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College Physics 2017

Raymond A. Serway, Chris Vuille, John hughes

Chapter 11

Energy in thermal Processes

Educators


Problem 1

Convert $3.50 \times 10^{3}$ cal to the equivalent number of (a) kilo-calories (also known as Calories, used to describe the energy content of food) and (b) joules.

Averell H.
Carnegie Mellon University

Problem 2

A medium-sized banana provides about 105 Calories of energy. (a) Convert 105 Cal to joules. (b) Suppose that amount of energy is transformed into kinetic energy of a 1.00 -kg object initially at rest. Calculate the final speed of the object. (c) If that same amount of energy is added to 3.79 $\mathrm{kg}$ (about 1 gal) of water at $20.0^{\circ} \mathrm{C}$ , what is the water's final temperature?

Salamat A.
Numerade Educator

Problem 3

A 75 -kg sprinter accelerates from rest to a speed of 11.0 $\mathrm{m} / \mathrm{s}$ in 5.0 s. (a) Calculate the mechanical work done by the sprinter during this time. (b) Calculate the average power
the sprinter must generate. (c) If the sprinter converts food energy to mechanical energy with an efficiency of $25 \%,$ at what average rate is he burning Calories? (d) What happens to the other 75$\%$ of the food energy being used?

Averell H.
Carnegie Mellon University

Problem 4

A 55-kg student eats a 540 -Calorie (540 kcal) jelly doughnut for breakfast. (a) How many joules of energy are the equivalent of one jelly doughnut? (b) How many stairs must the student climb to perform an amount of mechanical work equivalent to the food energy in one jelly doughnut? Assume the height of a single stair is 15 $\mathrm{cm} .(\mathrm{c})$ loughnut? Assume only 25$\%$ efficient in converting chemical energy to mechanical energy, how many stairs must the woman climb to work off her breakfast?

Salamat A.
Numerade Educator

Problem 5

A person's basal metabolic rate (BMR) is the rate at which energy is expended while resting in a neutrally temperate environment. A typical BMR is $7.00 \times 10^{6} \mathrm{J} / \mathrm{day}$ . Convert
this $\mathrm{BMR}$ to units of (a) watts and (b) kilocalories (or Calories) per hour. (c) Suppose a $1.00-\mathrm{kg}$ object's gravitation potential energy is increased at a rate equal to this typical BMR. Find the rate of change of the object's height in $\mathrm{m} / \mathrm{s}$ .

Averell H.
Carnegie Mellon University

Problem 6

The temperature of a silver bar rises by $10.0^{\circ} \mathrm{C}$ when it absorbs 1.23 $\mathrm{kJ}$ of energy by heat. The mass of the bar is 525 $\mathrm{g}$ . Determine the specific heat of silver from these data.

Salamat A.
Numerade Educator

Problem 7

The highest recorded waterfall in the world is found at Angel Falls in Venezuela. Its longest single waterfall has a height of 807 $\mathrm{m}$ . If water at the top of the falls is at $15.0^{\circ} \mathrm{C}$ , what is the maximum temperature of the water at the bottom of the falls?
ssume all the kinetic energy of the water as it reaches the bottom goes into raising the water's temperature.

Averell H.
Carnegie Mellon University

Problem 8

An aluminum rod is 20.0 $\mathrm{cm}$ long at $20.0^{\circ} \mathrm{C}$ and has a mass of
0.350 $\mathrm{kg} .$ If $1.00 \times 10^{4} \mathrm{J}$ of energy is added to the rod by heat,
what is the change in length of the rod?

Salamat A.
Numerade Educator

Problem 9

Lake Erie contains roughly $4.00 \times 10^{11} \mathrm{m}^{3}$ of water. (a) How much energy is required to raise the temperature of that volume of water from $11.0^{\circ} \mathrm{C}$ to $12.0^{\circ} \mathrm{C}$ (b) How many years would it take to supply this amount of energy by using the $1.00 \times 10^{4}$ -MW exhaust energy of an electric power plant?

Averell H.
Carnegie Mellon University

Problem 10

A $3.00-\mathrm{g}$ copper coin at $25.0^{\circ} \mathrm{C}$ drops 50.0 $\mathrm{m}$ to the ground. (a) Assuming 60.0$\%$ of the change in gravitational potential energy of the coin-Earth system goes into increasing the internal energy of the coin, determine the coin's final temperature. (b) Does the result depend on the mass of the coin? Explain.

Salamat A.
Numerade Educator

Problem 11

$\mathrm{A} 5.00-\mathrm{g}$ lead bullet traveling at $3.00 \times 10^{2} \mathrm{m} / \mathrm{s}$ is stopped by a large tree. If half the kinetic energy of the bullet is trans- formed into internal energy and remains with the bullet while the other half is transmitted to the tree, what is the increase in temperature of the bullet?

Averell H.
Carnegie Mellon University

Problem 12

The apparatus shown in Figure P11.12 was used by Joule to measure the mechanical equivalent of
heat. Work is done on the water by a rotating paddle wheel, which is driven by two blocks falling at a
constant speed. The temperature of the stirred water increases due to the friction between the water and the paddles. If the energy lost in the bearings and through the walls is neglected, then the loss in potential energy associated with the blocks equals the work done by the paddle wheel on the water. If each block has a mass of 1.50 $\mathrm{kg}$ and the insulated tank is filled with 0.200 $\mathrm{kg}$ of water, what is the increase in temperature of the water after the blocks fall through a distance of 3.00 $\mathrm{m} ?$

Salamat A.
Numerade Educator

Problem 13

A 0.200 -kg aluminum cup contains 800 . g of water in thermal equilibrium with the cup at $80 .^{\circ} \mathrm{C}$ . The combination of cup and water is cooled uniformly so that the temperature decreases by $1.5^{\circ} \mathrm{C}$ per minute. At what rate is energy being removed? Express your answer in watts.

