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College Physics 2017

Raymond A. Serway, Chris Vuille, John hughes

Chapter 12

The Laws of Thermodynamics

Educators


Problem 1

An ideal gas is enclosed in a cylinder with a movable piston on top of it. The piston has a mass of $8.00 \times 10^{3} \mathrm{g}$ and an area of 5.00 $\mathrm{cm}^{2}$ and is free to slide up and down, keeping the pressure of the gas constant. (a) How much work is done on the gas as the temperature of 0.200 $\mathrm{mol}$ of the gas is raised from $20.0^{\circ} \mathrm{C}$ to $3.00 \times 10^{2 \circ} \mathrm{C}$ ? (b) What does the sign of your answer to part (a) indicate?

Averell H.
Carnegie Mellon University

Problem 2

Sketch a $P V$ diagram and find the work done by the gas during the following stages. (a) A gas is expanded from a volume of 1.0 $\mathrm{L}$ to 3.0 $\mathrm{L}$ at a constant pressure of 3.0 $\mathrm{atm} .$ (b) The gas is then cooled at constant volume until the pressure falls to 2.0 atm. (c) The gas is then compressed at a constant pressure of 2.0 atm from a volume of 3.0 $\mathrm{L}$ to 1.0 $\mathrm{L}$ . Note: Be careful of signs. (d) The gas is heated until its pressure increases from 2.0 atm to 3.0 $\mathrm{atm}$ at a constant volume. (e) Find the net work done during the complete cycle.

Salamat A.
Numerade Educator

Problem 3

Gas in a container is at a pressure of 1.5 atm and a volume of 4.0 $\mathrm{m}^{3} .$ What is the work done on the gas (a) if it expands at constant pressure to twice its initial volume, and (b) if it is compressed at constant pressure to one-quarter its initial volume?

Averell H.
Carnegie Mellon University

Problem 4

Find the numeric value of the work done on the gas in (a) Figure $P 12.4 \mathrm{a}$ and (b) Figure $\mathrm{P} 12.4 \mathrm{b}$ .

Salamat A.
Numerade Educator

Problem 5

A gas expands from $I$ to $F$ along the three paths indicated in Figure $\mathrm{P} 12.5$ . Calculate the work done on the gas along paths (a) $I A F,$ (b) $I F,$ and $(\mathrm{c})$ IBF.

Averell H.
Carnegie Mellon University

Problem 6

A gas follows the PV diagram in Figure P12.6. Find the work done on the gas along the paths (a) AB, (b) BC, (c) CD, (d) DA, and (e) ABCDA.

Salamat A.
Numerade Educator

Problem 7

A sample of helium behaves as an ideal gas as it is heated at constant pressure from 273 $\mathrm{K}$ to 373 $\mathrm{K}$ . If 20.0 $\mathrm{J}$ of work is done by the gas during this process, what is the mass of helium present?

Averell H.
Carnegie Mellon University

Problem 8

(a) Find the work done by an ideal gas as it expands from point $A$ to point $B$ along the path shown in Figure $P 12.8 . \quad(b)$ How much work is done by the gas if it compressed from $B$ to $A$ along the same path?

Salamat A.
Numerade Educator

Problem 9

One mole of an ideal gas initially at a temperature of $T_{i}=0^{\circ} \mathrm{C}$ undergoes an expansion at a constant pressure of 1.00 atm to four times its original volume. (a) Calculate the new temperature $T_{f}$ of the gas. (b) Calculate the work done on the gas during the expansion.

Averell H.
Carnegie Mellon University

Problem 10

(a) Determine the work done on a fluid that expands from $i$ to $f$ as indicated in Figure $P 12.10$ (b) How much work is done on the fluid if it is compressed from $f$ to $i$ along the same path?

Salamat A.
Numerade Educator

Problem 11

A balloon holding 5.00 moles of helium gas absorbs 925 $\mathrm{J}$ of thermal energy while doing 102 $\mathrm{J}$ of work expanding to a larger volume. (a) Find the change in the balloon's internal energy. (b) Calculate the change in temperature of the gas.

