Solved: An ideal gas expands at a constant total pressure

An ideal gas expands isothermally, performing of work in the process. Calculate (a) the change in internal energy of the gas, and (b) the heat absorbed during this expansion.

One liter of air is cooled at constant pressure until its volume is halved, and then it is allowed to expand isothermally back to its original volume. Draw the process on a PV diagram

Sketch a PV diagram of the following process: 2.5 L of ideal gas at atmospheric pressure is cooled at constant pressure to a volume of 1.0 L, and then expanded isothermally back to 2.5 L, whereupon the pressure is increased at constant volume until the original pressure is reached.

(II) A gas is enclosed in a cylinder fitted with a light frictionless piston and maintained at atmospheric pressure. When 254 kcal of heat is added to the gas, the volume is observed to increase slowly from \(12.0~ \mathrm m^3\) to \(16.2~ \mathrm m^3\). Calculate \((a)\) the work done by the gas and \((b)\) the change in internal energy of the gas. Equation Transcription: Text Transcription: 12.0 m^3 16.2 m^3

A 1.0-L volume of air initially at 3.5 atm of (gauge) pressure is allowed to expand isothermally until the pressure is 1.0 atm. It is then compressed at constant pressure to its initial volume, and lastly is brought back to its original pressure by heating at constant volume. Draw the process on a PV diagram, including numbers and labels for the axes.

The pressure in an ideal gas is cut in half slowly, while being kept in a container with rigid walls. In the process, 465 kJ of heat left the gas. (a) How much work was done during this process? (b) What was the change in internal energy of the gas during this process?

In an engine, an almost ideal gas is compressed adiabatically to half its volume. In doing so, 2630 J of work is done on the gas. (a) How much heat flows into or out of the gas? (b) What is the change in internal energy of the gas? (c) Does its temperature rise or fall?

An ideal gas expands at a constant total pressure of 3.0 atm from 410 mL to 690 mL. Heat then flows out of the gas at constant volume, and the pressure and temperature are allowed to drop until the temperature reaches its original value. Calculate (a) the total work done by the gas in the process, and (b) the total heat flow into the gas

8.5 moles of an ideal monatomic gas expand adiabatically, performing 8300 J of work in the process. What is the change in temperature of the gas during this expansion?

Consider the following two-step process. Heat is allowed to flow out of an ideal gas at constant volume so that its pressure drops from 2.2 atm to 1.4 atm. Then the gas expands at constant pressure, from a volume of 5.9 L to 9.3 L, where the temperature reaches its original value. See Fig.15–22. Calculate \((a)\) the total work done by the gas in the process, \((b)\) the change in internal energy of the gas in the process, and \((c)\) the total heat flow into or out of the gas.

Use the conservation of energy to explain why the temperature of a well-insulated gas increases when it is compressed—say, by pushing down on a piston—whereas the temperature decreases when the gas expands. Show your reasoning.

The PV diagram in Fig. 15–23 shows two possible states of a system containing 1.75 moles of a monatomic ideal gas. (\(P_1=P_2=425 ~\mathrm {N/m^2},~ V_1=2.00~ \mathrm m^3,~ V_2=8.00~ \mathrm m^3\).) \((a)\) Draw the process which depicts an isobaric expansion from state 1 to state 2, and label this process A. \((b)\) Find the work done by the gas and the change in internal energy of the gas in process A. \((c)\) Draw the two-step process which depicts an isothermal expansion from state 1 to the volume \(V_2\) followed by an isovolumetric increase in temperature to state 2, and label this process B. \((d)\) Find the change in internal energy of the gas for the two-step process B. Equation Transcription: Text Transcription: P_1=P_2=425 N/m^2, V_1=2.00 m^3, V_2=8.00 m^3 V_2

When a gas is taken from a to c along the curved path in Fig. 15–24, the work done by the gas is \(W=-35~\mathrm J\) and the heat added to the gas is \(Q=-175~\mathrm J\). Along path abc, the work done by the gas is \(W=-56~\mathrm J\) (That is, 56 J of work is done on the gas.) \((a)\) What is \(Q\) for path abc? \((b)\) If \(P_\mathrm c=\frac{1}{2}P_\mathrm b\), what is \(W\) for path cda? \((c)\) What is \(Q\) for path cda? \((d)\) What is \(U_\mathrm a-U_\mathrm c\)? \((e)\) If \(U_\mathrm d-U_\mathrm c=42~\mathrm J\), what is \(Q\) for path da? Equation Transcription: Text Transcription: W=-35 J Q=-175 J W=-56 J P_c=frac{1}{2}P_b U_a-U_c U_d-U_c=42 J

(I) How much energy would the person of Example 15–7 transform if instead of working 11.0 h she took a noontime break and ran at 15 km/h for 1.0 h?

(I) Calculate the average metabolic rate of a 65-kg person who sleeps 8.0 h, sits at a desk 6.0 h, engages in light activity 6.0 h, watches TV 2.0 h, plays tennis 1.5 h, and runs 0.50 h daily.