Averell H.
Carnegie Mellon University

Problem 14

A 1.5 -kg copper block is given an initial speed of 3.0 $\mathrm{m} / \mathrm{s}$ on a
rough horizontal surface. Because of friction, the block finally comes to rest. (a) If the block absorbs 85$\%$ of its initial kinetic energy as internal energy, calculate its increase in temperature. (b) What happens to the remaining energy?

Salamat A.
Numerade Educator

Problem 15

A swimming pool filled with water has dimensions of 5.00 $\mathrm{m}$ $\times 10.0 \mathrm{m} \times 1.78 \mathrm{m} .$ (a) Find the mass of water in the pool. (b) Find the thermal energy required to heat the pool water from $15.5^{\circ} \mathrm{C}$ to $26.5^{\circ} \mathrm{C}$ . (c) Calculate the cost of heating the pool from $15.5^{\circ} \mathrm{C}$ to $26.5^{\circ} \mathrm{C}$ if electrical energy costs $\$ 0.100$ per kilowat-hour.

Averell H.
Carnegie Mellon University

Problem 16

In the summer of 1958 in St. Petersburg, Florida, a new sidewalk was poured near the childhood home of one of the authors. No expansion joints were supplied, and by mid- July, the sidewalk had been completely destroyed by thermal expansion and had to be replaced, this time with the important addition of expansion joints! This event is modeled here. A slab of concrete 4.00 $\mathrm{cm}$ thick, 1.00 $\mathrm{m}$ long, and 1.00 $\mathrm{m}$ wide is poured for a sidewalk at an ambient temperature of $25.0^{\circ} \mathrm{C}$ and allowed to set.
The slab is exposed to direct sunlight and placed in a series of such slabs without proper expansion joints, so linear expansion is prevented. (a) Using the linear expansion equation (Eq. 10.4$)$ , eliminate $\Delta L$ from the equation for compressive stress and strain (Eq. 9.3$) .(\mathrm{b})$
Use the expression found in part (a) to eliminate $\Delta T$ from Equation $11.3,$ obtaining a symbolic equation for thermal energy transfer $Q$ (c) Compute the mass of the concrete slab
given that its density is $2.40 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3} .$ (d) Concrete has an ultimate compressive strength of $2.00 \times 10^{7}$ Pa, specific heat of $880 \mathrm{J} / \mathrm{kg} \cdot^{\circ} \mathrm{C},$ and Young's modulus of $2.1 \times 10^{10} \mathrm{Pa}$ . How much thermal energy must be transferred to the slab to reach this compressive stress? (e) What temperature change is required? (f) If the Sun delivers $1.00 \times 10^{3} \mathrm{W}$ of power to the top surface of the slab and if half the energy, on the average, is absorbed and retained, how long does it take the slab to reach the point at which it is in danger of cracking due to compressive stress?

Salamat A.
Numerade Educator

Problem 17

What mass of water at $25.0^{\circ} \mathrm{C}$ must be allowed to come to thermal equilibrium with a $1.85-\mathrm{kg}$ cube of aluminum initially at $1.50 \times 10^{2 \circ} \mathrm{C}$ to lower the temperature of the aluminum to $65.0^{\circ} \mathrm{C}$ Assume any water turned to steam subsequently recondenses.

Averell H.
Carnegie Mellon University

Problem 18

Lead pellets, each of mass $1.00 \mathrm{g},$ are heated to $200 .^{\circ} \mathrm{C} .$ How many pellets must be added to 0.500 $\mathrm{kg}$ of water that is initially at $20.0^{\circ} \mathrm{C}$ to make the equilibrium temperature $25.0^{\circ} \mathrm{C}$ ? Neglect any energy transfer to or from the container.

Salamat A.
Numerade Educator

Problem 19

An aluminum cup contains 225 g of water and a $40-\mathrm{g}$ copper stirrer, all at $27^{\circ} \mathrm{C}$ . A 400 - g sample of silver at an initial temperature of $87^{\circ} \mathrm{C}$ is placed in the water. The stirrer is used to stir the mixture until it reaches its final equilibrium temperature of $32^{\circ} \mathrm{C}$ . Calculate the mass of the aluminum cem.

Averell H.
Carnegie Mellon University

Problem 20

A large room in a house holds 975 $\mathrm{kg}$ of dry air at $30.0^{\circ} \mathrm{C} . \mathrm{A}$
woman opens a window briefly and a cool breeze brings in an additional 50.0 $\mathrm{kg}$ of dry air at $18.0^{\circ} \mathrm{C}$ . At what temperature will the two air masses come into thermal equilibrium, assuming they form a closed system? (The specific heat of dry air is 1006 $\mathrm{J} / \mathrm{kg} \cdot^{\circ} \mathrm{C}$ , although that value will cancel out of the calorimetry equation.)