Averell H.
Carnegie Mellon University

Problem 12

A chemical reaction transfers 1250 $\mathrm{J}$ of thermal energy into an ideal gas while the system expands by $2.00 \times 10^{-2} \mathrm{m}^{3}$ at a constant pressure of $1.50 \times 10^{5} \mathrm{Pa}$ . Find the change in the internal energy.

Salamat A.
Numerade Educator

Problem 13

The only form of energy possessed by molecules of a monatomic ideal gas is translational kinetic energy. Using the results from the discussion of kinetic theory in Section $10.5,$ show that the internal energy of a monatomic ideal gas at pressure $P$ and occupying volume $V$ may be written as $U=\frac{3}{2} P V$.

Averell H.
Carnegie Mellon University

Problem 14

A cylinder of volume 0.300 $\mathrm{m}^{3}$ contains 10.0 $\mathrm{mol}$ of neon gas at $20.0^{\circ}$ C. Assume neon behaves as an ideal gas. (a) What is the pressure of the gas? (b) Find the internal energy of the gas, (c) Suppose the gas expands at constant pressure to a volume of 1.000 $\mathrm{m}^{3} .$ How much work is done on the gas? (d) What is the temperature of the gas at the new volume? (e) Find the internal energy of the gas when its volume is 1.000 $\mathrm{m}^{3} .(1)$ Compute the change in the internal energy during the expansion. (g) Compute $\Delta U-W$ (h) Must thermal energy be transferred to the gas during the constant pressure expansion or be taken away? (i) Compute $Q$ , the thermal energy transfer. (j) What symbolic relationship between $Q,$ $\Delta U,$ and $W$ is suggested by the values obtained?

Salamat A.
Numerade Educator

Problem 15

A gas expands from $I$ to $F$ in Figure $\mathrm{P} 12.5 .$ The energy added to the gas by heat is 418 J when the gas goes from $I$ to $F$ along the diagonal path. (a) What is the change in internal energy of the gas? (b) How much energy must be added to the gas by heat for the indirect path $L A F$ to give the same change in internal energy?

Averell H.
Carnegie Mellon University

Problem 16

In a running event, a sprinter does $4.8 \times 10^{5} \mathrm{J}$ of work and her internal energy decreases by $7.5 \times 10^{5} \mathrm{J}$ . (a) Determine the heat transferred between her body and surroundings during this event. (b) What does the sign of your answer to part (a) indicate?

Salamat A.
Numerade Educator

Problem 17

A gas is compressed at a constant pressure of 0.800 atm from 9.00 $\mathrm{L}$ to 2.00 $\mathrm{L}$ . In the process, $400 . \mathrm{J}$ of energy leaves the gas by heat. (a) What is the work done on the gas? (b) What is the change in its internal energy?

Averell H.
Carnegie Mellon University

Problem 18

A quantity of a monatomic ideal gas undergoes a process in which both its pressure and volume are doubled as shown in Figure P12.18. What is the energy absorbed by heat into the gas during this process? the gas during this process? Hint: The internal energy of sure $P$ and occupying volume $V$ is given by $U=\frac{3}{2} P V$

Salamat A.
Numerade Educator

Problem 19

A gas is enclosed in a container fitted with a piston of cross-sectional area 0.150 $\mathrm{m}^{2}$ . The pressure of the gas is maintained at $6.00 \times 10^{3} \mathrm{Pa}$ as the piston moves inward 20.0 $\mathrm{cm} .$ (a) Calculate the work done by the gas. (b) If the internal energy of the gas decreases by $8.00 \mathrm{J},$ find the amount of energy removed from the system by heat during the compression.

Averell H.
Carnegie Mellon University

Problem 20

A monatomic ideal gas undergoes the thermodynamic process shown in the $P V$ diagram of Figure $P 12.20 .$ Determine whether each of the values $\Delta U$ $Q$ and $W$ for the gas is positive, negative, or zero. Hint: The internal energy of a monatomic ideal gas at pressure $P$ and occupying volume $V$ is
given by $U=\frac{3}{2} P V .$

Salamat A.
Numerade Educator

Problem 21

An ideal gas is compressed from a volume of $V_{i}=5.00 \mathrm{L}$ to a volume of $V_{j}=3.00 \mathrm{L}$ while in thermal contact with a heat reservoir at $T=295 \mathrm{K}$ as in Figure $\mathrm{P} 12.21$ . During the compression process, the piston moves down a distance of $d=$ 0.130 $\mathrm{m}$ under the action of an average external force of $F=$ 25.0 $\mathrm{kN}$ . Find (a) the work done on the gas, (b) the change in internal energy of the gas, and (c) the thermal energy exchanged between the gas and the reservoir. (d) If the gas is thermally insulated so no thermal energy could be exchanged, what would happen to the temperature of the gas during the compression?

