Chapter 12

A temperature of absolute zero occurs at \(-273.15^{\circ} \mathrm{C}\). What is this temperature on the Fahrenheit scale?

Suppose you are hiking down the Grand Canyon. At the top, the temperature early in the morning is a cool \(3^{\circ} \mathrm{C}\). By late afternoon, the temperature at the bottom of the canyon has warmed to a sweltering \(34^{\circ} \mathrm{C}\). What is the difference between the higher and lower temperatures in (a) Fahrenheit degrees and (b) kelvins?

On the moon the surface temperature ranges from \(375 \mathrm{~K}\) during the day to \(1.00 \times 10^{2} \mathrm{~K}\) at night. What are these temperatures on the (a) Celsius and (b) Fahrenheit scales?

Dermatologists often remove small precancerous skin lesions by freezing them quickly with liquid nitrogen, which has a temperature of \(77 \mathrm{~K}\). What is this temperature on the (a) Celsius and (b) Fahrenheit scales?

What's your normal body temperature? It may not be \(98.6^{\circ} \mathrm{F}\), the oft-quoted average that was determined in the nineteenth century. A more recent study has reported an average temperature of \(98.2^{\circ} \mathrm{F}\). What is the difference between these averages, expressed in Celsius degrees?

On the Rankine temperature scale, which is sometimes used in engineering applications, the ice point is at \(491.67^{\circ} \mathrm{R}\) and the steam point is at \(671.67^{\circ} \mathrm{R}\). Determine a relationship (analogous to Equation 12.1 ) between the Rankine and Fahrenheit temperature scales.

When the temperature of a coin is raised by \(75 \mathrm{C}^{\circ},\) the coin's diameter increases by \(2.3 \times 10^{-5} \mathrm{~m}\). If the original diameter of the coin is \(1.8 \times 10^{-2} \mathrm{~m}\), find the coefficient of linear expansion.

Multiple-Concept Example 4 reviews the concepts that are involved in this problem. A ruler is accurate when the temperature is \(25^{\circ} \mathrm{C}\). When the temperature drops to \(-14^{\circ} \mathrm{C}\), the ruler shrinks and no longer measures distances accurately. However, the ruler can be made to read correctly if a force of magnitude \(1.2 \times 10^{3} \mathrm{~N}\) is applied to each end so as to stretch it back to its original length. The ruler has a cross-sectional area of \(1.6 \times 10^{-5} \mathrm{~m}^{2},\) and it is made from a material whose coefficient of linear expansion is \(2.5 \times 10^{-5}\left(\mathrm{C}^{0}\right)^{-1}\). What is Young's modulus for the material from which the ruler is made?

A steel bicycle wheel (without the rubber tire) is rotating freely with an angular speed of \(18.00 \mathrm{rad} / \mathrm{s}\). The temperature of the wheel changes from -100.0 to \(+300.0{ }^{\circ} \mathrm{C}\). No net external torque acts on the wheel, and the mass of the spokes is negligible. (a) Does the angular speed increase or decrease as the wheel heats up? Why? (b) What is the angular speed at the higher temperature?

At a temperature of \(0{ }^{\circ} \mathrm{C}\), the mass and volume of a fluid are \(825 \mathrm{~kg}\) and \(1.17 \mathrm{~m}^{3}\). The coefficient of volume expansion is \(1.26 \times 10^{-3}\left(\mathrm{C}^{\circ}\right)^{-1}\). (a) What is the density of the fluid at this temperature? (b) What is the density of the fluid when the temperature has risen to \(20.0^{\circ} \mathrm{C}\) ?