Salamat A.
Numerade Educator

Problem 21

An aluminum calorimeter with a mass of 0.100 $\mathrm{kg}$ contains 0.250 kg of water. The calorimeter and water are in thermal equilibrium at $10.0^{\circ} \mathrm{C}$ . Two metallic blocks are placed into the water. One is a $50.0-\mathrm{g}$ piece of copper at $80.0^{\circ} \mathrm{C}$ . The
other has a mass of 70.0 $\mathrm{g}$ and is originally at a temperature of $100 .^{\circ} \mathrm{C}$ . The entire system stabilizes at a final temperature of $20.0^{\circ} \mathrm{C}$ . (a) Determine the specific heat of the unknown sample. (b) Using the data in Table 11.1, can you make a positive iden-
tification of the unknown material? Can you identify a possible material? (c) Explain your answers for part (b).

Averell H.
Carnegie Mellon University

Problem 22

A $1.50-\mathrm{kg}$ iron horseshoe initially at $600^{\circ} \mathrm{C}$ is dropped into a bucket containing 20.0 $\mathrm{kg}$ of water at $25.0^{\circ} \mathrm{C} .$ What is the final temperature of the water-horseshoe system? Ignore the heat capacity of the container and assume a negligible amount of water boils away.

Salamat A.
Numerade Educator

Problem 23

A student drops two metallic objects into a $120-\mathrm{g}$ steel container holding 150 $\mathrm{g}$ of water at $25^{\circ} \mathrm{C}$ . One object is a $200-\mathrm{g}$ cube of copper that is initially at $85^{\circ} \mathrm{C},$ and the other is a chunk of aluminum that is initially at $5.0^{\circ} \mathrm{C}$ . To the surprise of the student, the water reaches a final temperature of $25^{\circ} \mathrm{C}$ precisely where it started. What is the mass of the aluminum chunk?

Averell H.
Carnegie Mellon University

Problem 24

When a driver brakes an automobile, the friction between the brake drums and the brake shoes converts the car's kinetic energy to thermal energy. If a 1500 -kg automobile traveling at 30 $\mathrm{m} / \mathrm{s}$ comes to a halt, how much does the temperature rise in each of the four 8.0 $\mathrm{kg}$ iron brake drums? (The specific heat of iron is 448 $\mathrm{J} / \mathrm{kg} \cdot^{\circ} \mathrm{C}$ )

Salamat A.
Numerade Educator

Problem 25

A Styrofoam cup holds 0.275 $\mathrm{kg}$ of water at $25.0^{\circ} \mathrm{C}$ . Find the final equilibrium temperature after a $0.100-\mathrm{kg}$ block of copper at $90.0^{\circ} \mathrm{C}$ is placed in the water. Neglect any thermal energy transfer with the Styrofoam cup.

Averell H.
Carnegie Mellon University

Problem 26

An unknown substance has a mass of 0.125 $\mathrm{kg}$ and an initial temperature of $95.0^{\circ} \mathrm{C}$ . The substance is then dropped into a calorimeter made of aluminum containing 0.285 $\mathrm{kg}$ of water initially at $25.0^{\circ} \mathrm{C}$ . The mass of the aluminum container is
0.150 $\mathrm{kg}$ , and the temperature of the calorimeter increases to 0.150 $\mathrm{kg}$ , and the temperature of $32.0^{\circ} \mathrm{C}$ . Assuming no thermal energy is transferred to the environment, calculate the specific heat of the unknown substance.

Salamat A.
Numerade Educator

Problem 27

Suppose $9.30 \times 10^{5} \mathrm{J}$ of energy are transferred to 2.00 $\mathrm{kg}$ of
ice at $0^{\circ} \mathrm{C}$ . (a) Calculate the energy required to melt all the ice into liquid water. (b) How much energy remains to raise the temperature of the liquid water? (c) Determine the final temperature of the liquid water in Celsius.

Averell H.
Carnegie Mellon University

Problem 28

How much thermal energy is required to boil 2.00 $\mathrm{kg}$ of water at $100.0^{\circ} \mathrm{C}$ into steam at $125^{\circ} \mathrm{C}$ ? The latent heat of vaporiza- tion of water is $2.26 \times 10^{6} \mathrm{J} / \mathrm{kg}$ and the specific heat of steam is 2010 $\mathrm{J} /\left(\mathrm{kg} \cdot^{\circ} \mathrm{C}\right)$

Salamat A.
Numerade Educator

Problem 29

A 75 -g ice cube at $0^{\circ} \mathrm{C}$ is placed in 825 $\mathrm{g}$ of water at $25^{\circ} \mathrm{C}$ . What is the final temperature of the mixture?

Averell H.
Carnegie Mellon University

Problem 30

A 50 .kg ice cube at $0^{\circ} \mathrm{C}$ is heated until 45 $\mathrm{g}$ has become water
at $100 .^{\circ} \mathrm{C}$ and 5.0 $\mathrm{g}$ has become steam at $100 .^{\circ} \mathrm{C}$ . How much energy was added to accomplish the transformation?

Salamat A.
Numerade Educator

Problem 31

A 100.-g cube of ice at $0^{\circ} \mathrm{C}$ is dropped into $1.0 \mathrm{~kg}$ of water that was originally at $80 .^{\circ} \mathrm{C}$. What is the final temperature of the water after the ice has melted?

Averell H.
Carnegie Mellon University

Problem 32

How much energy is required to change a $40 . \mathrm{g}$ ice cube from ice at $-10 .^{\circ} \mathrm{C}$ to steam at $110 .^{\circ} \mathrm{C}$ ?