A 65-kg person decides to lose weight by sleeping one hour less per day, using the time for light activity. How much weight (or mass) can this person expect to lose in 1 year, assuming no change in food intake? Assume that 1 kg of fat stores about 40,000 kJ of energy.

(II) \((a)\) How much energy is transformed by a typical 65-kg person who runs at 15 km/h for 30 min/day in one week (Table 15–2)? \((b)\) How many food calories would the person have to eat to make up for this energy loss?

A heat engine exhausts 8200 J of heat while performing 2600 J of useful work. What is the efficiency of this engine?

(I) What is the maximum efficiency of a heat engine whose operating temperatures are \(560^\circ\mathrm C\) and \(345^\circ\mathrm C\)? Equation Transcription: Text Transcription: 560^oC 345^oC

(I) The exhaust temperature of a heat engine is \(230^\circ \mathrm C\). What is the high temperature if the Carnot efficiency is 34%? Equation Transcription: Text Transcription: 230^oC

A heat engine does 9200 J of work per cycle while absorbing 25.0 kcal of heat from a high- temperature reservoir. What is the efficiency of this engine?

A heat engine’s high temperature \(T_\mathrm H\) could be ambient temperature, because liquid nitrogen at 77 K could be \(T_\mathrm L\) and is cheap. What would be the efficiency of a Carnot engine that made use of heat transferred from air at room temperature (293 K) to the liquid nitrogen “fuel”(Fig.15–25)? Equation Transcription: Text Transcription: T_H T_L

Which will improve the efficiency of a Carnot engine more: a \(10~ \mathrm C^\circ\) increase in the high-temperature reservoir, or a \(10~ \mathrm C^\circ\) decrease in the low-temperature reservoir? Give detailed results. Can you state a generalization? Equation Transcription: Text Transcription: 10C^o 10C^o

A certain power plant puts out 580 MW of electric power. Estimate the heat discharged per second, assuming that the plant has an efficiency of 32%

A nuclear power plant operates at 65% of its maximum theoretical (Carnot) efficiency between temperatures of \(660^\circ \mathrm C\) and \(330^\circ \mathrm C\). If the plant produces electric energy at the rate of 1.4 GW, how much exhaust heat is discharged per hour? Equation Transcription: Text Transcription: 660^oC 330^oC

A heat engine exhausts its heat at \(340^\circ \mathrm C\) and has a Carnot efficiency of 36%. What exhaust temperature would enable it to achieve a Carnot efficiency of 42%? Equation Transcription: Text Transcription: 340^oC

(II) A Carnot engine’s operating temperatures are \(210^{\circ}C\) and \(45^{\circ}C\). The engine’s power output is 910W. Calculate the rate of heat output.

A four-cylinder gasoline engine has an efficiency of 0.22 and delivers 180 J of work per cycle per cylinder. If the engine runs at 25 cycles per second (1500 rpm), determine (a) the work done per second, and (b) the total heat input per second from the gasoline. (c) If the energy content of gasoline is 130 MJ per gallon, how long does one gallon last?

A Carnot engine performs work at the rate of 520 kW with an input of 950 kcal of heat per second. If the temperature of the heat source is \(520^\circ \mathrm C\), at what temperature is the waste heat exhausted? Equation Transcription: Text Transcription: 520^oC

(II) A heat engine uses a heat source at \(580^{\circ}C\) and has an ideal (Carnot) efficiency of 22%. To increase the ideal efficiency to 42%, what must be the temperature of the heat source?

A typical compact car experiences a total drag force of about 350 N at 55 mi/h. If this car gets 32 miles per gallon of gasoline at this speed, and a liter of gasoline (1 gal = 3.8 L) releases about \(3.2 \times 10^7~\mathrm J\) when burned, what is the car’s efficiency? Equation Transcription: Text Transcription: 3.2 x 10^7 J

If an ideal refrigerator keeps its contents at \(2.5^\circ \mathrm C\) when the house temperature is \(22^\circ \mathrm C\), what is its COP? Equation Transcription: Text Transcription: 2.5^oC 22^oC

The low temperature of a freezer cooling coil is \(-8^\circ \mathrm C\) and the discharge temperature is \(33^\circ \mathrm C\). What is the maximum theoretical coefficient of performance? Equation Transcription: Text Transcription: -8^oC 33^oC

What is the temperature inside an ideal refrigerator freezer that operates with a COP=7.0 in a \(22^\circ \mathrm C\) room?

