporation of sweat is an important mechanism for temperature control in some warmblooded animals. (a) What mass of water must evaporate from the skin of a 70.0 \(\mathrm{kg}\) man to cool his body 1.00 \(\mathrm{C}^{\circ}\) . The heat of vaporization of water at body temperature \(\left(37^{\circ} \mathrm{C}\right)\) is \(2.42 \times 10^{6} \mathrm{J} / \mathrm{kg} .\) The specific heat capacity of a typical human body is 3480 \(\mathrm{J} /(\mathrm{kg} \cdot \mathrm{K}) .\) (b) What volume of water must the man drink to replenish the evaporated water? Compare this result with the volume of a soft-drink can, which is 355 \(\mathrm{cm}^{3} .\)

. An ice-cube tray contains 0.350 \(\mathrm{kg}\) of water at \(18.0^{\circ} \mathrm{C}\) . How much heat must be removed from the water to cool it to \(0.00^{\circ} \mathrm{C}\) and freeze it? Express your answer in joules and in calories.

\(\bullet\) 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 \(\mathrm{W}\) 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 \(2.42 \times 10^{6} \mathrm{J} / \mathrm{kg}\) (b) The evaporated water must, of course, be replenished, or the person will dehydrate. How many 750 \(\mathrm{mL}\) 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.)

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

\(\bullet\) A 0.500 kg chunk of an unknown metal that has been in boiling water for several minutes is quickly dropped into an insulating Styrofoam TM beaker containing 1.00 kg of water at room temperature \(\left(20.0^{\circ} \mathrm{C}\right) .\) After waiting and gently stirring for 5.00 minutes, you observe that the water's temperature has reached a constant value of \(22.0^{\circ} \mathrm{C}\) (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 capacity of the metal? (b) Which is more useful for storing energy from heat, 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 capacity you calculated in part (a) be in error? Would it be too large, too small, or still correct? Explain your reasoning.

\(\bullet\) A copper pot with a mass of 0.500 \(\mathrm{kg}\) contains 0.170 \(\mathrm{kg}\) of water, and both are at a temperature of \(20.0^{\circ} \mathrm{C} . \mathrm{A} 0.250 \mathrm{kg}\) block of iron at \(85.0^{\circ} \mathrm{C}\) is dropped into the pot. Find the final temperature of the system, assuming no heat loss to the surroundings.

\(\bullet\) In a physics lab experiment, a student immersed 200 one-cent coins (each having a mass of 3.00 g) in boiling water. After they reached thermal equilibrium, she quickly fished them out and dropped them into 0.240 \(\mathrm{kg}\) of water at \(20.0^{\circ} \mathrm{C}\) in an insulated container of negligible mass. What was the final temperature of the coins? [One-cent coins are made of a metal alloy-mostly zinc-with a specific heat capacity of 390 \(\mathrm{J} /(\mathrm{kg} \cdot \mathrm{K}) . ]\)

\(\bullet\) A laboratory technician drops an 85.0 g solid sample of unknown material at a temperature of \(100.0^{\circ} \mathrm{C}\) into a calorimeter. The calorimeter can is made of 0.150 \(\mathrm{kg}\) of copper and contains 0.200 \(\mathrm{kg}\) of water, and both the can and water are initially at \(19.0^{\circ} \mathrm{C}\) . The final temperature of the system is measured to be \(26.1^{\circ} \mathrm{C}\) . Compute the specific heat capacity of the sample. (Assume no heat loss to the surroundings.)

\(\bullet\) An insulated beaker with negligible mass contains 0.250 \(\mathrm{kg}\) of water at a temperature of \(75.0^{\circ} \mathrm{C}\) . How many kilograms of ice at a temperature of \(-20.0^{\circ} \mathrm{C}\) must be dropped in the water so that the final temperature of the system will be \(30.0^{\circ} \mathrm{C}\) ?

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 heat exchange with the surroundings, what mass of ice was added?

\(\cdot\) A slab of a thermal insulator with a cross-sectional area of 100 \(\mathrm{cm}^{2}\) is 3.00 cm thick. Its thermal conductivity is 0.075 \(\mathrm{W} /(\mathrm{m} \cdot \mathrm{K}) .\) If the temperature difference between opposite faces is \(80 \mathrm{C}^{\circ},\) how much heat flows the slab in 1 day?

