Principles of Physics a Calculus Based Text

Raymond A. Serway, John W. Jewett, Jr.

Chapter 18

Heat Engines, Entropy, and the Second Law of Thermodynamics - all with Video Answers

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Chapter Questions

Problem 1

A heat engine takes in $360 \mathrm{J}$ of energy from a hot reservoir and performs $25.0 \mathrm{J}$ of work in each cycle. Find (a) the efficiency of the engine and (b) the energy expelled to the cold reservoir in each cycle.

Penny Riley

Penny Riley

Numerade Educator

Problem 2

A multicylinder gasoline engine in an airplane, operating at $2.50 \times 10^{3}$ rev $/$ min, takes in energy $7.89 \times 10^{3} \mathrm{J}$ and exhausts $4.58 \times 10^{3} \mathrm{J}$ for each revolution of the crankshaft.
(a) How many liters of fuel does it consume in $1.00 \mathrm{h}$ of operation if the heat of combustion of the fuel is equal to $4.03 \times 10^{7} \mathrm{J} / \mathrm{L} ?(\mathrm{b})$ What is the mechanical power output of the engine? Ignore friction and express the answer in horsepower. (c) What is the torque exerted by the crankshaft on the load? (d) What power must the exhaust and cooling system transfer out of the engine?

Sanjeev Kumar

Numerade Educator

Problem 3

Suppose a heat engine is connected to two energy reservoirs, one a pool of molten aluminum $\left(660^{\circ} \mathrm{C}\right)$ and the other a block of solid mercury $\left(-38.9^{\circ} \mathrm{C}\right) .$ The engine runs by freezing $1.00 \mathrm{g}$ of aluminum and melting $15.0 \mathrm{g}$ of mercury during each cycle. The heat of fusion of aluminum is $3.97 \times 10^{5} \mathrm{J} / \mathrm{kg}$; the heat of fusion of mercury is $1.18 \times 10^{4} \mathrm{J} / \mathrm{kg} .$ What is the efficiency of this engine?

Henrique Saito

Numerade Educator

Problem 4

A gun is a heat engine. In particular, it is an internal combustion piston engine that does not operate in a cycle, but comes apart during its adiabatic expansion process. A certain gun consists of $1.80 \mathrm{kg}$ of iron. It fires one 2.40 -g bullet at $320 \mathrm{m} / \mathrm{s}$ with an energy efficiency of $1.10 \% .$ Assume the body of the gun absorbs all the energy exhaust-the other $98.9 \%$ -and increases uniformly in temperature for a short time interval before it loses any energy by heat into the environment. Find its temperature increase.

Henrique Saito

Numerade Educator

Problem 5

A particular heat engine has a mechanical power output of $5.00 \mathrm{kW}$ and an efficiency of $25.0 \% .$ The engine expels $8.00 \times 10^{3} \mathrm{J}$ of exhaust energy in each cycle. Find (a) the energy taken in during each cycle and (b) the time interval for each cycle.

Henrique Saito

Numerade Educator

Problem 6

An electric generating station is designed to have an electric output power of $1.40 \mathrm{MW}$ using a turbine with twothirds the efficiency of a Carnot engine. The exhaust energy is transferred by heat into a cooling tower at $110^{\circ} \mathrm{C}$. (a) Find the rate at which the station exhausts energy by heat as a function of the fuel combustion temperature $T_{h^{*}}$ (b) If the firebox is modified to run hotter by using more advanced combustion technology, how does the amount of energy exhaust change? (c) Find the exhaust power for $T_{h}=800^{\circ} \mathrm{C}$.
(d) Find the value of $T_{h}$ for which the exhaust power would be only half as large as in part (c). (e) Find the value of $T_{h}$ for which the exhaust power would be one-fourth as large as in part (c).

Sanjeev Kumar

Numerade Educator

Problem 7

One of the most efficient heat engines ever built is a coal-fired steam turbine in the Ohio River valley, operating between $1870^{\circ} \mathrm{C}$ and $430^{\circ} \mathrm{C}$. (a) What is its maximum theoretical efficiency? (b) The actual efficiency of the engine 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 its hot reservoir?