At a temperature of \(0^{\circ} \mathrm{C}\), the mass and volume of a fluid are \(825 \mathrm{~kg}\) and \(1.17 \mathrm{~m}^{3}\). The coefficient of volume expansion is \(1.26 \times 10^{-3}\left(\mathrm{C}^{\circ}\right)^{-1}\) (a) What is the density of the fluid at this temperature? (b) What is the density of the fluid when the temperature has risen to \(20.0^{\circ} \mathrm{C} ?\)

During an all-night cram session, a student heats up a one-half liter \(\left(0.50 \times 10^{-3} \mathrm{~m}^{3}\right)\) glass (Pyrex) beaker of cold coffee. Initially, the temperature is \(18^{\circ} \mathrm{C}\), and the beaker is filled to the brim. A short time later when the student retums, the temperature has risen to \(92^{\circ} \mathrm{C}\). The coefficient of volume expansion of coffee is the same as that of water. How much coffee (in cubic meters) has spilled out of the beaker?

Consult Interactive LearningWare 12.1 at for help in solving this problem. During an all-night cram session, a student heats up a one-half liter \(\left(0.50 \times 10^{-3} \mathrm{~m}^{3}\right)\) glass (Pyrex) beaker of cold coffee. Initially, the temperature is \(18^{\circ} \mathrm{C},\) and the beaker is filled to the brim. A short time later when the student returns, the temperature has risen to \(92^{\circ} \mathrm{C}\). The coefficient of volume expansion of coffee is the same as that of water. How much coffee (in cubic meters) has spilled out of the beaker?

Many hot-water heating systems have a reservoir tank connected directly to the pipeline, so as to allow for expansion when the water becomes hot. The heating system of a house has \(76 \mathrm{~m}\) of copper pipe whose inside radius is \(9.5 \times 10^{-3} \mathrm{~m}\). When the water and pipe are heated from 24 to \(78^{\circ} \mathrm{C}\), what must be the minimum volume of the reservoir tank to hold the overflow of water?

Interactive LearningWare 12.1 at provides some useful background for this problem. Many hot-water heating systems have a reservoir tank connected directly to the pipeline, so as to allow for expansion when the water becomes hot. The heating system of a house has \(76 \mathrm{~m}\) of copper pipe whose inside radius is \(9.5 \times 10^{-3} \mathrm{~m}\). When the water and pipe are heated from 24 to \(78{ }^{\circ} \mathrm{C}\), what must be the minimum volume of the reservoir tank to hold the overflow of water?

At the bottom of an old mercury-in-glass thermometer is a \(45-\mathrm{mm}^{3}\) reservoir filled with mercury. When the thermometer was placed under your tongue, the warmed mercury would expand into a very narrow cylindrical channel, called a capillary, whose radius was \(1.7 \times 10^{-2} \mathrm{~mm}\). Marks were placed along the capillary that indicated the temperature. Ignore the thermal expansion of the glass and determine how far (in \(\mathrm{mm}\) ) the mercury would expand into the capillary when the temperature changed by \(1.0 \mathrm{C}^{\circ}\).

A solid aluminum sphere has a radius of \(0.50 \mathrm{~m}\) and a temperature of \(75^{\circ} \mathrm{C}\). The sphere is then completely immersed in a pool of water whose temperature is \(25^{\circ} \mathrm{C}\). The sphere cools, while the water temperature remains nearly at \(25^{\circ} \mathrm{C}\), because the pool is very large. The sphere is weighed in the water immediately after being submerged (before it begins to cool) and then again after cooling to \(25^{\circ} \mathrm{C}\). (a) Which weight is larger? Why? (b) Use Archimedes' principle to find the magnitude of the difference between the weights.

When you take a bath, how many kilograms of hot water \(\left(49.0^{\circ} \mathrm{C}\right)\) must you mix with cold water \(\left(13.0^{\circ} \mathrm{C}\right)\) so that the temperature of the bath is \(36.0^{\circ} \mathrm{C} ?\) The total mass of water (hot plus cold) is \(191 \mathrm{~kg}\). Ignore any heat flow between the water and its external surroundings.