Salamat A.
Numerade Educator

Problem 33

A 75 $\mathrm{kg}$ cross-country skier glides over snow as in Figure P11.33. The coefficient of friction between skis and snow is 0.20 . Assume all the snow beneath her skis is at $0^{\circ} \mathrm{C}$
and that all the internal energy generated by friction is added to snow, which sticks to her skis until it melts. How far would she have to ski to melt 1.0 kg of snow?

Averell H.
Carnegie Mellon University

Problem 34

Into a 0.500 -kg aluminum container at $20.0^{\circ} \mathrm{C}$ is placed 6.00 $\mathrm{kg}$ of ethyl alcohol at $30.0^{\circ} \mathrm{C}$ and 1.00 $\mathrm{kg}$ ice at $-10.0^{\circ} \mathrm{C} .$ Assume the system is insulated from its environment. (a) Identify all five thermal energy transfers that occur as the system goes to a final equilibrium temperature $T$ . Use the form "substance at $X^{\circ} \mathrm{C}$ to substance at $Y^{\circ} \mathrm{C}^{\prime \prime}$ (b) Construct a table similar to the table in Example $11.5 .(\mathrm{c})$ Sum all terms in the right-most column of the table and set the
sum equal to zero. (d) Substitute information from the table into the equation found in part (c) and solve for the final equilibrium temperature, $T .$

Check back soon!

Problem 35

A $40 .$ -g block of ice is cooled to $-78^{\circ} \mathrm{C}$ and is then added to 560 $\mathrm{g}$ of water in an 80 -g copper calorimeter at a temperature of $25^{\circ} \mathrm{C}$ . Determine the final temperature of the system consisting of the ice, water, and calorimeter. (If not all the ice melts, determine how much ice is left.) Remember that the ice must first warm to $0^{\circ} \mathrm{C}$ , melt, and then continue warming as water. (The specific heat of ice is $0.500 \mathrm{cal} / \mathrm{g} \cdot^{\circ} \mathrm{C}=2090 \mathrm{J} / \mathrm{kg} \cdot^{\circ} \mathrm{C} .$ .

Averell H.
Carnegie Mellon University

Problem 36

When you jog, most of the food energy you burn above your basal metabolic rate (BMR) ends up as internal energy that would raise your body temperature if it were not eliminated.
The evaporation of perspiration is the primary mechanism for eliminating this energy. Determine the amount of water you lose to evaporation when running for $30 .$ minutes at a rate that
uses $4.00 \times 10^{2} \mathrm{kcal} / \mathrm{h}$ above your BMR. (That amount is often
considered to be the "maximum fat-burning" energy output.) The metabolism of 1.0 grams of fat generates approximately 9.0 kcal of energy and produces approximately 1.0 grams of water.
(The hydrogen atoms in the fat molecule are transferred to oxygen to form water.) What fraction of your need for water will be provided by fat metabolism? (The latent heat of vaporization of water at room temperature is $2.5 \times 10^{6} \mathrm{J} / \mathrm{kg} .$ )

Salamat A.
Numerade Educator

Problem 37

A high-end gas stove usually has at least one burner rated at 14000 $\mathrm{Btu} / \mathrm{h}$ . (a) If you place a 0.25 $\mathrm{kg}$ aluminum pot containing 2.0 liters of water at $20 .^{\circ} \mathrm{C}$ on this burner, how long will it take to bring the water to a boil, assuming all the heat from the burner goes into the pot? (b) Once boiling begins, how much time is required to boil all the water out of the pot?

Averell H.
Carnegie Mellon University

Problem 38

A 60.0 -kg runner expends $3.00 \times 10^{2} \mathrm{W}$ of power while running a marathon. Assuming 10.0$\%$ of the energy is delivered to the muscle tissue and that the excess energy is removed from the body primarily by sweating, determine the volume of bodily fluid (assume it is water) lost per hour. (At $37.0^{\circ} \mathrm{C},$ the latent heat of vaporization of water is $2.41 \times 10^{6} \mathrm{J} / \mathrm{kg.}$ )

Salamat A.
Numerade Educator

Problem 39

Steam at $100 .^{\circ} \mathrm{C}$ is added to ice at $0^{\circ} \mathrm{C}$ . (a) Find the amount of
ice melted and the final temperature when the mass of steam is $10 . \mathrm{g}$ and the mass of ice is $50 . \mathrm{g} .(\mathrm{b})$ Repeat with steam of mass 1.0 $\mathrm{g}$ and ice of mass $50 . \mathrm{g} .$

Check back soon!

Problem 40

The excess internal energy of metabolism is exhausted through a variety of channels, such as through radiation and evaporation of perspiration. Consider another pathway for energy loss: moisture in exhaled breath. Suppose you breathe out 22.0 breaths per minute, each with a volume of 0.600 $\mathrm{L}$ .Suppose also that you inhale dry air and exhale air at $37^{\circ} \mathrm{C}$ containing water vapor with a vapor pressure of 3.20 $\mathrm{kPa}$ . The vapor comes from the evaporation of liquid water in your body. Model the water vapor as an ideal gas. Assume its latent heat of evaporation at $37^{\circ} \mathrm{C}$ is the same as its heat of vaporization at 100 . C. Calculate the rate at which you lose energy by exhaling humid air.

