Physics for Scientists and Engineers with Modern Physics

Paul Tipler, Gene Mosca

Chapter 20

Heat and the First Law of Thermodynamics - all with Video Answers

Educators

+ 4 more educators

Chapter Questions

Problem 1

On his honeymoon James Joule traveled from England to Switzerland. He attempted to verify his idea of the interconvertibility of mechanical energy and internal energy by measuring the increase in temperature of water that fell in a waterfall. If water at the top of an alpine waterfall has a temperature of 10.0°C and then falls 50.0 m (as at Niagara
1. Falls), what maximum temperature at the bottom of the falls could Joule expect? He did not succeed in measuring the temperature change, partly because evaporation cooled the falling water, and also because his thermometer was not sufficiently sensitive.

Shahab Ullah

Shahab Ullah

Numerade Educator

Problem 2

Consider Joule’s apparatus described in Figure 20.1. The mass of each of the two blocks is 1.50 kg, and the insulated tank is filled with 200 g of water. What is the increase in the temperature of the water after the blocks fall through a distance of 3.00 m?

Shoukat Ali

Other Schools

Problem 3

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.

John Palmer

Numerade Educator

Problem 4

A 50.0 -g sample of copper is at $25.0^{\circ} \mathrm{C}$ . If 1200 $\mathrm{J}$ of energy is added to it by heat, what is the final temperature of the copper?

Shoukat Ali

Other Schools

Problem 5

Systematic use of solar energy can yield a large saving in the cost of winter space heating for a typical house in the north central United States. If the house has good insulation, you may model it as losing energy by heat steadily at the rate 6 000 W on a day in April when the average exterior temperature is $4^{\circ} \mathrm{C}$ , and when the conventional heating system is not used at all. The passive solar energy collector can consist simply of very large windows in a room facing south. Sunlight shining in during the daytime is absorbed by the floor, interior walls, and objects in the room, raising their temperature to $38^{\circ} \mathrm{C} .$ As the sun goes down, insulating draperies or shutters are closed over the windows. During the period between $5 : 00$ P.M. and $7 : 00$ A.M. the temperature of the house will drop, and a sufficiently large "thermal mass" is required to keep it from dropping too far. The thermal mass can be a large quantity of stone (with specific heat 850 $\mathrm{J} / \mathrm{kg} \cdot^{\circ} \mathrm{C} )$ in the floor and the interior walls exposed to sunlight. What mass of stone is required if the temperature is not to drop below $18^{\circ} \mathrm{C}$ overnight?

Shahab Ullah

Numerade Educator

Problem 6

The Nova laser at Lawrence Livermore National Laboratory in California is used in studies of initiating controlled nuclear fusion (Section 45.4$)$ . It can deliver a power of $1.60 \times 10^{13} \mathrm{W}$ over a time interval of 2.50 $\mathrm{ns}$ . Compare its energy output in one such time interval to the energy required to make a pot of tea by warming 0.800 $\mathrm{kg}$ of water from $20.0^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$ .

Shahab Ullah

Numerade Educator

Problem 7

A 1.50 -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? (Ignore the heat capacity of the container, and assume that a negligible amount of water boils away.)

Shahab Ullah

Numerade Educator

Problem 8

An aluminum cup of mass 200 $\mathrm{g}$ contains 800 $\mathrm{g}$ of water in thermal equilibrium at $80.0^{\circ} \mathrm{C}$ . The combination of cup and water is cooled uniformly so that the temperature decreases by $1.50^{\circ} \mathrm{C}$ per minute. At what rate is energy being removed by heat? Express your answer in watts.

Banhishikha Sinha

Banhishikha Sinha

Numerade Educator

Problem 9

An aluminum calorimeter with a mass of 100 g contains 250 g 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 block 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) Guess the material of the un- known, using the data in Table $20.1 .$

Emily Anderson

Numerade Educator

Problem 10

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

Shahab Ullah

Numerade Educator

Problem 11

A combination of 0.250 $\mathrm{kg}$ of water at $20.0^{\circ} \mathrm{C}, 0.400 \mathrm{kg}$ of
aluminum at $26.0^{\circ} \mathrm{C},$ and 0.100 $\mathrm{kg}$ of copper at $100^{\circ} \mathrm{C}$ is mixed in an insulated container and allowed to come to thermal equilibrium. Ignore any energy transfer to or from the container and determine the final temperature of the mixture.