Averell H.
Carnegie Mellon University

Problem 22

A system consisting of 0.025 6 moles of a diatomic ideal gas is taken from state A to state C along the path in Figure P12.22. (a) How much work is done on the gas during this process? (b) What is the lowest temperature of the gas during this process, and where does it occur? (c) Find the change in internal energy of the gas and (d) the energy delivered to the gas in going from A to C. Hint: For part (c), adapt the equation in the remarks of Example 12.9 to a diatomic ideal gas.

Salamat A.
Numerade Educator

Problem 23

An ideal monatomic gas expands isothermally from 0.500 $\mathrm{m}^{3}$ to 1.25 $\mathrm{m}^{3}$ at a constant temperature of 675 $\mathrm{K}$ . If the initial pressure is $1.00 \times 10^{5} \mathrm{Pa}$ , find (a) the work done on the gas, (b) the thermal energy transfer $Q$ , and (c) the change in the internal energy.

Averell H.
Carnegie Mellon University

Problem 24

An ideal gas expands at constant pressure. (a) Show that $P \Delta V=n R \Delta T$ . (b) If the gas is monatomic, start from the definition of internal energy and show that $\Delta U=\frac{3}{2} W_{\text { ew }},$ where $W_{\text { cny is the work done by the gas on its environment. }}(c)$ For the same monatomic ideal gas, show with the first law that $Q=\frac{5}{2} W_{\text { env }}$ (d) Is it possible for an ideal gas to expand at constant pressure while exhausting thermal energy? Explain.

Salamat A.
Numerade Educator

Problem 25

An ideal monatomic gas contracts in an isobaric process from 1.25 $\mathrm{m}^{9}$ to 0.500 $\mathrm{m}^{3}$ at a constant pressure of $1.50 \times 10^{5} \mathrm{Pa} .$ If the initial temperature is 425 $\mathrm{K}$ , find (a) the work done on the gas, (b) the change in internal energy, (c) the energy transfer $Q,$ and $(\mathrm{d})$ the final temperature.

Averell H.
Carnegie Mellon University

Problem 26

An ideal diatomic gas expands adiabatically from 0.750 $\mathrm{m}^{3}$ to 1.50 $\mathrm{m}^{3}$ . If the initial pressure and temperature are $1.50 \times 10^{5}$ Pa and 325 $\mathrm{K}$ , respectively, find (a) the number of moles in the gas, (b) the final gas pressure, (c) the final gas temperature, and (d) the work done on the gas.

Salamat A.
Numerade Educator

Problem 27

An ideal monatomic gas is contained in a vessel of constant volume 0.200 $\mathrm{m}^{3}$ . The initial temperature and pressure of the gas are $300 . \mathrm{K}$ and 5.00 $\mathrm{atm}$ , respectively. The goal of this problem is to find the temperature and pressure of the gas after 16.0 $\mathrm{kJ}$ of thermal energy is supplied to the gas. (a) Use the ideal gas law and initial conditions to calculate the number of moles of gas in the vessel. (b) Find the specific heat of the gas. (c) What is the work done by the gas during this process? (d) Use the first law of thermodynamics to find the change in internal energy of the gas. (e) Find the change in temperature of the gas. (f) Calculate the final temperature of the gas. (g) Use the ideal gas expression to find the final pressure of the gas.

Averell H.
Carnegie Mellon University

Problem 28

Consider the cyclic process described by Figure $P 12.28$ . If $Q$ is negative for the process $B C$ and $\Delta U$ is negative for the process $C A,$ determine the signs of $Q, W,$ and $\Delta U$ associated with each process.

Salamat A.
Numerade Educator

Problem 29

A 5.0 -kg block of aluminum is heated from $20^{\circ} \mathrm{C}$ to $90^{\circ} \mathrm{C}$ at atmospheric pressure. Find (a) the work done by the aluminum, (b) the amount of energy transferred to it by heat, and (c) the increase in its internal energy.