Salamat A.
Numerade Educator

Problem 41

A $3.00-$ g lead bullet at $30.0^{\circ} \mathrm{C}$ is fired at a speed of 2.40 $\times 10^{2} \mathrm{m} / \mathrm{s}$ into a large, fixed block of ice at $0^{\circ} \mathrm{C}$ , in which it
becomes embedded. (a) Describe the energy transformations that occur as the bullet is cooled. What is the final temperature of the bullet? (b) What quantity of ice melts?

Averell H.
Carnegie Mellon University

Problem 42

A glass windowpane in a home is 0.62 $\mathrm{cm}$ thick and has dimensions of 1.0 $\mathrm{m} \times 2.0 \mathrm{m}$ . On a certain day, the indoor temperature is $25^{\circ} \mathrm{C}$ and the outdoor temperature is $0^{\circ} \mathrm{C}$ . (a) What is the rate at which energy is transferred by heat through the glass? (b) How much energy is lost through the window in one day, assuming the temperatures inside and outside remain constant?

Salamat A.
Numerade Educator

Problem 43

A pond with a flat bottom has a surface area of 820 $\mathrm{m}^{2}$ and a depth of 2.0 $\mathrm{m} .$ On a warm day, the surface water is at a temperature of $25^{\circ} \mathrm{C}$ , while the bottom of the pond is at $12^{\circ} \mathrm{C}$ . Find the rate at which energy is transferred by conduction from the surface to the bottom of the pond.

Averell H.
Carnegie Mellon University

Problem 44

The thermal conductivities of human tissues vary greatly. Fat and skin have conductivities of about 0.20 $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ and 0.020 $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ , respectively, while other tissues inside the body have conductivities of about 0.50 $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ Assume that between the core region of the body and the skin surface lies a skin layer of $1.0 \mathrm{mm},$ fat layer of $0.50 \mathrm{cm},$ and 3.2 $\mathrm{cm}$ of other tissues.(a)Find the $R$ -factor for each of these layers, and the equivalent $R$ -factor for all layers taken together, retaining two digits. (b) Find the rate of energy loss when the core temperature is $37^{\circ} \mathrm{C}$ and the exterior temperature is $0^{\circ} \mathrm{C}$ . Assume that both a protective laver of clothing and an insulating layer of unmoving air are absent, and a body area of 2.0 $\mathrm{m}^{2}$

Salamat A.
Numerade Educator

Problem 45

A steam pipe is covered with $1.50-\mathrm{cm}$ -thick insulating material of thermal conductivity 0.200 $\mathrm{cal} / \mathrm{cm} \cdot^{\circ} \mathrm{C} \cdot$ s. How much energy is lost every second when the steam is at $200 .$ ' $\mathrm{Cand~the~}$ surrounding air is at $20.0^{\circ} \mathrm{C}$ ? The pipe has a circumference of 800 . cm and a length of 50.0 $\mathrm{m} .$ Neglect losses through the ends of the pipe.

Averell H.
Carnegie Mellon University

Problem 46

The average thermal conductivity of the walls (including windows) and roof of a house in Figure $\mathrm{P} 11.46$ is $4.8 \times 10^{-4}$ $\mathrm{kW} / \mathrm{m} \cdot^{\circ} \mathrm{C},$ and their average thickness is 21.0 $\mathrm{cm} .$ The house is heated with natural gas, with a heat of combustion (energy released per cubic meter of gas burned) of 9300 $\mathrm{kcal} / \mathrm{m}^{3}$ . How many cubic meters of gas must be burned each day to maintain an inside temperature of $25.0^{\circ} \mathrm{C}$ if the outside temperature is $0.0^{\circ} \mathrm{C}^{2}$ Disregard surface air layers, radiation, and energy loss by heat through the ground.

Salamat A.
Numerade Educator

Problem 47

Consider two cooking pots of the same dimensions, each containing the same amount of water at the same initial temperature. The bottom of the first pot is made of copper, while the bottom of the second pot is made of aluminum. Both pots are placed on a hot surface having a temperature of $145^{\circ} \mathrm{C}$ . The water in the copper-bottomed pot boils away completely in 425 s. How long does it take the water in the aluminum-bottomed pot to boil away completely?

Averell H.
Carnegie Mellon University

Problem 48

A thermopane window consists of two glass panes, each 0.50 $\mathrm{cm}$ thick, with a $1.0-\mathrm{cm}$ -thick sealed layer of air in between. (a) If the inside surface temperature is $23^{\circ} \mathrm{C}$ and the outside surface temperature is $0.0^{\circ} \mathrm{C},$ determine the rate of energy transfer through 1.0 $\mathrm{m}^{2}$ of the window. (b) Compare your answer to $(\mathrm{a})$ with the rate of energy transfer through 1.0 $\mathrm{m}^{2}$ of a single $1.0-\mathrm{cm}$ -thick pane of glass. Disregard surface air layers.

Salamat A.
Numerade Educator

Problem 49

A copper rod and an aluminum rod of equal diameter are joined end to end in good thermal contact. The temperature of the free end of the copper rod is held constant at $100 .^{\circ} \mathrm{C}$ and that of the far end of the aluminum rod is held at $0^{\circ} \mathrm{C}$ . If the copper rod is 0.15 $\mathrm{m}$ long, what must be the length of the aluminum rod so that the temperature at the junction is $50 .^{\circ} \mathrm{C}$ ?