Sanjeev Kumar

Numerade Educator

Problem 8

Euppose you build a two-engine device with the exhaust energy output from one heat engine supplying the input energy for a second heat engine. We say that the two engines are running in series. Let $e_{1}$ and $e_{2}$ represent the efficiencies of the two engines. (a) The overall efficiency of the two-engine device is defined as the total work output divided by the energy put into the first engine by heat. Show that the overall efficiency $e$ is given by $$e=e_{1}+e_{2}-e_{1} e_{2}$$ What If? For parts (b) through (e) that follow, assume the two engines are Carnot engines. Engine 1 operates between temperatures $T_{h}$ and $T_{i} .$ The gas in engine 2 varies in temperature between $T_{i}$ and $T_{c \cdot}$ In terms of the temperatures, (b) what is the efficiency of the combination engine?
(c) Does an improvement in net efficiency result from the use of two engines instead of one? (d) What value of the intermediate temperature $T_{i}$ results in equal work being done by each of the two engines in series? (e) What value of $T_{i}$ results in each of the two engines in series having the same efficiency?

Sanjeev Kumar

Numerade Educator

Problem 9

A Carnot engine has a power output of $150 \mathrm{kW}$. The engine operates between two reservoirs at $20.0^{\circ} \mathrm{C}$ and $500^{\circ} \mathrm{C}$.
(a) How much energy enters the engine by heat per hour?
(b) How much energy is exhausted by heat per hour?

Henrique Saito

Numerade Educator

Problem 10

A Carnot engine has a power output $P .$ The engine operates between two reservoirs at temperature $T_{c}$ and $T_{h^{*}}$
(a) How much energy enters the engine by heat in a time interval $\Delta t ?$ (b) How much energy is exhausted by heat in the time interval $\Delta t$ ?

Sanjeev Kumar

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Problem 11

An ideal gas is taken through a Carnot cycle. The isothermal expansion occurs at $250^{\circ} \mathrm{C},$ and the isothermal compression takes place at $50.0^{\circ} \mathrm{C}$. The gas takes in $1.20 \times 10^{3} \mathrm{J}$ of energy from the hot reservoir during the isothermal expansion. Find (a) the energy expelled to the cold reservoir in each cycle and (b) the net work done by the gas in each cycle.

Henrique Saito

Numerade Educator

Problem 12

Why is the following situation impossible? An inventor comes to a patent office with the claim that her heat engine, which employs water as a working substance, has a thermodynamic efficiency of $0.110 .$ Although this efficiency is low compared with typical automobile engines, she explains that her engine operates between an energy reservoir at room temperature and a water-ice mixture at atmospheric pressure and therefore requires no fuel other than that to make the ice. The patent is approved, and working prototypes of the engine prove the inventor's efficiency claim.

Henrique Saito

Numerade Educator

Problem 13

A power plant operates at a $32.0 \%$ efficiency during the summer when the seawater used for cooling is at $20.0^{\circ} \mathrm{C}$. The plant uses $350^{\circ} \mathrm{C}$ steam to drive turbines. If the plant's efficiency changes in the same proportion as the ideal efficiency, what would be the plant's efficiency in the winter, when the seawater is at $10.0^{\circ} \mathrm{C}$ ?

Sanjeev Kumar

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Problem 14

An electric power plant that would make use of the temperature gradient in the ocean has been proposed. 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 $\mathrm{km}) \cdot(\mathrm{a})$ What is the maximum efficiency of such a system?
(b) If the electric power output of the plant is $75.0 \mathrm{MW}$, how much energy is taken in from the warm reservoir per hour?
(c) In view of your answer to part (a), explain whether you think such a system is worthwhile. Note that the "fuel" is free.

Henrique Saito

Numerade Educator

Problem 15

Argon enters a turbine at a rate of $80.0 \mathrm{kg} / \mathrm{min},$ a temperature of $800^{\circ} \mathrm{C},$ and a pressure of $1.50 \mathrm{MPa}$. It expands adiabatically as it pushes on the turbine blades and exits at pressure 300 kPa.(a) Calculate its temperature at exit.
(b) Calculate the (maximum) power output of the turning turbine. (c) The turbine is one component of a model closed-cycle gas turbine engine. Calculate the maximum efficiency of the engine.