Averell H.
Carnegie Mellon University

Problem 30

One mole of gas initially at a pressure of 2.00 atm and a volume of 0.300 L has an internal energy equal to 91.0 J. In its final state, the gas is at a pressure of 1.50 atm and a volume of $0.800 \mathrm{L},$ and its internal energy equals 182 $\mathrm{J}$ . For the paths $I A F, \quad I B F,$ and $I F$ in Figure $P 12.30$ , calculate (a) the work done on the gas and (b) the net energy transferred to the gas by heat in the process.

Salamat A.
Numerade Educator

Problem 31

A gas increases in pressure from 2.00 atm to 6.00 atm at a constant volume of 1.00 $\mathrm{m}^{3}$ and then expands at constant pressure to a volume of 3.00 $\mathrm{m}^{3}$ before returning to its initial state as shown in Figure $\mathrm{P} 12.31$ How much work is done in one cycle?

Averell H.
Carnegie Mellon University

Problem 32

An ideal gas expands at a constant pressure of $6.00 \times 10^{5} \mathrm{Pa}$ from a volume of
1.00 $\mathrm{m}^{3}$ to a volume of 4.00 $\mathrm{m}^{3}$ and then is compressed to one-third that pressure and a volume of 2.50 $\mathrm{m}^{3}$ as shown in Figure $\mathrm{P} 12.32$ before returning to its initial state. How much work is done in taking a gas through one cycle of the process shown in the figure?

Salamat A.
Numerade Educator

Problem 33

A heat engine operates between a reservoir at $25^{\circ} \mathrm{C}$ and one at $375^{\circ} \mathrm{C}$ . What is the maximum efficiency possible for this engine?

Averell H.
Carnegie Mellon University

Problem 34

A heat engine is being designed to have a Carnot efficiency of 65% when operating between two heat reservoirs. (a) If the temperature of the cold reservoir is $20^{\circ} \mathrm{C},$ what must be the temperature of the hot reservoir? (b) Can the actual efficiency of the engine be equal to 65$\% ?$ Explain.

Salamat A.
Numerade Educator

Problem 35

The work done by an engine equals one-fourth the energy it absorbs from a reservoir. (a) What is its thermal efficiency? (b) What fraction of the energy absorbed is expelled to the cold reservoir?

Averell H.
Carnegie Mellon University

Problem 36

In each cycle of its operation, a heat engine expels 2400 $\mathrm{J}$ of energy and performs 1800 $\mathrm{J}$ of mechanical work. (a) How much thermal energy must be added to the engine in each cycle? (b) Find the thermal efficiency of the engine.

Salamat A.
Numerade Educator

Problem 37

One of the most efficient engines ever built is a coal-fired steam turbine engine in the Ohio River valley, driving an electric generator as it operates between $1870^{\circ} \mathrm{C}$ and $430^{\circ} \mathrm{C} .$ (a) What is its maximum theoretical efficiency? (b) Its actual efficiency is 42.0$\%$ . How much mechanical power does the engine deliver if it absorbs $1.40 \times 10^{5} \mathrm{J}$ of energy each second from the hot reservoir.

Averell H.
Carnegie Mellon University

Problem 38

A lawnmower engine ejects $1.00 \times 10^{4} \mathrm{J}$ each second while running with an efficiency of $0.200 .$ Find the engine's horsepower rating, using the conversion factor $1 \mathrm{hp}=746 \mathrm{W}$

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Problem 39

An engine absorbs 1.70 kJ from a hot reservoir at $277^{\circ} \mathrm{C}$ and expels 1.20 $\mathrm{kJ}$ to a cold reservoir at $27^{\circ} \mathrm{C}$ in each cycle. (a) What is the engine's efficiency? (b) How much work is done by the engine in each cycle? (c) What is the power output of the engine if each cycle lasts 0.300 $\mathrm{s?}$

Averell H.
Carnegie Mellon University

Problem 40

A heat pump has a coefficient of performance of 3.80 and operates with a power consumption of $7.03 \times 10^{3} \mathrm{W}$ . (a) How much energy does the heat pump deliver into a home during 8.00 $\mathrm{h}$ of continuous operation? (b) How much energy does it extract from the outside air in 8.00 $\mathrm{h}$ ?