Blood can carry excess energy from the interior to the surface of the body, where the energy is dispersed in a number of ways. While a person is exercising, \(0.6 \mathrm{~kg}\) of blood flows to the surface of the body and releases \(2000 \mathrm{~J}\) of energy. The blood arriving at the surface has the temperature of the body interior, \(37.0^{\circ} \mathrm{C}\). Assuming that blood has the same specific heat capacity as water, determine the temperature of the blood that leaves the surface and returns to the interior.

An ice chest at a beach party contains 12 cans of soda at \(5.0^{\circ} \mathrm{C}\). Each can of soda has a mass of \(0.35 \mathrm{~kg}\) and a specific heat capacity of \(3800 \mathrm{~J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right)\). Someone adds a \(6.5-\mathrm{kg}\) watermelon at \(27{ }^{\circ} \mathrm{C}\) to the chest. The specific heat capacity of watermelon is nearly the same as that of water. Ignore the specific heat capacity of the chest and determine the final temperature \(T\) of the soda and watermelon.

When resting, a person has a metabolic rate of about \(3.0 \times 10^{5}\) joules per hour. The person is submerged neck-deep into a tub containing \(1.2 \times 10^{3} \mathrm{~kg}\) of water at \(21.00{ }^{\circ} \mathrm{C}\). If the heat from the person goes only into the water, find the water temperature after half an hour.

Review Interactive Solution \(\underline{12.43}\) at for help in approaching this problem. When resting, a person has a metabolic rate of about \(3.0 \times 10^{5}\) joules per hour. The person is submerged neck-deep into a tub containing \(1.2 \times 10^{3} \mathrm{~kg}\) of water at \(21.00{ }^{\circ} \mathrm{C}\). If the heat from the person goes only into the water, find the water temperature after half an hour.

A piece of glass has a temperature of \(83.0^{\circ} \mathrm{C}\). Liquid that has a temperature of \(43.0^{\circ} \mathrm{C}\) is poured over the glass, completely covering it, and the temperature at equilibrium is \(53.0^{\circ}\) C. The mass of the glass and the liquid is the same. Ignoring the container that holds the glass and liquid and assuming that the heat lost to or gained from the surroundings is negligible, determine the specific heat capacity of the liquid.

At a fabrication plant, a hot metal forging has a mass of \(75 \mathrm{~kg}\) and a specific heat capacity of \(430 \mathrm{~J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right) .\) To harden it, the forging is immersed in \(710 \mathrm{~kg}\) of oil that has a temperature of \(32^{\circ} \mathrm{C}\) and a specific heat capacity of \(2700 \mathrm{~J} /\left(\mathrm{kg} \dot{\mathrm{c}} \mathrm{C}^{\circ}\right) .\) The final temperature of the oil and forging at thermal equilibrium is \(47^{\circ} \mathrm{C}\). Assuming that heat flows only between the forging and the oil, determine the initial temperature of the forging.

Multiple-Concept Example 11 deals with a situation that is similar, but not identical, to that here. When \(4200 \mathrm{~J}\) of heat are added to a \(0.15-\mathrm{m}\) -long silver bar, its length increases by \(4.3 \times 10^{-3} \mathrm{~m}\). What is the mass of the bar?

A \(0.35-\mathrm{kg}\) coffee mug is made from a material that has a specific heat capacity of \(920 \mathrm{~J} /\) \(\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right)\) and contains \(0.25 \mathrm{~kg}\) of water. The cup and water are at \(15^{\circ} \mathrm{C}\). To make a cup of coffeee, a small electric heater is immersed in the water and brings it to a boil in three minutes. Assume that the cup and water always have the same temperature and determine the minimum power rating of this heater.

An electric hot water heater takes in cold water at \(13.0^{\circ} \mathrm{C}\) and delivers hot water. The hot water has a constant temperature of \(45.0^{\circ} \mathrm{C}\) when the "hot" faucet is left open all the time, and the volume flow rate is \(5.0 \times 10^{-6} \mathrm{~m}^{3 / \mathrm{s}}\). What is the minimum power rating of the hot water heater?