Henrique Saito

Numerade Educator

Problem 16

At point $A$ in a Carnot cycle, 2.34 mol of a monatomic ideal gas has a pressure of $1400 \mathrm{kPa}$, a volume of $10.0 \mathrm{L}$, and a temperature of $720 \mathrm{K}$. The gas expands isothermally to point $B$ and then expands adiabatically to point $C,$ where its volume is 24.0 L. An isothermal compression brings it to point $D$, where its volume is 15.0 L. An adiabatic process returns the gas to point $A$. (a) Determine all the unknown pressures, volumes, and temperatures as you fill in the following table:

Dominador Tan

Numerade Educator

Problem 17

A refrigerator has a coefficient of performance equal to$5.00 .$ The refrigerator takes in $120 \mathrm{J}$ of energy from a cold reservoir in each cycle. Find (a) the work required in each cycle and (b) the energy expelled to the hot reservoir.

Sanjeev Kumar

Numerade Educator

Problem 18

A refrigerator has a coefficient of performance of $3.00 .$ The ice tray compartment is at $-20.0^{\circ} \mathrm{C}$, and the room temperature is $22.0^{\circ} \mathrm{C} .$ The refrigerator can convert $30.0 \mathrm{g}$ of water at $22.0^{\circ} \mathrm{C}$ to $30.0 \mathrm{g}$ of ice at $-20.0^{\circ} \mathrm{C}$ each minute. What input power is required? Give your answer in watts.

Sanjeev Kumar

Numerade Educator

Problem 19

If a $35.0 \%$ -efficient Carnot heat engine (Active Fig. 18.1 ) is run in reverse so as to form a refrigerator (Active Fig. 18.7 ), what would be this refrigerator's coefficient of performance?

Keshav Singh

Numerade Educator

Problem 20

In $1993,$ the U.S. government instituted a requirement that all room air conditioners sold in the United States must have an energy efficiency ratio (EER) of 10 or higher. The EER is defined as the ratio of the cooling capacity of the air conditioner, measured in British thermal units per hour, or $\mathrm{Btu} / \mathrm{h},$ to its electrical power requirement in watts. (a) Convert the EER of 10.0 to dimensionless form, using the convert the EER of 10.0 to dimensionless form, using the conversion $1 \mathrm{Btu}=1055 \mathrm{J} .$ (b) What is the appropriate name for this dimensionless quantity? (c) In the 1970 s, it was common to find room air conditioners with EERs of 5 or lower. State how the operating costs compare for 10000 -Btu/h air conditioners with EERs of 5.00 and $10.0 .$ Assume each air
conditioner operates for 1500 h during the summer in a city where electricity costs $17.0 \not c$ per kWh.

Sanjeev Kumar

Numerade Educator

Problem 21

What is the coefficient of performance of a refrigerator that operates with Carnot efficiency between temperatures $-3.00^{\circ} \mathrm{C}$ and $+27.0^{\circ} \mathrm{C} ?$

Henrique Saito

Numerade Educator

Problem 22

An ideal refrigerator or ideal heat pump is equivalent to a Carnot engine running in reverse. That is, energy $\left|Q_{c}\right|$ is taken in from a cold reservoir and energy $\left|Q_{h}\right|$ is rejected to a hot reservoir. (a) Show that the work that must be supplied to run the refrigerator or heat pump is $$W=\frac{T_{h}-T_{c}}{T_{c}}\left|Q_{c}\right|$$ (b) Show that the coefficient of performance (COP) of the ideal refrigerator is $$\mathrm{COP}=\frac{T_{c}}{T_{h}-T_{c}}$$

Sanjeev Kumar

Numerade Educator

Problem 23

What is the maximum possible coefficient of performance of a heat pump that brings energy from outdoors at $-3.00^{\circ} \mathrm{C}$ into a $22.0^{\circ} \mathrm{C}$ house? Note: The work done to run the heat pump is also available to warm the house.

Henrique Saito

Numerade Educator

Problem 24

A heat pump has a coefficient of performance of 3.80 and operates with a power consumption of $7.03 \times 10^{5} \mathrm{W}$. (a) How much energy does it deliver into a home during $8.00 \mathrm{h}$ of continuous operation? (b) How much energy does it extract from the outside air?

Sanjeev Kumar

Numerade Educator

Problem 25

A heat pump used for heating shown in Figure P18.25 is essentially an air conditioner installed backward. It extracts energy from colder air outside and deposits it in a warmer room. Suppose the ratio of the actual energy entering the room to the work done by the device's motor is $10.0 \%$ of the theoretical maximum ratio. Determine the energy entering the room per joule of work done by the motor given that the inside temperature is $20.0^{\circ} \mathrm{C}$ and the outside temperature is $-5.00^{\circ} \mathrm{C}.$

Sanjeev Kumar

Numerade Educator

Problem 26

How much work does an ideal Carnot refrigerator require to remove $1.00 \mathrm{J}$ of energy from liquid helium at $4.00 \mathrm{K}$ and expel this energy to a room-temperature (293-K) environment?