John Palmer

Numerade Educator

Problem 12

If water with a mass $m_{h}$ at temperature $T_{h}$ is poured into an aluminum cup of mass $m_{A 1}$ containing mass $m_{c}$ of water at $T_{c},$ where $T_{h}>T_{c},$ what is the equilibrium temperature of the system?

Shahab Ullah

Numerade Educator

Problem 13

A water heater is operated by solar power. If the solar collector has an area of 6.00 $\mathrm{m}^{2}$ and the intensity delivered by sunlight is $550 \mathrm{W} / \mathrm{m}^{2},$ how long does it take to increase the temperature of 1.00 $\mathrm{m}^{3}$ of water from $20.0^{\circ} \mathrm{C}$ to $60.0^{\circ} \mathrm{C} ?$

Shahab Ullah

Numerade Educator

Problem 14

Two thermally insulated vessels are connected by a narrow tube fitted with a valve that is initially closed. One vessel, of volume 16.8 $\mathrm{L}$ , contains oxygen at a temperature of volume and a pressure of 1.75 $\mathrm{atm}$ . The other vessel, of volume 22.4 $\mathrm{L}$ , contains oxygen at a temperature of 450 $\mathrm{K}$ and a pressure of 2.25 $\mathrm{atm}$ . When the valve is opened, the gases in the two vessels mix, and the temperature and pressure become uniform throughout. (a) What is the final temperature? (b) What is the final pressure?

Emily Anderson

Numerade Educator

Problem 15

How much energy is required to change a $40.0-\mathrm{g}$ ice cube from ice at $-10.0^{\circ} \mathrm{C}$ to steam at $110^{\circ} \mathrm{C} ?$

Vipender Yadav

Numerade Educator

Problem 16

A 50.0 -g copper calorimeter contains 250 $\mathrm{g}$ of water at $20.0^{\circ} \mathrm{C} .$ How much steam must be condensed into the water if the final temperature of the system is to reach $50.0^{\circ} \mathrm{C} ?$

Rob Ball

Numerade Educator

Problem 17

A 3.00 -g lead bullet at $30.0^{\circ} \mathrm{C}$ is fired at a speed of 240 $\mathrm{m} / \mathrm{s}$ into a large block of ice at $0^{\circ} \mathrm{C},$ in which it becomes em- bedded. What quantity of ice melts?

Shahab Ullah

Numerade Educator

Problem 18

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 10.0 $\mathrm{g}$ and the mass of ice is 50.0 $\mathrm{g}$ . (b) What If? Repeat when the mass of steam is 1.00 $\mathrm{g}$ and the mass of ice is 50.0 $\mathrm{g}$ .

John Palmer

Numerade Educator

Problem 19

A $1.00-\mathrm{kg}$ block of copper at $20.0^{\circ} \mathrm{C}$ is dropped into a large vessel of liquid nitrogen at 77.3 $\mathrm{K}$ . How many kilograms of nitrogen boil away by the time the copper reaches 77.3 $\mathrm{K}$ ? (The specific heat of copper is 0.0920 $\mathrm{cal} / \mathrm{g} \cdot^{\circ} \mathrm{C}$ . The latent heat of vaporization of nitrogen is $48.0 \mathrm{cal} / \mathrm{g} . )$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 20

Assume that a hailstone at $0^{\circ} \mathrm{C}$ falls through air at a uniform temperature of $0^{\circ} \mathrm{C}$ and lands on a sidewalk also at this temperature. From what initial height must the hail- stone fall in order to entirely melt on impact?

Shahab Ullah

Numerade Educator

Problem 21

In an insulated vessel, 250 $\mathrm{g}$ of ice at $0^{\circ} \mathrm{C}$ is added to 600 $\mathrm{g}$ of water at $18.0^{\circ} \mathrm{C}$ . (a) What is the final temperature of the system? (b) How much ice remains when the system reaches equilibrium?

Rob Ball

Numerade Educator

Problem 22

Review problem. Two speeding lead bullets, each of mass $5.00 \mathrm{g},$ and at temperature $20.0^{\circ} \mathrm{C},$ collide head-on at speeds of 500 $\mathrm{m} / \mathrm{s}$ each. Assuming a perfectly inelastic collision and no loss of energy by heat to the atmosphere, describe the final state of the two-bullet system.

Emily Anderson

Numerade Educator

Problem 23

A sample of ideal gas is expanded to twice its original volume of 1.00 $\mathrm{m}^{3}$ in a quasi-static process for which $P=\alpha V^{2},$ with $\alpha=5.00 \mathrm{atm} / \mathrm{m}^{6},$ as shown in Figure $\mathrm{P} 20.23$ . How much work is done on the expanding gas?