Averell H.
Carnegie Mellon University

Problem 50

A Styrofoam box has a surface area of 0.80 $\mathrm{m}^{2}$ and a wall thickness of 2.0 $\mathrm{cm} .$ The temperature of the inner surface is $5.0^{\circ} \mathrm{C},$ and the outside temperature is $25^{\circ} \mathrm{C}$ . If it takes 8.0 $\mathrm{h}$ for 5.0 $\mathrm{kg}$ of ice to melt in the container, determine the thermal conductivity of the Styrofoam.

Salamat A.
Numerade Educator

Problem 51

A rectangular glass window pane on a house has a width of $1.0 \mathrm{m},$ a height of $2.0 \mathrm{m},$ and a thickness of 0.40 $\mathrm{cm} .$ Find the energy transferred through the window by conduction in 12 hours on a day when the inside temperature of the house is $22^{\circ} \mathrm{C}$ and the outside temperature is $2.0^{\circ} \mathrm{C}$ Take surface air layers into consideration.

Averell H.
Carnegie Mellon University

Problem 52

A granite ball of radius 2.00 $\mathrm{m}$ and emissivity 0.450 is heated to $135^{\circ} \mathrm{C}$ . (a) Convert the given temperature to Kelvin. (b) What is the surface area of the ball? (c) If the ambient temperature is $25.0^{\circ} \mathrm{C},$ what net power does the ball radiate?

Salamat A.
Numerade Educator

Problem 53

Measurements on two stars indicate that Star $\mathrm{X}$ has a surface temperature of $5727^{\circ} \mathrm{C}$ and Star $\mathrm{Y}$ has a surface temperature of $11727^{\circ} \mathrm{C}$ . If both stars have the same radius, what is the ratio of the luminosity (total power output) of Star Y to the luminosity of Star $\mathrm{X}$ . Both stars can be considered to have an emissivity of $1.0 .$

Averell H.
Carnegie Mellon University

Problem 54

The filament of a $75-\mathrm{W}$ light bulb is at a temperature of 3300 $\mathrm{K}$ . Assuming the filament has an emissivity $e=1.0,$ find its surface area.

Salamat A.
Numerade Educator

Problem 55

The bottom of a copper kettle has a $10.0-\mathrm{cm}$ radius and is 2.00 $\mathrm{mm}$ thick. The temperature of the outside surface is $102^{\circ} \mathrm{C},$ and the water inside the kettle is boiling at 1 $\mathrm{atm}$ of pressure. Find the rate at which energy is being transferred through the bottom of the kettle.

Averell H.
Carnegie Mellon University

Problem 56

A family comes home from a long vacation with laundry to do and showers to take. The water heater has been turned off during the vacation. If the heater has a capacity of 50.0 gallons and a $4800-\mathrm{W}$ heating element, how much time is required to raise the temperature of the water from $20.0^{\circ} \mathrm{C}$ to $60.0^{\circ} \mathrm{C}$ ? Assume the heater is well insulated and no water is withdrawn from the tank during that time.

Salamat A.
Numerade Educator

Problem 57

A $0.040 . \mathrm{kg}$ ice cube floats in 0.200 $\mathrm{kg}$ of water in a 0.100 -kg copper cup; all are at a temperature of $0^{\circ} \mathrm{C}$ . A piece of lead at $98^{\circ} \mathrm{C}$ is dropped into the cup, and the final equilibrium temperature is $12^{\circ} \mathrm{C}$ . What is the mass of the lead?

Averell H.
Carnegie Mellon University

Problem 58

The surface area of an unclothed person is 1.50 $\mathrm{m}^{2}$ and his skin temperature is $33.0^{\circ} \mathrm{C}$ . The person is located in a dark room with a temperature of $20.0^{\circ} \mathrm{C},$ and the emissivity of the skin is $e=0.95 .(\mathrm{a})$ At what rate is energy radiated by the body? (b) What is the significance of the sign of your answer?

Salamat A.
Numerade Educator

Problem 59

A student measures the following data in a calorimetry experiment designed to determine the specific heat of aluminum:
$$
\begin{array}{l}{\text { Initial temperature of water }} \\ {\begin{array}{ll}{\text { and calorimeter: }} & {70.0^{\circ} \mathrm{C}} \\ {\text { Mass of water: }} & {0.400 \mathrm{kg}} \\ {\text { Mass of calorimeter: }} & {0.040 \mathrm{kg}} \\ {\text { Specific heat of calorimeter: }} & {0.63 \mathrm{kJ} / \mathrm{kg} \cdot \circ \mathrm{C}} \\ {\text { Initial temperature of aluminum: }} & {27.0^{\circ} \mathrm{C}} \\ {\text { Mass of aluminum: }} & {0.200 \mathrm{kg}} \\ {\text { Final temperature of mixture: }} & {66.3^{\circ} \mathrm{C}}\end{array}}\end{array}
$$
Use these data to determine the specific heat of aluminum. Explain whether your result is within 15$\%$ of the value listed in Table $11.1 .$

Averell H.
Carnegie Mellon University

Problem 60

Overall, 80$\%$ of the energy used by the body must be eliminated as excess thermal energy and needs to be dissipated. The mechanisms of elimination are radiation, evaporation of sweat $(2430 \mathrm{kJ} / \mathrm{kg}),$ evaporation from the lungs $(38 \mathrm{kJ} / \mathrm{h}),$ conduction, and convection.
A person working out in a gym has a metabolic rate of 2500 $\mathrm{kJ} / \mathrm{h}$ . His body temperature is $37^{\circ} \mathrm{C},$ and the outside temperature $24^{\circ} \mathrm{C}$ . Assume the skin has an area of 2.0 $\mathrm{m}^{2}$ and emissivity of 0.97 . (a) At what rate is his excess thermal energy dissipated by radiation? (b) If he eliminates 0.40 $\mathrm{kg}$ of
perspiration during that hour, at what rate is thermal energydissipated by evaporation of sweat? (c) At what rate is energy eliminated by evaporation from the lungs? (d) At what rate must the remaining excess energy be eliminated through conduction and convection?