Sanjeev Kumar

Numerade Educator

Problem 27

In making raspberry jelly, $900 \mathrm{g}$ of raspberry juice is combined with $930 \mathrm{g}$ of sugar. The mixture starts at room temperature, $23.0^{\circ} \mathrm{C},$ and is slowly heated on a stove until it reaches $220^{\circ} \mathrm{F}$. It is then poured into heated jars and allowed to cool. Assume that the juice has the same specific heat as water. The specific heat of sucrose is $0.299 \mathrm{cal} / \mathrm{g} \cdot^{\circ} \mathrm{C}$ Consider the heating process. (a) Which of the following terms describe(s) this process: adiabatic, isobaric, isothermal, isovolumetric, cyclic, reversible, isentropic?
(b) How much energy does the mixture absorb? (c) What is the minimum change in entropy of the jelly while it is heated?

Sanjeev Kumar

Numerade Educator

Problem 28

An ice tray contains 500 g of liquid water at $0^{\circ} \mathrm{C}$. Calculate the change in entropy of the water as it freezes slowly and completely at $0^{\circ} \mathrm{C}$.

Sanjeev Kumar

Numerade Educator

Problem 29

Calculate the change in entropy of 250 g of water warmed slowly from $20.0^{\circ} \mathrm{C}$ to $80.0^{\circ} \mathrm{C} .$ ( Suggestion: Note that $d Q=m c d T .)$

Sanjeev Kumar

Numerade Educator

Problem 30

What change in entropy occurs when a 27.9 -g ice cube at $-12^{\circ} \mathrm{C}$ is transformed into steam at $115^{\circ} \mathrm{C}$ ?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 31

Prepare a table like Table 18.1 by using the same procedure
(a) for the case in which you draw three marbles from your bag rather than four and (b) for the case in which you draw five marbles rather than four.

Sanjeev Kumar

Numerade Educator

Problem 32

(a) Prepare a table like Table 18.1 for the following occurrence. You toss four coins into the air simultaneously and then record the results of your tosses in terms of the numbers of heads $(\mathrm{H})$ and tails $(\mathrm{T})$ that result. For example, HHTH and HTHH are two possible ways in which three heads and one tail can be achieved. (b) On the basis of your table, what is the most probable result recorded for a toss? In terms of entropy, (c) what is the most ordered macrostate, and (d) what is the most disordered?

Sanjeev Kumar

Numerade Educator

Problem 33

When an aluminum bar is 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.

Sanjeev Kumar

Numerade Educator

Problem 34

When a metal bar is 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 $Q$. In this irreversible process, find expressions for the change in entropy of (a) the hot reservoir, (b) the cold reservoir, and (c) the Universe, neglecting any change in entropy of the metal rod.

Sanjeev Kumar

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Problem 35

If you roll two dice, what is the total number of ways in which you can obtain (a) a 12 and (b) a 7 ?

Sheh Lit Chang

University of Washington

Problem 36

The temperature at the surface of the Sun is approximately $5800 \mathrm{K},$ and the temperature at the surface of the Earth is approximately 290 K. What entropy change of the Universe occurs when $1.00 \times 10^{3} \mathrm{J}$ of energy is transferred by radiation from the Sun to the Earth?

Sanjeev Kumar

Numerade Educator

Problem 37

A $2.00-\mathrm{L}$ container has a center partition that divides it into two equal parts as shown in Figure $P 18.37 .$ The left side contains $0.0440 \mathrm{mol}$ of $\mathrm{H}_{2}$ gas, and the right side contains $0.0440 \mathrm{mol}$ of $\mathrm{O}_{2}$ gas. Both gases are at room temperature and at atmospheric pressure. The partition is removed, and the gases are allowed to mix. What is the entropy increase of the system?

Sanjeev Kumar

Numerade Educator

Problem 38

A 1.00 -kg iron horseshoe is taken from a forge at $900^{\circ} \mathrm{C}$ and dropped into $4.00 \mathrm{kg}$ of water at $10.0^{\circ} \mathrm{C}$. Assuming that no energy is lost by heat to the surroundings, determine the total entropy change of the horseshoe-plus-water system.