John Palmer

Numerade Educator

Problem 24

(a) Determine the work done on a fluid that expands from i to $f$ as indicated in Figure $\mathrm{P} 20.24$ . (b) What If? How much work is performed on the fluid if it is compressed from $f$ to $i$ along the same path?

Vishal Gupta

Numerade Educator

Problem 25

An ideal gas is enclosed in a cylinder with a movable piston on top of it. The piston has a mass of 8000 $\mathrm{g}$ and an area of 5.00 $\mathrm{cm}^{2}$ and is free to slide up and down, keeping the pressure of the gas constant. How much work is done on the gas as the temperature of 0.200 mol of the gas is raised from $20.0^{\circ} \mathrm{C}$ to $300^{\circ} \mathrm{C}$ ?

Vipender Yadav

Numerade Educator

Problem 26

An ideal gas is enclosed in a cylinder that has a movable piston on top. The piston has a mass $m$ and an area $A$ and is free to slide up and down, keeping the pressure of the gas constant. How much work is done on the gas as the temperature of $n$ mol of the gas is raised from $T_{1}$ to $T_{2}$ ?

Vipender Yadav

Numerade Educator

Problem 27

One mole of an ideal gas is heated slowly so that it goes from the $P V$ state $\left(P_{i}, V_{i}\right)$ to $\left(3 P_{i}, 3 V_{i}\right)$ in such a way that the pressure is directly proportional to the volume. (a) How much work is done on the gas in the process? (b) How is the temperature of the gas related to its volume during this process?

Shahab Ullah

Numerade Educator

Problem 28

An ideal gas under goes a process with the following parameters: $Q=10.0 \mathrm{~J}$, $W=12.0 \mathrm{~J},$ and $\Delta T=-2.00^{\circ} \mathrm{C}$ .

Shahab Ullah

Numerade Educator

Problem 29

A thermodynamic system undergoes a process in which its internal energy decreases by 500 $\mathrm{J}$ . At the same time, 220 $\mathrm{J}$ of work is done on the system. Find the energy transferred to or from it by heat.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 30

A gas is taken through the cyclic process described in Figure $\mathrm{P} 20.30 .$ ( a) Find the net energy transferred to the system by heat during one complete cycle. (b) What If? If the cycle is reversed - that is, the process follows the path $A C B A-$ what is the net energy input per cycle by heat?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 31

Consider the cyclic process depicted in Figure $\mathrm{P} 20.30 .$ If $Q$ is negative for the process $B C$ and $\Delta E_{\text { in }}$ is negative for the process $C A,$ what are the signs of $Q, W,$ and $\Delta E_{\text { int }}$ that are associated with each process?

John Palmer

Numerade Educator

Problem 32

A sample of an ideal gas goes through the process shown in Figure $\mathrm{P} 20.32$ . From $A$ to $B,$ the process is adiabatic; from $B$ to $C,$ it is isobaric with 100 $\mathrm{kJ}$ of energy entering the system by heat. From $C$ to $D,$ the process is isothermal; from $D$ to $A,$ it is isobaric with 150 $\mathrm{kJ}$ of energy leaving the system by heat. Determine the difference in internal energy $E_{\mathrm{int}, B}-E_{\mathrm{int}, A}$

Narayan Hari

Numerade Educator

Problem 33

A sample of an ideal gas is in a vertical cylinder fitted with a piston. As 5.79 $\mathrm{kJ}$ of energy is transferred to the gas by heat to raise its temperature, the weight on the piston is adjusted so that the state of the gas changes from point $A$ to point $B$ along the semicircle shown in Figure $\mathrm{P} 20.33$ . Find the change in internal energy of the gas.

Emily Anderson

Numerade Educator

Problem 34

One mole of an ideal gas does 3000 $\mathrm{J}$ of work on its surroundings as it expands isothermally to a final pressure of 1.00 $\mathrm{atm}$ and volume of 25.0 $\mathrm{L}$ . Determine (a) the initial volume and (b) the temperature of the gas.

Vipender Yadav

Numerade Educator

Problem 35

An ideal gas initially at 300 $\mathrm{K}$ undergoes an isobaric expansion at 2.50 $\mathrm{kPa}$ . If the volume increases from 1.00 $\mathrm{m}^{3}$ to 3.00 $\mathrm{m}^{3}$ and 12.5 $\mathrm{kJ}$ is transferred to the gas by heat, what are (a) the change in its internal energy and (b) its final temperature?