Salamat A.
Numerade Educator

Problem 41

A freezer has a coefficient of performance of $6.30 .$ The freezer is advertised as using $457 \mathrm{kW}-\mathrm{h} / \mathrm{y}$ . (a) On average, how much energy does the freezer use in a single day? (b) On average, how much thermal energy is removed from the freezer each day? (c) What maximum mass of water at $20.0^{\circ} \mathrm{C}$ could the freezer freeze in a single day? Note: One kilowat-hour (kW-h) is an amount of energy equal to operating a $1-\mathrm{k} \mathrm{W}$ appliance for one hour.

Averell H.
Carnegie Mellon University

Problem 42

Two heat engines are operated in series so that part of the energy expelled from engine $A$ is absorbed by engine $B$ with $\left|Q_{h B}\right|=0.750\left|Q_{c A}\right| .$ Engines $A$ and $B$ have efficiencies $e_{A}=$ $e_{B}=0.250$ and engine A performs work $W_{A}=275 \mathrm{J}$ . Find the overall efficiency of the two-engine combination, given by $e=\frac{W_{\mathrm{A}}+W_{\mathrm{B}}}{ | Q_{\mathrm{AA}}}$

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Problem 43

In one cycle a heat engine absorbs 500 $\mathrm{J}$ from a high-temperature reservoir and expels 300 $\mathrm{J}$ to a low-temperature reservoir. If the efficiency of this engine is 60$\%$ of the efficiency of a Carnot engine, what is the ratio of the low temperature to the high temperature in the Carnot engine?

Averell H.
Carnegie Mellon University

Problem 44

A power plant has been proposed that would make use of the temperature gradient in the ocean. The system is to operate between $20.0^{\circ} \mathrm{C}$ (surface water temperature) and $5.00^{\circ} \mathrm{C}$ (water temperature at a depth of about 1 $\mathrm{km} )$ (a) What is the maximum efficiency of such a system? (b) If the useful power output of the plant is $75.0 \mathrm{MW},$ how much energy is absorbed per hour? (c) In view of your answer to part (a), do you think such a system is worthwhile (considering that there is no charge for fuel)?

Salamat A.
Numerade Educator

Problem 45

A certain nuclear power plant has an electrical power output of 435 MW. The rate at which energy must be supplied to the plant is 1420 $\mathrm{MW}$ . (a) What is the thermal efficiency of the power plant? (b) At what rate is thermal energy expelled by the plant?

Averell H.
Carnegie Mellon University

Problem 46

A heat engine operates in a Carnot cycle between $80.0^{\circ} \mathrm{C}$ and $350^{\circ} \mathrm{C}$ . It absorbs 21000 $\mathrm{J}$ of energy per cycle from the hot reservoir. The duration of each cycle is 1.00 $\mathrm{s}$ . (a) What is the mechanical power output of this engine? (b) How much energy does it expel in each cycle by heat?

Salamat A.
Numerade Educator

Problem 47

A Styrofoam cup holding 125 $\mathrm{g}$ of hot water at $1.00 \times 10^{2 \circ} \mathrm{C}$ cools to room temperature, $20.0^{\circ} \mathrm{C}$ . What is the change in entropy of the room? (Neglect the specific heat of the cup and any change in temperature of the room.)

Averell H.
Carnegie Mellon University

Problem 48

A 65-g ice cube is initially at 0.0°C. (a) Find the change in entropy of the cube after it melts completely at 0.0°C. (b) What is the change in entropy of the environment in this process? Hint: The latent heat of fusion for water is $3.33 \times 10^{5} {J} / {kg}$

Salamat A.
Numerade Educator

Problem 49

A freezer is used to freeze 1.0 $\mathrm{L}$ of water completely into ice. The water and the freezer remain at a constant temperature of $T=0^{\circ} \mathrm{C}$ . Determine (a) the change in the entropy of the water and (b) the change in the entropy of the freezer.

Averell H.
Carnegie Mellon University

Problem 50

A freezer is used to freeze 1.0 L of water completely into ice. The water and the freezer remain at a constant temperature of T 5 0°C. Determine (a) the change in the entropy of the water and (b) the change in the entropy of the freezer.