To help prevent frost damage, fruit growers sometimes protect their crop by spraying it with water when overnight temperatures are expected to go below the freezing mark. When the water turns to ice during the night, heat is released into the plants, thereby giving them a measure of protection against the falling temperature. Suppose a grower sprays \(7.2 \mathrm{~kg}\) of water at \(0^{\circ} \mathrm{C}\) onto a fruit tree. (a) How much heat is released by the water when it freezes? (b) How much would the temperature of a \(180-\mathrm{kg}\) tree rise if it absorbed the heat released in part (a)? Assume that the specific heat capacity of the tree is \(2.5 \times 10^{3} \mathrm{~J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right)\) and that no phase change occurs within the tree itself.

Assume that the pressure is one atmosphere and determine the heat required to produce \(2.00 \mathrm{~kg}\) of water vapor at \(100.0^{\circ} \mathrm{C}\), starting with (a) \(2.00 \mathrm{~kg}\) of water at \(100.0^{\circ} \mathrm{C}\) and ( \(\mathrm{b}\) ) \(2.00 \mathrm{~kg}\) of liquid water at \(0.0^{\circ} \mathrm{C}\).

A person eats a container of strawberry yogurt. The Nutritional Facts label states that it contains 240 Calories ( 1 Calorie \(=4186 \mathrm{~J}\) ). What mass of perspiration would one have to lose to get rid of this energy? At body temperature, the latent heat of vaporization of water is \(2.42 \times 10^{6} \mathrm{~J} / \mathrm{kg} .\)

A woman finds the front windshield of her car covered with ice at \(-12.0^{\circ} \mathrm{C}\). The ice has a thickness of \(4.50 \times 10^{-4} \mathrm{~m},\) and the windshield has an area of \(1.25 \mathrm{~m}^{2}\). The density of ice is \(917 \mathrm{~kg} / \mathrm{m}^{3}\). How much heat is required to melt the ice?

A thermos contains \(150 \mathrm{~cm}^{3}\) of coffee at \(85^{\circ} \mathrm{C}\). To cool the coffee, you drop two \(11-\mathrm{g}\) ice cubes into the thermos. The ice cubes are initially at \(0^{\circ} \mathrm{C}\) and melt completely. What is the final temperature of the coffee? Treat the coffee as if it were water.

Equal masses of two different liquids have the same temperature of \(25.0^{\circ} \mathrm{C}\). Liquid A has a freezing point of \(-68.0{ }^{\circ} \mathrm{C}\) and a specific heat capacity of \(1850 \mathrm{~J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right) .\) Liquid \(\mathrm{B}\) has a freezing point of \(-96.0^{\circ} \mathrm{C}\) and a specific heat capacity of \(2670 \mathrm{~J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right) .\) The same amount of heat must be removed from each liquid in order to freeze it into a solid at its respective freezing point. Determine the difference \(L_{\mathrm{f}, \mathrm{A}}-L_{\mathrm{f}, \mathrm{B}}\) between the latent heats of fusion for these liquids.

Interactive Solution \(12.6112 .61\) at provides a model for solving problems such as this. A \(42-\mathrm{kg}\) block of ice at \(0{ }^{\circ} \mathrm{C}\) is sliding on a horizontal surface. The initial speed of the ice is \(7.3 \mathrm{~m} / \mathrm{s}\) and the final speed is \(3.5 \mathrm{~m} / \mathrm{s}\). Assume that the part of the block that melts has a very small mass and that all the heat generated by kinetic friction goes into the block of ice, and determine the mass of ice that melts into water at \(0{ }^{\circ} \mathrm{C}\).