Vipender Yadav

Numerade Educator

Problem 36

A $1.00-\mathrm{kg}$ block of aluminum is heated at atmospheric pressure so that its temperature increases from $22.0^{\circ} \mathrm{C}$ to $40.0^{\circ} \mathrm{C}$ . Find $(\mathrm{a})$ the work done on the aluminum, (b) the energy added to it by heat, and (c) the change in its internal energy.

Shahab Ullah

Numerade Educator

Problem 37

How much work is done on the steam when 1.00 mol of water at $100^{\circ} \mathrm{C}$ boils and becomes 1.00 $\mathrm{mol}$ of steam at $100^{\circ} \mathrm{C}$ at 1.00 atm pressure? Assuming the steam to behave as an ideal gas, determine the change in internal energy of the material as it vaporizes.

Shahab Ullah

Numerade Educator

Problem 38

An ideal gas initially at $P_{i}, V_{i},$ and $T_{i}$ is taken through a cycle as in Figure $\mathrm{P} 20.38$ (a) Find the net work done on the gas per cycle. (b) What is the net energy added by heat to the system per cycle? (c) Obtain a numerical value for the net work done per cycle for 1.00 mol of gas initially at $0^{\circ} \mathrm{C}$ .

John Palmer

Numerade Educator

Problem 39

A 2.00 -mol sample of helium gas initially at 300 $\mathrm{K}$ and 0.400 atm is compressed isothermally to 1.20 atm. Noting that the helium behaves as an ideal gas, find (a) the final volume of the gas, (b) the work done on the gas, and (c) the energy transferred by heat.

Vipender Yadav

Numerade Educator

Problem 40

In Figure $\mathrm{P} 20.40$ , the change in internal energy of a gas that is taken from $A$ to $C$ is $+800 \mathrm{J}$ . The work done on the gas along path $A B C$ is $-500 \mathrm{J}$ . (a) How much energy must be added to the system by heat as it goes from $A$ through $B$ to $C ?$ (b) If the pressure at point $A$ is five times that of point $C,$ what is the work done on the system in going from $C$ to $D P$ (c) What is the energy exchanged with the surroundings by heat as the cycle goes from $C$ to $A$ along the green path? (d) If the change in internal energy in going from point $D$ to point $A$ is $+500 \mathrm{J}$ , how much energy must be added to the system by heat as it goes from point $C$ to point $D ?$

Emily Anderson

Numerade Educator

Problem 41

A box with a total surface area of 1.20 $\mathrm{m}^{2}$ and a wall thickness of 4.00 $\mathrm{cm}$ is made of an insulating material. A $10.0-\mathrm{W}$ electric heater inside the box maintains the inside temperature at $15.0^{\circ} \mathrm{C}$ above the outside temperature. Find the thermal conductivity $k$ of the insulating material.

Shahab Ullah

Numerade Educator

Problem 42

A glass window pane has an area of 3.00 $\mathrm{m}^{2}$ and a thickness of $0.600 \mathrm{cm} .$ If the temperature difference between its faces is $25.0^{\circ} \mathrm{C},$ what is the rate of energy transfer by conduction through the window?

Shahab Ullah

Numerade Educator

Problem 43

A bar of gold is in thermal contact with a bar of silver of the same length and area (Fig. P20.43). One end of the compound bar is maintained at $80.0^{\circ} \mathrm{C}$ while the opposite end is at $30.0^{\circ} \mathrm{C}$ . When the energy transfer reaches steady state, what is the temperature at the junction

Vipender Yadav

Numerade Educator

Problem 44

A thermal window with an area of 6.00 $\mathrm{m}^{2}$ is constructed of two layers of glass, each 4.00 $\mathrm{mm}$ thick, and separated from each other by an air space of $5.00 \mathrm{mm} .$ If the inside surface is at $20.0^{\circ} \mathrm{C}$ and the outside is at $-30.0^{\circ} \mathrm{C},$ what is the rate of energy transfer by conduction through the window?