Salamat A.
Numerade Educator

Problem 51

A 70.0 -kg log falls from a height of 25.0 $\mathrm{m}$ into a lake. If the log, the lake, and the air are all at $300 . \mathrm{K}$ , find the change in entropy of the Universe during this process.

Averell H.
Carnegie Mellon University

Problem 52

A sealed container holding 0.500 $\mathrm{kg}$ of liquid nitrogen at its boiling point of 77.3 $\mathrm{K}$ is placed in a large room at $21.0^{\circ} \mathrm{C}$ . Energy is transferred from the room to the nitrogen as the liquid nitrogen boils into a gas and then warms to the room's temperature. (a) Assuming the room's temperature remains essentially unchanged at $21.0^{\circ} \mathrm{C},$ calculate the energy transferred from the room to the nitrogen. (b) Estimate the change in entropy of the room. Liquid nitrogen has a latent heat of vaporization of $2.01 \times 10^{5} \mathrm{J} / \mathrm{kg}$ . The specific heat of $\mathrm{N}_{2}$ gas at constant pressure is $c_{\mathrm{N}_{z}}=1.04 \times 10^{3} \mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$ .

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Problem 53

The surface of the Sun is approximately at $5.70 \times 10^{3} \mathrm{K}$ and the temperature of the Earth's surface is approximately $290 . \mathrm{K}$ . What entropy change occurs when $1.00 \times 10^{3} \mathrm{J}$ of energy is transferred by heat from the Sun to the Earth?

Averell H.
Carnegie Mellon University

Problem 54

When an aluminum bar is temporarily connected between a hot reservoir at 725 $\mathrm{K}$ and a cold reservoir at $310 \mathrm{K}, 2.50 \mathrm{kJ}$ of energy is transferred by heat from the hot reservoir to the cold reservoir. In this irreversible process, calculate the change in entropy of (a) the hot reservoir, (b) the cold reservoir, and (c) the Universe, neglecting any change in entropy of the aluminum rod. (d) Mathematically, why did the result for the Universe in part (c) have to be positive?

Salamat A.
Numerade Educator

Problem 55

Prepare a table like Table 12.3 for the following occurrence: You toss four coins into the air simultaneously and record all the possible results of the toss in terms of the numbers of heads and tails that can result. (For example, HHTH and HTHH are two possible ways in which three heads and one tail can be achieved.) (a) On the basis of your table, what is the most probable result of a toss? In terms of entropy, (b) what is the most ordered state, and (c) what is the most disordered?

Averell H.
Carnegie Mellon University

Problem 56

This is a symbolic version of Problem 54. When a metal bar is temporarily connected between a hot reservoir at $T_{h}$ and a cold reservoir at $T_{c}$ the energy transferred by heat from the hot reservoir to the cold reservoir is $Q_{i k}$ In this irreversible process, find expressions for the change in entropy of (a) the hot reservoir, (b) the cold reservoir, and (c) the Universe.

Salamat A.
Numerade Educator

Problem 57

On a typical day, a 65-kg man sleeps for 8.0 h, does light chores for 3.0 h, walks slowly for 1.0 h, and jogs at moderate pace for 0.5 h. What is the change in his internal energy for all these activities?

Averell H.
Carnegie Mellon University

Problem 58

A weightlifter has a basal metabolic rate of 80.0 W. As he is working out, his metabolic rate increases by about 650 W. (a) How many hours does it take him to work off a 450-Calorie bagel if he stays in bed all day? (b) How long does it take him if he’s working out? (c) Calculate the amount of
mechanical work necessary to lift a 120-kg barbell 2.00 m. (d) He drops the barbell to the floor and lifts it repeatedly. How many times per minute must he repeat this process to do an amount of mechanical work equivalent to his metabolic rate increase of 650 W during exercise? (e) Could he actually do repetitions at the rate found in part (d) at the given metabolic level? Explain.

Salamat A.
Numerade Educator

Problem 59

BIO Sweating is one of the main mechanisms with which the body dissipates heat. Sweat evaporates with a latent heat of 2 430 kJ/kg at body temperature, and the body can produce as much as 1.5 kg of sweat per hour. If sweating were the only heat dissipation mechanism, what would be the maximum sustainable metabolic rate, in watts, if 80% of the energy used by the body goes into waste heat?