You are asked to design a cylindrical steel rod 50.0 \(\mathrm{cm}\) long, with a circular cross section, that will conduct 150.0 \(\mathrm{J} / \mathrm{s}\) from a furnace at \(400.0^{\circ} \mathrm{C}\) to a container of boiling water under 1 atmosphere of pressure. What must the rod's diameter be?

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 \(37^{\circ} \mathrm{C}\) and that the outer surface of the skin is at \(30.0^{\circ} \mathrm{C}\) . Skin varies in thickness from 0.50 \(\mathrm{mm}\) to a few millimeters on the palms and soles, so we shall assume an average thickness of \(0.75 \mathrm{mm} . \mathrm{A} 165 \mathrm{lb}, 6 \mathrm{ft}\) person has a surface area of about 2.0 \(\mathrm{m}^{2}\) and loses heat at a net rate of 75 \(\mathrm{W}\) while resting. On the basis of our assumptions, what is the thermal conductivity of this person's skin?

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

\bullet A carpenter builds an exterior house wall with a layer of wood 3.0 \(\mathrm{cm}\) thick on the outside and a layer of Styrofoam"" insulation 2.2 \(\mathrm{cm}\) thick on the inside wall surface. The wood has a thermal conductivity of \(0.080 \mathrm{W} /(\mathrm{m} \cdot \mathrm{K}),\) and the Sty- rofoam TM has a thermal conductivity of 0.010 \(\mathrm{W} /(\mathrm{m} \cdot \mathrm{K})\) . The interior surface temperature is \(19.0^{\circ} \mathrm{C},\) and the exterior surface temperature is \(-10.0^{\circ} \mathrm{C}\) . (a) What is the temperature at the plane where the wood meets the Styrofoamm? (b) What is the rate of heat flow per square meter through this wall?

\(\bullet\) One end of an insulated metal rod is maintained at \(100^{\circ} \mathrm{C}\) , while the other end is maintained at \(0^{\circ} \mathrm{C}\) by an ice-water mixture. The rod is 60.0 \(\mathrm{cm}\) long and has a cross- sectional area of 1.25 \(\mathrm{cm}^{2} .\) The heat conducted by the rod melts 8.50 \(\mathrm{g}\) of ice in 10.0 min. Find the thermal conductivity \(k\) of the metal.

A box-shaped wood stove has dimensions of 0.75 \(\mathrm{m} \times\) \(1.2 \mathrm{m} \times 0.40 \mathrm{m},\) an emissivity of \(0.85,\) and a surface temperature of \(205^{\circ} \mathrm{C}\) . Calculate its rate of radiation into the surrounding space.

How large is the sun? By measuring the spectrum of wave- lengths of light from our sun, we know that its surface temperature is 5800 \(\mathrm{K}\) . By measuring the rate at which we receive its energy on earth, we know that it is radiating a total of \(3.92 \times 10^{26} \mathrm{J} / \mathrm{s}\) and behaves nearly like an ideal blackbody. Use this information to calculate the diameter of our sun.

\(\bullet\) 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 \(\mathrm{kg}(165 \mathrm{lb})\) person of height 1.83 \(\mathrm{m}\) (6 ft) would have a body surface area of approximately 2.0 \(\mathrm{m}^{2}\) . (a) What is the net amount of heat this person could radiate per second into a room at \(18^{\circ} \mathrm{C}\) (about \(65^{\circ} \mathrm{F}\) ) if his skin's surface temperature is \(30^{\circ} \mathrm{C}\) ? (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.

\(\cdot\) The emissivity of tungsten is \(0.35 .\) A tungsten sphere with a radius of 1.50 \(\mathrm{cm}\) is suspended within a large evacuated enclo- sure whose walls are at 290 \(\mathrm{K}\) . What power input is required to maintain the sphere at a temperature of 3000 \(\mathrm{K}\) if heat conduction along the supports is negligible?

\(\cdot\) Size of a lightbulb filament. The operating temperature of a tungsten filament in an incandescent lightbulb is \(2450 \mathrm{K},\) and its emissivity is \(0.35 .\) Find the surface area of the filament of a 150 \(\mathrm{W}\) bulb if all the electrical energy consumed by the bulb is radiated by the filament as light. (In reality, only a small fraction of the radiation appears as visible light.)

An 8.50 kg block of ice at \(0^{\circ} \mathrm{C}\) is sliding on a rough horizontal icehouse floor (also at \(0^{\circ} \mathrm{C} )\) at 15.0 \(\mathrm{m} / \mathrm{s} .\) Assume that half of any heat generated goes into the floor and the rest goes into the ice. (a) How much ice melts after the speed of the ice has been reduced to 10.0 \(\mathrm{m} / \mathrm{s} ?\) (b) What is the maximum amount of ice that will melt?