Salamat A.
Numerade Educator

Problem 61

Liquid helium has a very low boiling point, $4.2 \mathrm{K},$ as well as a very low latent heat of vaporization, $2.00 \times 10^{4} \mathrm{J} / \mathrm{kg}$ . If energy is transferred to a container of liquid helium at the boiling point from an immersed electric heater at a rate of $10.0 \mathrm{W},$
how long does it take to boil away 2.00 $\mathrm{kg}$ of the liquid?

Averell H.
Carnegie Mellon University

Problem 62

A class of 10 students taking an exam has a power output per student of about 200 $\mathrm{W}$ . Assume the initial temperature of the room is $20^{\circ} \mathrm{C}$ and that its dimensions are 6.0 $\mathrm{m}$ by 15.0 $\mathrm{m}$ by 3.0 $\mathrm{m}$ . What is the temperature of the room at the end of 1.0 $\mathrm{h}$ if all the energy remains in the air in the room and none is added by an outside source? The specific heat of air is $837 \mathrm{J} / \mathrm{kg} \cdot^{\circ} \mathrm{C},$ and its density is about $1.3 \times 10^{-3} \mathrm{g} / \mathrm{cm}^{3}$

Salamat A.
Numerade Educator

Problem 63

A bar of gold (Au) is in thermal contact with a bar of silver (Ag) of the same length and area (Fig.
P11.63). One end of the compound bar is maintained at $80.0^{\circ} \mathrm{C}$ and the opposite end is at $30.0^{\circ} \mathrm{C}$ and the opposite end is at $30.0^{\circ} \mathrm{C}$ Find the temperature at the junction when the energy flow reaches a steady state.

Averell H.
Carnegie Mellon University

Problem 64

An iron plate is held against an iron wheel so that a sliding frictional force of $50 . \mathrm{N}$ acts between the two pieces of metal. The relative speed at which the two surfaces slide over each other is
$40 . \mathrm{m} / \mathrm{s}$ , (a) Calculate the rate at which mechanical energy is converted to internal energy. (b) The plate and the wheel have masses of 5.0 $\mathrm{kg}$ each, and each receives 50$\%$ of the internal energy. If the system is run as described for $10 . \mathrm{s}$ and each object is then allowed to reach a uniform internal temperature, what is the resultant temperature increase?

Salamat A.
Numerade Educator

Problem 65

An automobile has a mass of $1500 \mathrm{kg},$ and its aluminum brakes have an overall mass of 6.00 $\mathrm{kg}$ . (a) Assuming all the internal energy transformed by friction when the car stops is deposited in the brakes and neglecting energy transfer, how many times could the car be braked to rest starting from 25.0 $\mathrm{m} / \mathrm{s}$ before the brakes would begin to melt? (Assume an initial temperature of $20.0^{\circ} \mathrm{C}$ ) (b) Identify some effects that are neglected in
part (a) but are likely to be important in a more realistic assessment of the temperature increase of the brakes.

Averell H.
Carnegie Mellon University

Problem 66

Three liquids are at temperatures of $10^{\circ} \mathrm{C}, 20^{\circ} \mathrm{C},$ and $30^{\circ} \mathrm{C},$ respectively. Equal masses of the first two liquids are mixed, and the equilibrium temperature is $17^{\circ} \mathrm{C}$ . Equal masses of the second and third are then mixed, and the equilibrium temperature is $28^{\circ} \mathrm{C}$ . Find the equilibrium temperature when equal masses of the first and third are mixed.

Salamat A.
Numerade Educator

Problem 67

Earth's surface absorbs an average of about $960 . \mathrm{W} / \mathrm{m}^{2}$ from the Sun's irradiance. The power absorbed is $P_{\mathrm{abs}}=\left(960 . \mathrm{W} / \mathrm{m}^{2}\right)$ $\left(A_{\mathrm{disc}}\right),$ where $A_{\mathrm{disc}}=\pi R_{E}^{2}$ is Earth's projected area. An equal amount of power is radiated so that Earth remains in thermal equilibrium with its environment at nearly 0 K. Estimate Earth's surface temperature by setting the radiated power from Stefan's law equal to the absorbed power and solving for the temperature in Kelvin. In Stefan's law, assume $e=1$ and take the area to be $A=4 \pi R_{E}^{2}$ , the surface area of a spherical Earth. (Note: Earth's atmosphere acts like a blanket and warms the planet to a global average about 30 $\mathrm{K}$ above the value calculated here.)