Shahab Ullah

Numerade Educator

Problem 45

A power transistor is a solid-state electronic device. Assume that energy entering the device at the rate of 1.50 $\mathrm{W}$ by electrical transmission causes the internal energy of the device to increase. The surface area of the transistor is so small that it tends to overheat. To prevent overheating, the transistor is attached to a larger metal heat sink with fins. The temperature of the heat sink remains constant at $35.0^{\circ} \mathrm{C}$ under steady-state conditions. The transistor is electrically insulated from the heat sink by a rectangular sheet of mica measuring 8.25 $\mathrm{mm}$ by $6.25 \mathrm{mm},$ and 0.0852 $\mathrm{mm}$ thick. The thermal conductivity of mica is equal to $0.0753 \mathrm{W} / \mathrm{m} \cdot^{\circ} \mathrm{C} .$ What is the operating temperature of the transistor?

Shahab Ullah

Numerade Educator

Problem 46

Calculate the $R$ value of (a) a window made of a single pane of flat glass $\frac{1}{8}$ in. thick, and $(b)$ a thermal window made of two single panes each $\frac{1}{8}$ in. thick and separated by a $\frac{1}{4}$ -in. air space. (c) By what factor is the transfer of energy by heat through the window reduced by using the thermal window instead of the single pane window?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 47

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 that the emissivity of the Sun is $0.965 .$

Shahab Ullah

Numerade Educator

Problem 48

A large hot pizza floats in outer space. What is the order of magnitude of (a) its rate of energy loss? (b) its rate of temperature change? List the quantities you estimate and the value you estimate for each.

Ajay Singhal

Numerade Educator

Problem 49

The tungsten filament of a certain $100-\mathrm{W}$ light bulb radiates 2.00 $\mathrm{W}$ of light. (The other 98 $\mathrm{W}$ is carried away by convection and conduction.) The filament has a surface area of 0.250 $\mathrm{mm}^{2}$ and an emissivity of 0.950 . Find the filament's temperature. (The melting point of tungsten is $3683 \mathrm{K} . )$

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 50

At high noon, the Sun delivers 1000 $\mathrm{W}$ to each square meter of a blacktop road. If the hot asphalt loses energy only by radiation, what is its equilibrium temperature?

Shahab Ullah

Numerade Educator

Problem 51

The intensity of solar radiation reaching the top of the Earth's atmosphere is $1340 \mathrm{W} / \mathrm{m}^{2} .$ The temperature of the Earth is affected by the so-called greenhouse effect of the atmosphere. That effect makes our planet's emissivity for visible light higher than its emissivity for infrared light. For comparison, consider a spherical object with no atmosphere, at the same distance from the Sun as the Earth. Assume that its emissivity is the same for all kinds of electromagnetic waves and that its temperature is uniform over its surface. Identify the projected area over which it absorbs sunlight and the surface area over which it radiates. Compute its equilibrium temperature. Chilly, isn't it? Your calculation applies to (a) the average temperature of

Jeff Vermeire

Numerade Educator

Problem 52

Liquid nitrogen with a mass of 100 $\mathrm{g}$ at 77.3 $\mathrm{K}$ is stirred into a beaker containing 200 $\mathrm{g}$ of $5.00^{\circ} \mathrm{C}$ water. If the nitrogen leaves the solution as soon as it turns to gas, how much water freezes? (The latent heat of vaporization of nitrogen is $48.0 \mathrm{cal} / \mathrm{g},$ and the latent heat of fusion of water is $79.6 \mathrm{cal} / \mathrm{g} . )$

Prashant Bana

Numerade Educator

Problem 53

A 75.0 -kg cross-country skier moves across the snow (Fig. $\mathrm{P} 20.53$ ). The coefficient of friction between the skis and the snow is 0.200 . Assume that all the snow beneath his skis is at $0^{\circ} \mathrm{C}$ and that all the internal energy generated by friction is added to the snow, which sticks to his skis until it melts. How far would he have to ski to melt 1.00 $\mathrm{kg}$ of snow?

Shahab Ullah

Numerade Educator

Problem 54

On a cold winter day you buy roasted chestnuts from a street vendor. Into the pocket of your down parka you put the change he gives you- coins constituting 9.00 $\mathrm{g}$ of copper at $-12.0^{\circ} \mathrm{C}$ . Your pocket already contains 14.0 $\mathrm{g}$ of sitver coins at $30.0^{\circ} \mathrm{C}$ . A short time later the temperature of the copper coins is $4.00^{\circ} \mathrm{C}$ and is increasing at a rate of $0.500^{\circ} \mathrm{C} / \mathrm{s}$ . At this time, (a) what is the temperature of the silver coins, and (b) at what rate is it changing?