Vipender Yadav

Numerade Educator

Problem 39

A 1500 -kg car is moving at $20.0 \mathrm{m} / \mathrm{s}$. The driver brakes to
a stop. The brakes cool off to the temperature of the surrounding air, which is nearly constant at $20.0^{\circ} \mathrm{C}$. What is the total entropy change?

Sanjeev Kumar

Numerade Educator

Problem 40

How fast are you personally making the entropy of the Universe increase right now? Compute an order-of-magnitude estimate, stating what quantities you take as data and the values you measure or estimate for them.

Sanjeev Kumar

Numerade Educator

Problem 41

A 1.00 -mol sample of $\mathrm{H}_{2}$ gas is contained in the left side
of the container shown in Figure $P 18.41,$ which has equal volumes on the left and right. The right side is evacuated. When the valve is opened, the gas streams into the right side. (a) What is the entropy change of the gas? (b) Does the temperature of the gas change? Assume the container is so large that the hydrogen behaves as an ideal gas.

Sanjeev Kumar

Numerade Educator

Problem 42

We found the efficiency of the atmospheric heat engine to be about $0.8 \% .$ Taking the intensity of incoming solar radiation to be $1370 \mathrm{W} / \mathrm{m}^{2}$ and assuming that $64 \%$ of this energy is absorbed in the atmosphere, find the "wind power," that is, the rate at which energy becomes available for driving the winds.

Sanjeev Kumar

Numerade Educator

Problem 43

(a) Find the kinetic energy of the moving air in a hurricane, modeled as a disk $600 \mathrm{km}$ in diameter and $11 \mathrm{km}$ thick, with wind blowing at a uniform speed of $60 \mathrm{km} / \mathrm{h}$. (b) Consider sunlight with an intensity of $1000 \mathrm{W} / \mathrm{m}^{2}$ falling perpendicularly on a circular area $600 \mathrm{km}$ in diameter. During what time interval would the sunlight deliver the amount of energy computed in part (a)?

Sanjeev Kumar

Numerade Educator

Problem 44

A firebox is at $750 \mathrm{K}$, and the ambient temperature is $300 \mathrm{K}$. The efficiency of a Carnot engine doing $150 \mathrm{J}$ of work as it transports energy between these constanttemperature baths is $60.0 \% .$ The Carnot engine must take in energy $150 \mathrm{J} / 0.600=250 \mathrm{J}$ from the hot reservoir and must put out $100 \mathrm{J}$ of energy by heat into the environment. To follow Carnot's reasoning, suppose some other heat engine S could have an efficiency of $70.0 \%$. (a) Find the energy input and exhaust energy output of engine $\mathrm{S}$ as it does $150 \mathrm{J}$ of work. (b) Let engine $\mathrm{S}$ operate as in part (a) and run the Carnot engine in reverse between the same reservoirs. The output work of engine $\mathrm{S}$ is the input work for the Carnot refrigerator. Find the total energy transferred to or from the firebox and the total energy transferred to or from the environment as both engines operate together. (c) Explain how the results of parts (a) and (b) show that the Clausius statement of the second law of thermodynamics is violated. (d) Find the energy input and work output of engine $\mathrm{S}$ as it puts out exhaust energy of $100 \mathrm{J}$. Let engine $\mathrm{S}$ operate as in part
(c) and contribute $150 \mathrm{J}$ of its work output to running the Carnot engine in reverse. Find (e) the total energy the firebox puts out as both engines operate together,
(f) the total work output, and (g) the total energy transferred to the environment. (h) Explain how the results show that the Kelvin-Planck statement of the second law is violated. Therefore, our assumption about the efficiency of engine $S$ must be false. (i) Let the engines operate together through one cycle as in part (d). Find the change in entropy of the Universe. (j) Explain how the result of part (i) shows that the entropy statement of the second law is violated.