Global warming. As the earth warms, sea level will rise due to melting of the polar ice and thermal expansion of the oceans. Estimates of the expected temperature increase vary, but 3.5 \(\mathrm{C}^{\circ}\) by the end of the century has been plausibly suggested. If we assume that the temperature of the oceans also increases by this amount, how much will sea level rise by the year 2100 due only to the thermal expansion of the water? Assume, reasonably, that the ocean basins do not expand appreciably. The average depth of the ocean is \(4000 \mathrm{m},\) and the coefficient of volume expansion of water at \(20^{\circ} \mathrm{C}\) is \(0.207 \times 10^{-3}\left(\mathrm{C}^{\circ}\right)^{-1}\)

\(\bullet\) 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 drawback is that energy is wasted because the tank loses heat when it is not in use, and 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 \(/ \min (9.46 \mathrm{L} / \mathrm{min}\) ) with the tap water being heated from \(50^{\circ} \mathrm{F}\left(10^{\circ} \mathrm{C}\right)\) to \(120^{\circ} \mathrm{F}\left(49^{\circ} \mathrm{C}\right)\) 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?

\(\bullet\) Burning fat by exercise. Each pound of fat contains 3500 food calories. When the body metabolizes food, 80\(\%\) of this energy goes to heat. Suppose you decide to run without stopping, an activity that produces 1290 \(\mathrm{W}\) of metabolic power for a typical person. (a) For how many hours must you run to burn up 1 lb of fat? Is this a realistic exercise plan? (b) If you followed your planned exercise program, how much heat would your body produce when you burn up a pound of fat (c) If you needed to get rid of all of this excess heat by evaporating water (i.e., sweating), how many liters would you need to evaporate? The heat of vaporization of water at body temperature is \(2.42 \times 10^{6} \mathrm{J} / \mathrm{kg}\)

Shivering. You have no doubt noticed that you usually shiver when you get out of the shower. Shivering is the body's way of generating heat to restore its internal temperature to the normal \(37^{\circ} \mathrm{C},\) and it produces approximately 290 \(\mathrm{W}\) of heat power per square meter of body area. \(\mathrm{A} 68 \mathrm{kg}(150 \mathrm{lb}), 1.78 \mathrm{m}\) (5 foot, 10 inch) person has approximately 1.8 \(\mathrm{m}^{2}\) of surface area. How long would this person have to shiver to raise his or her body temperature by \(1.0 \mathrm{C}^{\circ},\) assuming that none of this heat is lost by the body? The specific heat capacity of the body is about 3500 \(\mathrm{J} /(\mathrm{kg} \cdot \mathrm{K})\) .

\(\bullet\) Pasta time! You are making pesto for your pasta and have a cylindrical measuring cup 10.0 \(\mathrm{cm}\) high made of ordinary glass \(\left(\beta=2.7 \times 10^{-5}\left(\mathrm{C}^{\circ}\right)^{-1}\right)\) and that is filled with olive oil \(\left(\beta=6.8 \times 10^{-4}\left(\mathrm{C}^{\circ}\right)^{-1}\right)\) to a height of 1.00 \(\mathrm{mm}\) below the top of the cup. Initially, the cup and oil are at a kitchen temperature of \(22.0^{\circ} \mathrm{C}\) 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?

A copper calorimeter can with mass 0.100 \(\mathrm{kg}\) contains 0.160 \(\mathrm{kg}\) of water and 0.018 \(\mathrm{kg}\) of ice in thermal equilibrium at atmospheric pressure. If 0.750 \(\mathrm{kg}\) of lead at a temperature of \(255^{\circ} \mathrm{C}\) is dropped into the can, what is the final temperature of the system if no heat is lost to the surroundings?

\(\bullet\) A piece of ice at \(0^{\circ} \mathrm{C}\) falls from rest into a lake whose tem- perature is \(0^{\circ} \mathrm{C},\) and 1.00\(\%\) of the ice melts. Compute the minimum height from which the ice has fallen.

. 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 \(\mathrm{m}^{3}\) of air in the room. The air has a specific heat capacity of 1020 \(\mathrm{J} /(\mathrm{kg} \cdot \mathrm{K})\) and a density of 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 min in this case?