Dominador Tan

Numerade Educator

Problem 55

An aluminum rod 0.500 $\mathrm{m}$ in length and with a crosssectional area of 2.50 $\mathrm{cm}^{2}$ is inserted into a thermally insulated vessel containing liquid helium at 4.20 $\mathrm{K}$ . The rod is initially at 300 $\mathrm{K}$ (a) If half of the rod is inserted into the helium, how many liters of helium boil off by the time the inserted half cools to 4.20 $\mathrm{K}$ ? (Assume the upper half does not yet cool.) (b) If the upper end of the rod is maintained

Emily Anderson

Numerade Educator

Problem 56

A copper ring (with mass of 25.0 g, coefficient of linear expansion of $1.70 \times 10^{-5}\left(^{\circ} \mathrm{C}\right)^{-1},$ and specific heat of $9.24 \times 10^{-2} \mathrm{cal} / \mathrm{g} \cdot^{\circ} \mathrm{C} )$ has a diameter of 5.00 $\mathrm{cm}$ at its temperature of $15.0^{\circ} \mathrm{C}$ . A spherical aluminum shell (with mass 10.9 $\mathrm{g}$ , coefficient of linear expansion $2.40 \times 10^{-5}$ $\left(^{\circ} \mathrm{C}\right)^{-1},$ and specific heat 0.215 $\mathrm{cal} / \mathrm{g} \cdot^{\circ} \mathrm{C} )$ has a diameter of 5.01 $\mathrm{cm}$ at a temperature higher than $15.0^{\circ} \mathrm{C} .$ The sphere is placed on top of the horizontal ring, and the two are allowed to come to thermal equilibrium without any exchange of energy with the surroundings. As soon as the sphere and ring reach thermal equilibrium, the sphere barely falls through the ring. Find (a) the equilibrium temperature, and (b) the initial temperature of the sphere.

Ummatul Choudary

Ummatul Choudary

Numerade Educator

Problem 57

A flow calorimeter is an apparatus used to measure the specific heat of a liquid. The technique of flow calorimetry involves measuring the temperature difference between the input and output points of a flowing stream of the liquid while energy is added by heat at a known rate. A liquid of density $\rho$ flows through the calorimeter with volume flow rate $R .$ At steady state, a temperature difference $\Delta T$ is established between the input and output points when energy is supplied at the rate $\mathscr{P} .$ What is the specific heat of the liquid?

Shahab Ullah

Numerade Educator

Problem 58

One mole of an ideal gas is contained in a cylinder with a movable piston. The initial pressure, volume, and temperature are $P_{i}, V_{i},$ and $T_{i},$ respectively. Find the work done on the gas for the following processes and show each process on a $P V$ diagram: (a) An isobaric compression in which the final volume is half the initial volume. (b) An isothermal compression in which the final pressure is four times the initial pressure. (c) An isovolumetric process in which the final pressure is three times the initial pressure.

Vipender Yadav

Numerade Educator

Problem 59

One mole of an ideal gas, initially at $300 \mathrm{K},$ is cooled at constant volume so that the final pressure is one fourth of the initial pressure. Then the gas expands at constant pressure until it reaches the initial temperature. Determine the work done on the gas.

Shahab Ullah

Numerade Educator

Problem 60

Review problem. Continue the analysis of Problem 60 in Chapter $19 .$ Following a collision between a large space-craft and an asteroid, a copper disk of radius 28.0 $\mathrm{m}$ and thickness $1.20 \mathrm{m},$ at a temperature of $850^{\circ} \mathrm{C},$ is floating in space, rotating about its axis with an angular speed of 25.0 rad/s. As the disk radiates infrared light, its temperature falls to $20.0^{\circ} \mathrm{C}$ . No external torque acts on the disk. (a) Find the change in kinetic energy of the disk. (b) Find the change in internal energy of the disk. (b) Find the amount of energy it radiates.

Dominador Tan

Numerade Educator

Problem 61

Review problem. A 670 -kg meteorite happens to be composed of aluminum. When it is far from the Earth, its temperature is $-15^{\circ} \mathrm{C}$ and it moves with a speed of 14.0 $\mathrm{km} / \mathrm{s}$ relative to the Earth. As it crashes into the planet, assume that the resulting additional internal energy is shared equally between the meteor and the planet, and that all of the material of the meteor rises momentarily to the same final temperature. Find this temperature. Assume that the specific heat of liquid and of gaseous aluminum is $1170 \mathrm{J} / \mathrm{kg} \cdot^{\circ} \mathrm{C} .$

Dominador Tan

Numerade Educator

Problem 62

An iron plate is held against an iron wheel so that a kinetic friction force of 50.0 $\mathrm{N}$ acts between the two pieces of metal. The relative speed at which the two surfaces slide over each other is 40.0 $\mathrm{m} / \mathrm{s}$ (a) Calculate the rate at which mechanical energy is converted to internal energy. (b) The plate and the wheel each have a mass of 5.00 $\mathrm{kg}$ , and each receives 50.0$\%$ of the internal energy. If the system is run reach a uniform internal temperature, what is the resultant temperature increase?