Sanjeev Kumar

Numerade Educator

Problem 45

Energy transfers by heat through the exterior walls and roof of a house at a rate of $5.00 \times 10^{3} \mathrm{J} / \mathrm{s}=5.00 \mathrm{kW}$
when the interior temperature is $22.0^{\circ} \mathrm{C}$ and the outside temperature is $-5.00^{\circ} \mathrm{C} .$ (a) Calculate the electric power required to maintain the interior temperature at $22.0^{\circ} \mathrm{C}$ if the power is used in electric resistance heaters that convert all the energy transferred in by electrical transmission into internal energy. (b) What If? Calculate the electric power required to maintain the interior temperature at $22.0^{\circ} \mathrm{C}$ if the power is used to drive an electric motor that operates the compressor of a heat pump that has a coefficient of performance equal to $60.0 \%$ of the Carnot-cycle value.

Sanjeev Kumar

Numerade Educator

Problem 46

Why is the following situation impossible? Two samples of water are mixed at constant pressure inside an insulated container: $1.00 \mathrm{kg}$ of water at $10.0^{\circ} \mathrm{C}$ and $1.00 \mathrm{kg}$ of water at $30.0^{\circ} \mathrm{C} .$ Because the container is insulated, there is no exchange of energy by heat between the water and the environment. Furthermore, the amount of energy that leaves the warm water by heat is equal to the amount that enters the cool water by heat. Therefore, the entropy change of the Universe is zero for this process.

Sanjeev Kumar

Numerade Educator

Problem 47

In $1816,$ Robert Stirling, a Scottish clergyman, patented the Stirling engine, which has found a wide variety of applications ever since. Fuel is burned externally to warm one of the engine's two cylinders. A fixed quantity of inert gas moves cyclically between the cylinders, expanding in the hot one and contracting in the cold one. Figure P18.47 represents a model for its thermodynamic cycle. Consider $n$ moles of an ideal monatomic gas being taken once through the cycle, consisting of two isothermal processes at temperatures $3 T_{i}$ and $T_{i}$ and two constant volume processes. Let us find the efficiency of this engine. (a) Find the energy transferred by heat into the gas during the isovolumetric process $A B .$ (b) Find the energy transferred by heat into the gas during the isothermal process $B C$. (c) Find the energy transferred by heat into the gas during the isovolumetric process $C D$
(d) Find the energy transferred by heat into the gas during the isothermal process $D A$. (e) Identify which of the results from parts (a) through (d) are positive and evaluate the energy input to the engine by heat. (f) From the first law of thermodynamics, find the work done by the engine. (g) From the results of parts (e) and (f), evaluate the efficiency of the engine. A Stirling engine is easier to manufacture than an internal combustion engine or a turbine. It can run on burning garbage. It can run on the energy transferred by sunlight and produce no material exhaust. Stirling engines are not currently used in automobiles due to long startup times and poor acceleration response.

Vipender Yadav

Numerade Educator

Problem 48

An idealized diesel engine operates in a cycle known as the airstandard diesel cycle shown
in Figure P18.48. Fuel is sprayed into the cylinder at the point of maximum compression, $B .$ Combustion occurs during the expansion $B \rightarrow C$, which is modeled as an isobaric process. Show that the efficiency of an engine operating in this idealized diesel cycle is $$e=1-\frac{1}{\gamma}\left(\frac{T_{D}-T_{A}}{T_{C}-T_{B}}\right)$$

Sanjeev Kumar

Numerade Educator

Problem 49

How much work is required, using an ideal Carnot refrigerator, to change $0.500 \mathrm{kg}$ of tap water at $10.0^{\circ} \mathrm{C}$ into ice at $-20.0^{\circ} \mathrm{C} ?$ Assume that the freezer compartment is held at $-20.0^{\circ} \mathrm{C}$ and that the refrigerator exhausts energy into a room at $20.0^{\circ} \mathrm{C}$

Sanjeev Kumar

Numerade Educator

Problem 50

A heat engine operates between two reservoirs at $T_{2}=600 \mathrm{K}$ and $T_{1}=350 \mathrm{K}$. It takes in $1.00 \times 10^{3} \mathrm{J}$ of
energy from the higher-temperature reservoir and performs $250 \mathrm{J}$ of work. Find (a) the entropy change of the Universe $\Delta S_{U}$ for this process and (b) the work $W$ that could have been done by an ideal Carnot engine operating between these two reservoirs. (c) Show that the difference between the amounts of work done in parts (a) and
(b) is $T_{1} \Delta S_{U}$