Averell H.
Carnegie Mellon University

Problem 60

BIO A woman jogging has a metabolic rate of 625 W. (a) Calculate her volume rate of oxygen consumption in L/s. (b) Estimate her required respiratory rate in breaths/min if her lungs inhale 0.600 L of air in each breath and air is 20.9% oxygen.

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Problem 61

BIO Suppose a highly trained athlete consumes oxygen at a rate of 70.0 mL/(min · kg) during a 30.0-min workout. If the athlete’s mass is 78.0 kg and their body functions as a heat engine with a 20.0% efficiency, calculate (a) their metabolic rate in kcal/min and (b) the thermal energy in kcal released
during the workout.

Averell H.
Carnegie Mellon University

Problem 62

A Carnot engine operates between the temperatures $T_{h}=$ $1.00 \times 10^{2 \circ} {C}$ and $T_{e}=20.0^{\circ} {C} .$ By what factor does the theoretical efficiency increase if the temperature of the hot reservoir is increased to $5.50 \times 10^{2 \circ} {C}$ ?

Salamat A.
Numerade Educator

Problem 63

A 1 500-kW heat engine operates at 25% efficiency. The heat energy expelled at the low temperature is absorbed by a stream of water that enters the cooling coils at 20.°C. If 60. L flows across the coils per second, determine the increase in temperature of the water.

Averell H.
Carnegie Mellon University

Problem 64

A Carnot engine operates between 100°C and 20°C. How much ice can the engine melt from its exhaust after it has done $5.0 \times 10^{4} {J}$ of work?

Salamat A.
Numerade Educator

Problem 65

A substance undergoes the cyclic process shown in Figure P12.65. Work output occurs along path
AB while work input is required along path BC, and no work is involved in the constant volume process CA. Energy transfers by heat occur during each process involved in the cycle. (a) What is the work output during process AB? (b) How much work input is required during process BC? (c) What is the net energy input Q during this cycle?

Averell H.
Carnegie Mellon University

Problem 66

When a gas follows path 123 on the PV diagram in Figure P12.66, 418 J of energy flows into the system by heat and 2167 J of work is done on the gas. (a) What is the change in the internal energy of the system? (b) How much energy Q flows into the system if the gas follows path 143? The work done on the gas along this path is 263.0 J. What net work would be done on or by the system if the system followed (c) path 12341 and (d) path 14321? (e) What is the change in internal energy of the system in the processes described in parts (c) and (d)?

Salamat A.
Numerade Educator

Problem 67

A $1.0 \times 10^{2}-$ kg steel support rod in a building has a length of 2.0 ${m}$ at a temperature of $20.0^{\circ} {C}$ . The rod supports a hanging load of $6.0 \times 10^{3} {kg} .$ Find (a) the work done on the rod as the temperature increases to 40.0°C, (b) the energy Q added to the rod (assume the specific heat of steel is the same as that for iron), and (c) the change in internal energy of the rod.

Averell H.
Carnegie Mellon University

Problem 68

An ideal gas initially at pressure $P_{0},$ volume $V_{0},$ and temperature $T_{0}$ is taken through the cycle described in Figure $P 12.68$
(a) Find the net work done by the gas per cycle in terms of $P_{0}$ and $V_{0} .$
(b) What is the net energy $Q$ added 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°C. Hint: Recall that the work done by the system equals the area under a PV curve.

Salamat A.
Numerade Educator

Problem 69

One mole of neon gas is heated from 300. K to 420. K at constant pressure. Calculate (a) the energy Q transferred to the gas, (b) the change in the internal energy of the gas, and (c) the work done on the gas. Note that neon has a molar specific heat of c 5 20.79 J/mol?K for a constant-pressure process.

Averell H.
Carnegie Mellon University

Problem 70

Every second at Niagara Falls, approximately $5.00 \times 10^{3} {m}^{3}$ of water falls a distance of 50.0 ${m}$ . What is the increase in entropy per second due to the falling water? Assume the mass of the
surroundings is so great that its temperature and that of the water stay nearly constant at 20.0°C. Also assume a negligible amount of water evaporates.