Shahab Ullah

Numerade Educator

Problem 63

A solar cooker consists of a curved reflecting surface that concentrates sunlight onto the object to be warmed (Fig. P20.63). The solar power per unit area reaching the Earth's surface at the location is 600 $\mathrm{W} / \mathrm{m}^{2}$ . The cooker faces the Sun and has a diameter of 0.600 $\mathrm{m}$ . Assume that 40.0$\%$ of the incident energy is transferred to 0.500 $\mathrm{L}$ of water in an open container, initially at $20.0^{\circ} \mathrm{C}$ . How long does it take to completely boil away the water? (Ignore the heat capacity of the container.)

Shahab Ullah

Numerade Educator

Problem 64

Water in an electric teakettle is boiling. The power absorbed by the water is 1.00 $\mathrm{kW}$ . Assuming that the pressure of vapor in the kettle equals atmospheric pressure, determine the speed of effusion of vapor from the kettle's spout, if the spout has a cross-sectional area of $2.00 \mathrm{cm}^{2} .$

Vipender Yadav

Numerade Educator

Problem 65

A cooking vessel on a slow burner contains 10.0 $\mathrm{kg}$ of water and an unknown mass of ice in equilibrium at $0^{\circ} \mathrm{C}$ at time $t=0 .$ The temperature of the mixture is measured at various times, and the result is plotted in Figure $\mathrm{P} 20.65 .$ During the first 50.0 $\mathrm{min}$ , the temperature increases to $2.00^{\circ} \mathrm{C}$ . Ignoring the heat capacity of the vessel, determine the initial mass of ice.

Vipender Yadav

Numerade Educator

Problem 66

(a) In air at $0^{\circ} \mathrm{C},$ a $1.60-\mathrm{kg}$ copper block at $0^{\circ} \mathrm{C}$ is set sliding at 2.50 $\mathrm{m} / \mathrm{s}$ over a sheet of ice at $0^{\circ} \mathrm{C}$ . Friction brings the block to rest. Find the mass of the ice that melts. To describe the process of slowing down, identify the energy input $Q,$ the work input $W,$ the change in internal energy $\Delta E_{\text { int }},$ and the change in mechanical energy $\Delta K$ for the block and also for the ice. (b) A $1.60-\mathrm{kg}$ block of ice at $0^{\circ} \mathrm{C}$ is set sliding at 2.50 $\mathrm{m} / \mathrm{s}$ over a sheet of copper at $0^{\circ} \mathrm{C}$ . Friction brings the block to rest. Find the mass of the ice that melts. Identify $Q, W, \Delta E_{\text { int }},$ and $\Delta K$ for the block and for the metal sheet during the process. (c) A thin $1.60-\mathrm{kg}$ slab of copper at $20^{\circ} \mathrm{C}$ is set sliding at 2.50 $\mathrm{m} / \mathrm{s}$ over an identical stationary slab at the same temperature. Friction quickly stops the motion. If no energy is lost to the environment by heat, find the change in temperature of both objects. Identify $Q, W, \Delta E_{\mathrm{int}},$ and $\Delta K$ for each object during the process.

Dominador Tan

Numerade Educator

Problem 67

The average thermal conductivity of the walls (including the windows) and roof of the house depicted in Figure $\mathrm{P} 20.67$ is $0.480 \mathrm{W} / \mathrm{m} \cdot^{\circ} \mathrm{C},$ and their average thickness is $21.0 \mathrm{cm} .$ The house is heated with natural gas having a heat of combustion (that is, the energy provided 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 $0.0^{\circ} \mathrm{C}$ ? Disregard radiation and the energy lost by heat through the ground.