Sanjeev Kumar

Numerade Educator

Problem 51

This problem complements Problem 20 in Chapter $10 .$ In the operation of a single-cylinder internal combustion piston engine, one charge of fuel explodes to drive the piston outward in the power stroke. Part of its energy output is stored in a turning flywheel. This energy is then used to push the piston inward to compress the next charge of fuel and air. In this compression process, assume an original volume of $0.120 \mathrm{L}$ of a diatomic ideal gas at atmospheric pressure is compressed adiabatically to one-eighth of its original volume. (a) Find the work input required to compress the gas. (b) Assume the flywheel is a solid disk of mass $5.10 \mathrm{kg}$ and radius $8.50 \mathrm{cm},$ turning freely without friction between the power stroke and the compression stroke. How fast must the flywheel turn immediately after the power stroke? This situation represents the minimum angular speed at which the engine can operate without stalling. (c) When the engine's operation is well above the point of stalling, assume the flywheel puts $5.00 \%$ of its maximum energy into compressing the next charge of fuel and air. Find its maxi-
mum angular speed in this case.

Sanjeev Kumar

Numerade Educator

Problem 52

On the $P V$ diagram for an ideal gas, one isothermal curve and one adiabatic curve pass through each point as shown in Figure P18.52. Prove that the slope of the adiabatic curve is steeper than the slope of the isotherm at that point by the factor $\gamma$

Sanjeev Kumar

Numerade Educator

Problem 53

A power plant, having a Carnot efficiency, produces $1.00 \mathrm{GW}$ of electrical power from turbines that take in steam at $500 \mathrm{K}$ and reject water at $300 \mathrm{K}$ into a flowing river. The water downstream is $6.00 \mathrm{K}$ warmer due to the output of the power plant. Determine the flow rate of the river.

Sanjeev Kumar

Numerade Educator

Problem 54

A power plant, having a Carnot efficiency, produces electric power $P$ from turbines that take in energy from steam at temperature $T_{h}$ and discharge energy at temperature $T_{c}$ through a heat exchanger into a flowing river. The water downstream is warmer by $\Delta$ T due to the output of the power plant. Determine the flow rate of the river.

Sanjeev Kumar

Numerade Educator

Problem 55

A biology laboratory is maintained at a constant temperature of $7.00^{\circ} \mathrm{C}$ by an air conditioner, which is vented to the air outside. On a typical hot summer day, the outside temperature is $27.0^{\circ} \mathrm{C}$ and the air-conditioning unit emits energy to the outside at a rate of $10.0 \mathrm{kW}$. Model the unit as having a coefficient of performance (COP) equal to $40.0 \%$ of the COP of an ideal Carnot device. (a) At what rate does the air conditioner remove energy from the laboratory? (b) Calculate the power required for the work input.
(c) Find the change in entropy of the Universe produced by the air conditioner in 1.00 h. (d) What If? The outside temperature increases to $32.0^{\circ} \mathrm{C}$. Find the fractional change in the COP of the air conditioner.

Sanjeev Kumar

Numerade Educator

Problem 56

An athlete whose mass is 70.0 kg drinks 16.0 ounces $(454 \mathrm{g})$ of refrigerated water. The water is at a temperature of $35.0^{\circ} \mathrm{F}$. (a) Ignoring the temperature change of the body that results from the water intake (so that the body is regarded as a reservoir always at $98.6^{\circ} \mathrm{F}$ ), find the entropy increase of the entire system. (b) What If? Assume the entire body is cooled by the drink and the average specific heat of a person is equal to the specific heat of liquid water. Ignoring any other energy transfers by heat and any metabolic energy release, find the athlete's temperature after she drinks the cold water given an initial body temperature of $98.6^{\circ} \mathrm{F}$. (c) Under these assumptions, what is the entropy increase of the entire system? (d) State how this result compares with the one you obtained in part (a).

Sanjeev Kumar

Numerade Educator

Problem 57

An ideal (Carnot) freezer in a kitchen has a constant temperature of $260 \mathrm{K}$, whereas the air in the kitchen has a constant temperature of $300 \mathrm{K}$. Suppose the insulation for the freezer is not perfect but rather conducts energy into the freezer at a rate of $0.150 \mathrm{W}$. Determine the average power required for the freezer's motor to maintain the constant temperature in the freezer.