Salamat A.
Numerade Educator

Problem 71

A cylinder containing 10.0 moles of a monatomic ideal gas expands from (A) to (B) along the path shown in Figure P12.71. (a) Find the temperature of the gas at point A and the temperature at point (B). (b) How much work is done by the gas during this expansion? (c) What is the change in internal energy of the gas? (d) Find the energy transferred to the gas by heat in this process.

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Problem 72

Two moles of molecular hydrogen $\left(\mathrm{H}_{2}\right)$ react with 1 mole of molecular oxygen $(\mathrm{O}_{2})$ to produce 2 moles of water $(\mathrm{H}_{2} \mathrm{O})$ together with an energy release of 241.8 $\mathrm{kJ} / \mathrm{mole}$ of water. Suppose a spherical vessel of radius 0.500 $\mathrm{m}$ contains 14.4 moles of $\mathrm{H}_{2}$ and 7.2 moles of $\mathrm{O}_{2}$ at $20.0^{\circ} \mathrm{C}$ . (a) What is the initial pressure in the vessel? (b) What is the initial internal energy of the gas? (c) Suppose a spark ignites the mixture and the gases burn completely into water vapor. How much energy is produced? (d) Find the temperature and pressure of the steam, assuming it’s an ideal gas. (e) Find the mass of steam and then calculate the steam’s density. (f) If a small hole were put in the sphere, what would be the initial exhaust velocity of the exhausted steam if spewed out into a vacuum? (Use Bernoulli’s equation.)

Salamat A.
Numerade Educator

Problem 73

Suppose you spend 30.0 minutes on a stair-climbing machine, climbing at a rate of 90.0 steps per minute, with each step 8.00 inches high. If you weigh 150. lb and the machine reports that 600. kcal have been burned at the end of the workout, what efficiency is the machine using in obtaining this result? If your actual efficiency is 0.18, how many kcal did you actually burn?

Averell H.
Carnegie Mellon University

Problem 74

Hydrothermal vents deep on the ocean floor spout water at temperatures as high as 570°C. This temperature is below the boiling point of water because of the immense pressure at that depth. Because the surrounding ocean temperature is at 4.0°C, an organism could use the temperature
gradient as a source of energy. (a) Assuming the specific heat of water under these conditions is 1.0 cal/g ? °C, how much energy is released when 1.0 L of water is cooled from 570°C to 4.0°C? (b) What is the maximum usable energy an organism can extract from this energy source? (Assume the organism has some internal type of heat engine acting between the two temperature extremes.) (c) Water from these vents contains hydrogen sulfide $(\mathrm{H}_{2} \mathrm{S})$ at a concentration of 0.90 mmole/L. Oxidation of 1.0 mole of $\mathrm{H}_{2} \mathrm{S}$ produces 310 kJ of energy. How much energy is available through$\mathrm{H}_{2} \mathrm{S}$ oxidation of 1.0 L of water?

Salamat A.
Numerade Educator

Problem 75

An electrical power plant has an overall efficiency of 15%. The plant is to deliver 150 MW of electrical power to a city, and its turbines use coal as fuel. The burning coal produces steam at 190°C, which drives the turbines. The steam is condensed into water at 25°C by passing through coils that are in contact with
river water. (a) How many metric tons of coal does the plant consume each day$(1 \text { metric ton }=1 \times 10^{3} \mathrm{kg})?$ (b) What is the total cost of the fuel per year if the delivery price is $8 per
metric ton? (c) If the river water is delivered at 20°C, at what minimum rate must it flow over the cooling coils so that its temperature doesn’t exceed 25°C? Note: The heat of combustion of coal is $7.8 \times 10^{3} \mathrm{cal} / \mathrm{g}$

Averell H.
Carnegie Mellon University

Problem 75

A diatomic ideal gas expands from a volume of $V_{A}=1.00 \mathrm{m}^{3}$ to $V_{B}=$ 3.00 $\mathrm{m}^{3}$ along the path shown in Figure P12.76. If the initial pressure is PA 5 $2.00 \times 10^{5} \mathrm{Pa}$ and there $2.00 \times 10^{5} \mathrm{Pa}$ and there

Averell H.
Carnegie Mellon University