Interactive Solution \(\underline{12.61} 12.61\) at provides a model for solving problems such as this. A \(42-\mathrm{kg}\) block of ice at \(0{ }^{\circ} \mathrm{C}\) is sliding on a horizontal surface. The initial speed of the ice is \(7.3 \mathrm{~m} / \mathrm{s}\) and the final speed is \(3.5 \mathrm{~m} / \mathrm{s}\). Assume that the part of the block that melts has a very small mass and that all the heat generated by kinetic friction goes into the block of ice, and determine the mass of ice that melts into water at \(0^{\circ} \mathrm{C}\).

An unknown material has a normal melting/ freezing point of \(-25.0^{\circ} \mathrm{C},\) and the liquid phase has a specific heat capacity of \(160 \mathrm{~J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right)\). One-tenth of a kilogram of the solid at \(-25.0^{\circ} \mathrm{C}\) is put into a \(0.150-\mathrm{kg}\) aluminum calorimeter cup that contains 0.100 kg of glycerin. The temperature of the cup and the glycerin is initially \(27.0^{\circ} \mathrm{C}\). All the unknown material melts, and the final temperature at equilibrium is \(20.0^{\circ} \mathrm{C}\). The calorimeter neither loses energy to nor gains energy from the external environment. What is the latent heat of fusion of the unknown material?

To help keep his barn warm on cold days, a farmer stores \(840 \mathrm{~kg}\) of solar-heated water \(\left(L_{\mathrm{f}}=3.35 \times 10^{5} \mathrm{~J} / \mathrm{kg}\right)\) in barrels. For how many hours would a \(2.0-\mathrm{kW}\) electric space heater have to operate to provide the same amount of heat as the water does when it cools from 10.0 to \(0.0^{\circ} \mathrm{C}\) and completely freezes?

A locomotive wheel is \(1.00 \mathrm{~m}\) in diameter. A \(25.0-\mathrm{kg}\) steel band has a temperature of \(20.0^{\circ} \mathrm{C}\) and a diameter that is \(6.00 \times 10^{-4} \mathrm{~m}\) less than that of the wheel. What is the smallest mass of water vapor at \(100^{\circ} \mathrm{C}\) that can be condensed on the steel band to heat it, so that it will fit onto the wheel? Do not ignore the water that results from the condensation.

The latent heat of vaporization of \(\mathrm{H}_{2} \mathrm{O}\) at body temperature \(\left(37.0^{\circ} \mathrm{C}\right)\) is \(2.42 \times 10^{6} \mathrm{~J} / \mathrm{kg} .\) To cool the body of a \(75-\mathrm{kg}\) jogger [average specific heat capacity \(\left.=3500 \mathrm{~J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right)\right]\) by \(1.5 \mathrm{C}^{\circ},\) how many kilograms of water in the form of sweat have to be evaporated?

A \(0.200-\mathrm{kg}\) piece of aluminum that has a temperature of \(-155^{\circ} \mathrm{C}\) is added to \(1.5 \mathrm{~kg}\) of water that has a temperature of \(3.0^{\circ} \mathrm{C}\). At equilibrium the temperature is \(0.0^{\circ} \mathrm{C}\). Ignoring the container and assuming that the heat exchanged with the surroundings is negligible, determine the mass of water that has been frozen into ice.

Suppose you are selling apple cider for two dollars a gallon when the temperature is \(4.0^{\circ} \mathrm{C} .\) The coefficient of volume expansion of the cider is \(280 \times 10^{6}\left(\mathrm{C}^{\circ}\right)^{-1} .\) If the expansion of the container is ignored, how much more money (in pennies) would you make per gallon by refilling the container on a day when the temperature is \(26^{\circ} \mathrm{C} ?\)

Ideally, when a thermometer is used to measure the temperature of an object, the temperature of the object itself should not change. However, if a significant amount of heat flows from the object to the thermometer, the temperature will change. A thermometer has a mass of \(31.0 \mathrm{~g},\) a specific heat capacity of \(c=815 \mathrm{~J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right),\) and a temperature of \(12.0^{\circ} \mathrm{C}\). It is immersed in \(119 \mathrm{~g}\) of water, and the final temperature of the water and thermometer is \(41.5^{\circ} \mathrm{C}\). What was the temperature of the water before the insertion of the thermometer?