Sanjeev Kumar

Numerade Educator

Problem 58

A 1.00 -mol sample of a monatomic ideal gas is taken through the cycle shown in Figure $\mathrm{P} 18.58$ At point $A,$ the pressure, volume, and temperature $\operatorname{are} P_{i}, V_{i},$ and $T_{i},$ respectively. In terms of $R$ and $T_{i},$ find $(\mathrm{a})$ the total en $-$ ergy entering the system by heat per cycle, (b) the total energy leaving the system by heat per cycle, and (c) the efficiency of an engine operating in this cycle.
(d) Explain how the efficiency compares with that of an engine operating in a Carnot cycle between the same temperature extremes.

Sanjeev Kumar

Numerade Educator

Problem 59

Calculate the increase in entropy of the Universe when you add $20.0 \mathrm{g}$ of $5.00^{\circ} \mathrm{C}$ cream to $200 \mathrm{g}$ of $60.0^{\circ} \mathrm{C}$ coffee. Assume that the specific heats of cream and coffee are both $4.20 \mathrm{J} / \mathrm{g} \cdot^{\circ} \mathrm{C}$

Sanjeev Kumar

Numerade Educator

Problem 60

A sample consisting of $n$ moles of an ideal gas undergoes a reversible isobaric expansion from volume $V_{i}$ to volume $3 V_{i}$. Find the change in entropy of the gas by calculating $\int_{i}^{f} \frac{d Q}{T},$ where $d Q=n C_{P} d T.$

Sanjeev Kumar

Numerade Educator

Problem 61

A 1.00 -mol sample of an ideal gas $(\gamma=1.40)$ is carried through the Carnot cycle described in Active Figure 18.6 At point $A,$ the pressure is 25.0 atm and the temperature is $600 \mathrm{K} .$ At point $C,$ the pressure is $1.00 \mathrm{atm}$ and the temperature is $400 \mathrm{K}$. (a) Determine the pressures and volumes at points $A, B, C,$ and $D .$ (b) Calculate the net work done per cycle.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 62

A system consisting of $n$ moles of an ideal gas with molar specific heat at constant pressure $C_{P}$ undergoes two reversible processes. It starts with pressure $P_{i}$ and volume $V_{i},$ expands isothermally, and then contracts adiabatically to reach a final state with pressure $P_{i}$ and volume $3 V_{i} .$ (a) Find its change in entropy in the isothermal process. (The entropy does not change in the adiabatic process.) (b) What If ? Explain why the answer to part (a) must be the same as the answer to Problem $60 .$ (You do not need to solve Problem 60 to answer this question.)

David Morabito

Numerade Educator

Problem 63

A 1.00 -mol sample of an ideal monatomic gas is taken through the cycle shown in Figure $P 18.63$ The process $A \rightarrow B$ is a reversible isothermal expansion. Calculate (a) the net work done by the gas, (b) the energy added to the gas by heat, (c) the energy exhausted from the gas by heat, and (d) the efficiency of the cycle. (e) Explain how the efficiency compares with that of a Carnot engine operating between the same temperature extremes.

Sanjeev Kumar

Numerade Educator

Problem 64

The Ottocycle in Figure $\mathrm{P} 18.64$ models the operation of the internal combustion engine in an automobile. A mixture of gasoline vapor and air is drawn into a cylinder as the piston moves down during the intake stroke $O \rightarrow A$ The piston moves up toward the closed end of the cylinder to compress the mixture adiabatically in process $A \rightarrow$ $B .$ The ratio $r=V_{1} / V_{2}$ is the compression ratio of the engine. At $B$, the gasoline is ignited by the spark plug and the pressure rises rapidly as it burns in process $B \rightarrow C$. In the power stroke $C \rightarrow D$, the combustion products expand adiabatically as they drive the piston down. The combustion products cool further in an isovolumetric process $D \rightarrow A$ and in the exhaust stroke $A \rightarrow O,$ when the exhaust gases are pushed out of the cylinder. Assume that a single value of the specific heat ratio characterizes both the fuel-air mixture and the exhaust gases after combustion. Prove that the efficiency of the engine is $1-r^{1-\gamma}.$

Sanjeev Kumar

Numerade Educator

Problem 65

Every second at Niagara Falls, some $5.00 \times 10^{3} \mathrm{m}^{3}$ of water falls a distance of $50.0 \mathrm{m}$. What is the increase in entropy of the Universe 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 $20.0^{\circ} \mathrm{C}$. Also assume a negligible amount of water evaporates.

Sanjeev Kumar

Numerade Educator