Averell H.
Carnegie Mellon University

Problem 68

A wood stove is used to heat a single room. The stove is cylindrical in shape, with a diameter of 40.0 $\mathrm{cm}$ and a length of $50.0 \mathrm{cm},$ and operates at a temperature of $400 .^{\circ} \mathrm{F}$ . (a) If the temperature of the room is $70.0^{\circ} \mathrm{F}$ , determine the amount of radiant energy delivered to the room by the stove each second if the emissivity is $0.920 .(\mathrm{b})$ If the room is a square with walls that are 8.00 $\mathrm{ft}$ high and 25.0 $\mathrm{ft}$ wide, determine the $R$ -value needed in the walls and ceiling to maintain the inside temperature at $70.0^{\circ} \mathrm{F}$ if the outside temperature is $32.0^{\circ} \mathrm{F}$ . Note that we are ignoring any heat conveyed by the stove via convection and any energy lost through the walls (and windows!) via convection or radiation.

Salamat A.
Numerade Educator

Problem 69

A "solar cooker" consists of a curved reflecting mirror that focuses sunlight onto the object to be heated (Fig. $\quad \mathrm{P} 11.69 )$ . The solar power per unit area reaching the Earth at the location of a 0.50 -m-diameter solar cooker is $600 . \mathrm{W} / \mathrm{m}^{2}$ Assuming 50$\%$ of the inci-
dent energy is converted to thermal energy, how long would it take to boil away 1.0 L of water initially at $20 .^{\circ} \mathrm{C}$ ? (Neglect the specific heat of the container.)

Averell H.
Carnegie Mellon University

Problem 70

For bacteriological testing of water supplies and in medical clinics, samples must routinely be incubated for 24 $\mathrm{h}$ at $37^{\circ} \mathrm{C}$ . A standard constant-temperature bath with electric heating and thermostatic control is not suitable in developing nations without continuously operating electric power lines. Peace Corps volunteer and MIT engineer Amy Smith invented a low-cost, low-maintenance incubator to fill the need. The device consists of a foam-insulated box containing several packets of a waxy material that melts at $37.0^{\circ} \mathrm{C}$ , interspersed among tubes, dishes, or bottles containing the test samples and growth
medium (food for bacteria). Outside the box, the waxy material is first melted by a stove or solar energy collector. Then it is put into the box to keep the test samples warm as it solidifies.
The heat of fusion of the phase-change material is 205 $\mathrm{kJ} / \mathrm{kg}$ .
Model the insulation as a panel with surface area 0.490 $\mathrm{m}^{2}$ thickness $9.50 \mathrm{cm},$ and conductivity 0.0120 $\mathrm{W} / \mathrm{m} \cdot^{\circ}$ . Assume the exterior temperature is $23.0^{\circ} \mathrm{C}$ for 12.0 $\mathrm{h}$ and $16.0^{\circ} \mathrm{C}$ for 12.0 $\mathrm{h}$ . (a) What mass of the waxy material is required to con-
duct the bacteriological test? (b) Explain why your calculation can be done without knowing the mass of the test samples or of the insulation.

Salamat A.
Numerade Educator

Problem 71

The surface of the Sun has a temperature of about 5800 $\mathrm{K}$ . The radius of the Sun is $6.96 \times 10^{8} \mathrm{m} .$ Calculate the total energy radiated by the Sun each second. Assume the emissivity of the Sun is 0.986 .

Averell H.
Carnegie Mellon University

Problem 72

The evaporation of perspiration is the primary mechanism for cooling the human body. Estimate the amount of water you will lose when you bake in the sun on the beach for an hour. Use a value of 1000 $\mathrm{W} / \mathrm{m}^{2}$ for the intensity of sun- light and note that the energy required to evaporate a liquid at a particular temperature is approximately equal to the sum point and the latent heat of vaporization (determined at the boiling point).of the energy required to raise its temperature to the boiling

Salamat A.
Numerade Educator

Problem 73

At time $t=0,$ a vessel contains a mixture of $10 . \mathrm{kg}$ of water and an unknown mass of ice in equilibrium at $0^{\circ} \mathrm{C}$ . The temperature of the mixture is measured over a period of an hour, with the following results: During the first $50 .$ min, the mixture remains at $0^{\circ} \mathrm{C} ;$ from $50 .$ min to $60 .$ min, the temperature increases steadily from $0^{\circ} \mathrm{C}$ to $2.0^{\circ} \mathrm{C}$ . Neglecting the heat capacity of the vessel, determine the mass of ice that was initially placed in it. Assume a constant power input to the container.

Averell H.
Carnegie Mellon University

Problem 74

An ice-cube tray is filled with 75.0 $\mathrm{g}$ of water. After the filled tray reaches an equilibrium temperature $20.0^{\circ} \mathrm{C},$ it is placed in a freezer set at $-8.00^{\circ} \mathrm{C}$ to make ice cubes. (a) Describe the processes that occur as energy is being removed from the water to make ice. (b) Calculate the energy that must be removed from the water to make ice cubes at $-8.00^{\circ} \mathrm{C}$ .

Salamat A.
Numerade Educator

Problem 75

An aluminum rod and an iron rod are joined end to end in good thermal contact. The two rods have equal lengths and radii. The free end of the aluminum rod is maintained at a temperature of $100 .^{\circ} \mathrm{C},$ and the free end of the iron rod is maintained at $0^{\circ} \mathrm{C}$ . (a) Determine the temperature of the maintace between the two rods. (b) If each rod is 15 $\mathrm{cm}$ long and each has a cross-sectional area of $5.0 \mathrm{cm}^{2},$ what quantity of energy is conducted across the combination in $30 .$ min?

Averell H.
Carnegie Mellon University