A copper-constantan thermocouple can generate a voltage of \(4.75 \times 10^{-3}\) volts when the temperature of the hot junction is \(110.0{ }^{\circ} \mathrm{C}\) and the reference junction is kept at \(0.0^{\circ}\) C. If the voltage is proportional to the difference in temperature between the junctions, what is the temperature of the hot junction when the voltage is \(1.90 \times 10^{-3}\) volts?

Occasionally, huge icebergs are found floating on the ocean's currents. Suppose one such iceberg is \(120 \mathrm{~km}\) long, \(35 \mathrm{~km}\) wide, and \(230 \mathrm{~m}\) thick. (a) How much heat would be required to melt this iceberg (assumed to be at \(0{ }^{\circ} \mathrm{C}\) ) into liquid water at \(0{ }^{\circ} \mathrm{C}\) ? The density of ice is \(917 \mathrm{~kg} / \mathrm{m}^{3}\). (b) The annual energy consumption by the United States in 1994 was \(9.3 \times 10^{19} \mathrm{~J}\). If this energy were delivered to the iceberg every year, how many years would it take before the ice melted?

Two grams of liquid water are at \(0{ }^{\circ} \mathrm{C}\), and another two grams are at \(100{ }^{\circ} \mathrm{C}\). Heat is removed from the water at \(0{ }^{\circ} \mathrm{C}\), completely freezing it at \(0{ }^{\circ} \mathrm{C}\). This heat is then used to vaporize some of the water at \(100{ }^{\circ} \mathrm{C}\). What is the mass (in grams) of the liquid water that remains?

Two grams of liquid water are at \(0^{\circ} \mathrm{C},\) and another two grams are at \(100^{\circ} \mathrm{C}\). Heat is removed from the water at \(0^{\circ} \mathrm{C}\), completely freezing it at \(0^{\circ} \mathrm{C}\). This heat is then used to vaporize some of the water at \(100{ }^{\circ} \mathrm{C}\). What is the mass (in grams) of the liquid water that remains?

The box of a well-known breakfast cereal states that one ounce of the cereal contains 110 Calories \((1\) food Calorie \(=4186 \mathrm{~J})\). If \(2.0 \%\) of this energy could be converted by a weight lifter's body into work done in lifting a barbell, what is the heaviest barbell that could be lifted a distance of \(2.1 \mathrm{~m}\) ?

A rock of mass \(0.20 \mathrm{~kg}\) falls from rest from a height of \(15 \mathrm{~m}\) into a pail containing \(0.35 \mathrm{~kg}\) of water. The rock and water have the same initial temperature. The specific heat capacity of the rock is \(1840 \mathrm{~J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right)\). Ignore the heat absorbed by the pail itself, and determine the rise in the temperature of the rock and water.

A rock of mass 0.20 kg falls from rest from a height of \(15 \mathrm{~m}\) into a pail containing \(0.35 \mathrm{~kg}\) of water. The rock and water have the same initial temperature. The specific heat capacity of the rock is \(1840 \mathrm{~J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right)\). Ignore the heat absorbed by the pail itself, and determine the rise in the temperature of the rock and water.

A steel ruler is calibrated to read true at \(20.0^{\circ} \mathrm{C}\). A draftsman uses the ruler at \(40.0^{\circ} \mathrm{C}\) to draw a line on a \(40.0^{\circ} \mathrm{C}\) copper plate. As indicated on the warm ruler, the length of the line is \(0.50 \mathrm{~m}\). To what temperature should the plate be cooled, such that the length of the line truly becomes \(0.50 \mathrm{~m}\) ?