Vipender Yadav

Numerade Educator

Problem 68

A pond of water at $0^{\circ} \mathrm{C}$ is covered with a layer of ice 4.00 $\mathrm{cm}$ thick. If the air temperature stays constant at $-10.0^{\circ} \mathrm{C},$ how long does it take for the ice thickness to increase to 8.00 $\mathrm{cm}^{2}$ . Suggestion: Utilize Equation 20.15 in the form
$$
\frac{d Q}{d t}=k A \frac{\Delta T}{x}
$$
and note that the incremental energy $d Q$ extracted from the water through the thickness $x$ of ice is the amount required to freeze a thickness $d x$ of ice. That is, $d Q=$ Le $A$ where $\rho$ is the density of the ice, $A$ is the area, and $L$ is the latent heat of fusion.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 69

An ideal gas is carried through a thermodynamic cycle consisting of two isobaric and two isothermal processes as shown in Figure $\mathrm{P} 20.69$ . Show that the net work done on the gas in the entire cycle is given by
$$W_{\text { net }}=-P_{1}\left(V_{2}-V_{1}\right) \ln \frac{P_{2}}{P_{1}}$$

John Palmer

Numerade Educator

Problem 70

The inside of a hollow cylinder is maintained at a temperature $T_{a}$ while the outside is at a lower temperature, $T_{b}$ (Fig. P20.70). The wall of the cylinder has a thermal conductivity $k$ . Ignoring end effects, show that the rate of energy conduction from the inner to the outer surface in the radial direction
is
$$
\frac{d Q}{d t}=2 \pi L k\left[\frac{T_{a}-T_{b}}{\ln (b / a)}\right]
$$
(Suggestions: The temperature gradient is $d T / d r$ . Note that a radial energy current passes through a concentric cylinder of area $2 \pi r L .$ )

Shahab Ullah

Numerade Educator

Problem 71

The passenger section of a jet airliner is in the shape of a cylindrical tube with a length of 35.0 $\mathrm{m}$ and an inner radius of 2.50 $\mathrm{m}$ . Its walls are lined with an insulating material 6.00 $\mathrm{cm}$ in thickness and having a thermal conductivity of $4.00 \times 10^{-5} \mathrm{cal} / \mathrm{s} \cdot \mathrm{cm} \cdot^{\circ} \mathrm{C}$ . A heater must maintain the interior temperature at $25.0^{\circ} \mathrm{C}$ while the outside temperature is $-35.0^{\circ} \mathrm{C}$ . What power must be supplied to the heater? (Use the result of Problem $70 . )$

Shahab Ullah

Numerade Educator

Problem 72

A student obtains the following data in a calorimetry experiment designed to measure the specific heat of aluminum:
$$\begin{array}{ll}{\text { Initial temperature of }} \\ {\text { water and calorimeter: }} & {70^{\circ} \mathrm{C}} \\ {\text { Mass of water: }} & {0.400 \mathrm{kg}} \\ {\text { Mass of calorimeter: }} & {0.040 \mathrm{kg}} \\ {\text { Specific heat of calorimeter: }} & {0.63 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}} \\ {\text { Initial temperature of aluminum: }} & {27^{\circ} \mathrm{C}} \\ {\text { Mass of aluminum: }} & {0.200 \mathrm{kg}} \\ {\text { Final temperature of mixture: }} & {66.3^{\circ} \mathrm{C}}\end{array}$$
Use these data to determine the specific heat of aluminum. Your result should be within 15% of the value listed in Table 20.1.

Shahab Ullah

Numerade Educator

Problem 73

During periods of high activity, the Sun has more sunspots than usual. Sunspots are cooler than the rest of the luminous layer of the Sun’s atmosphere (the photosphere). Paradoxically, the total power output of the active Sun is not lower than average but is the same or slightly higher than average. Work out the details of the following crude model of this phenomenon. Consider a patch of the photosphere with an area of $5.10 \times 10^{14} \mathrm{m}^{2} .$ Its emissivity is $0.965 .$ (a) Find the power it radiates if its temperature is uniformly 5800 $\mathrm{K}$ , corresponding to the quiet Sun. (b) To represent a sunspot, assume that 10.0$\%$ of the area is at 4800 $\mathrm{K}$ and the other 90.0$\%$ is at 5890 $\mathrm{K}$ . That is, a section with the surface area of the Earth is 1000 $\mathrm{K}$ cooler than before and a section nine times as large is 90 $\mathrm{K}$ warmer. Find the average temperature of the patch. (c) Find the power output of the patch. Compare it with the answer to part (a). (The next sunspot maximum is expected around the year $2012 .$ )

Shahab Ullah

Numerade Educator