A heat pump is used to keep a house warm at \(22^\circ \mathrm C\). How much work is required of the pump to deliver 3100 J of heat into the house if the outdoor temperature is \((a)~0^\circ \mathrm C\), \((b)~-15^\circ \mathrm C\)? Assume a COP of 3.0. \((c)\) Redo for both temperatures, assuming an ideal (Carnot) coefficient of performance \(\mathrm{COP}=T_\mathrm H/(T_\mathrm H-T_\mathrm L)\). Equation Transcription: Text Transcription: 22^oC 0^oC -15^oC COP=T_H/(T_H-T_L)

\((a)\) What is the coefficient of performance of an ideal heat pump that extracts heat from \(6^\circ \mathrm C\) air outside and deposits heat inside a house at \(24^\circ \mathrm C\)? \((b)\) If this heat pump operates on 1200 W of electrical power, what is the maximum heat it can deliver into the house each hour? See Problem 35. Equation Transcription: Text Transcription: 6^oC 24^oC

(II) What volume of water at \(0^{\circ}C\) can a freezer make into ice cubes in 1.0 h, if the coefficient of performance of the cooling unit is 6.0 and the power input is 1.2 kilowatt?

How much less per year would it cost a family to operate a heat pump that has a coefficient of performance of 2.9 than an electric heater that costs $2000 to heat their home for a year? If the conversion to the heat pump costs $15,000, how long would it take the family to break even on heating costs? How much would the family save in 20 years?

(I) What is the change in entropy of 320 g of steam at \(100^{\circ}C\) when it is condensed to water at \(100^{\circ}C\)?

1.0 kg of water is heated from \(\0^\circ\mathrm C) to \(100^\circ\mathrm C\). Estimate the change in entropy of the water. Equation Transcription: Text Transcription: 0^oC 100^oC

What is the change in entropy of \(1.00~ \mathrm m^3\) of water at \(0^\circ \mathrm C\) when it is frozen to ice at \(0^\circ \mathrm C\)? Equation Transcription: Text Transcription: 1.00 m^3 0^oC 0^oC

(II) A 5.8-kg box having an initial speed of 4.0 m/s slides along a rough table and comes to rest. Estimate the total change in entropy of the universe. Assume all objects are at room temperature (293 K).

(II) If \(1.00~ \mathrm m^3\) of water at \(0^\circ \mathrm C\) is frozen and cooled to \(-8.0^\circ \mathrm C\) by being in contact with a great deal of ice at \(-8.0^\circ \mathrm C\), estimate the total change in entropy of the process. Equation Transcription: Text Transcription: 1.00 m^3 0^oC -8.0^oC -8.0^oC

(II) An aluminum rod conducts 8.40 cal/s from a heat source maintained at \(225^\circ \mathrm C\) to a large body of water at \(22^\circ \mathrm C\). Calculate the rate at which entropy increases in this process.

(II) A 2.8-kg piece of aluminum at \(28.5^\circ \mathrm C\) is placed in 1.0 kg of water in a Styrofoam container at room temperature \((20.0^\circ \mathrm C)\). Estimate the net change in entropy of the system. Equation Transcription: Text Transcription: 28.5^oC (20.0^oC)

A falling rock has kinetic energy KE just before striking the ground and coming to rest. What is the total change in entropy of rock plus environment as a result of this collision?

1.0 kg of water at \(35^\circ \mathrm C\) is mixed with 1.0 kg of water at \(45^\circ \mathrm C\) in a well-insulated container. Estimate the net change in entropy of the system. Equation Transcription: Text Transcription: 35^oC 45^oC

A real heat engine working between heat reservoirs at 970 K and 650 K produces 550 J of work per cycle for a heat input of 2500 J. (a) Compare the efficiency of this real engine to that of an ideal (Carnot) engine. (b) Calculate the total entropy change of the universe per cycle of the real engine, and (c) also if the engine is ideal (Carnot).

Calculate the probabilities, when you throw two dice, of obtaining (a) a 4, and (b) a 10

Suppose that you repeatedly shake six coins in your hand and drop them on the floor. Construct a table showing the number of microstates that correspond to each macrostate. What is the probability of obtaining (a) three heads and three tails, and (b) six heads?

A bowl contains many red, orange, and green jelly beans, in equal numbers. You are to make a line of 3 jelly beans by randomly taking 3 beans from the bowl. (a) Construct a table showing the number of microstates that correspond to each macrostate. Then determine the probability of (b) all 3 beans red, and (c) 2 greens, 1 orange

Rank the following five-card hands in order of increasing probability: (a) four aces and a king; (b) six of hearts, eight of diamonds, queen of clubs, three of hearts, jack of spades; (c) two jacks, two queens, and an ace; and (d) any hand having no two equal-value cards (no pairs, etc.). Discuss your ranking in terms of microstates and macrostate

Solar cells (Fig. 15–26) can produce about 40 W of electricity per square meter of surface area if directly facing the Sun. How large an area is required to supply the needs of a house that requires 24 kWh/day? Would this fit on the roof of an average house? (Assume the Sun shines about 9 h/day.)

(II) Energy may be stored by pumping water to a high reservoir when demand is low and then releasing it to drive turbines during peak demand. Suppose water is pumped to a lake 115 m above the turbines at a rate of \(1.00 \times 10^5~ \mathrm {kg/s}\) for 10.0 h at night. \((a)\) How much energy (kWh) is needed to do this each night? \((b)\) If all this energy is released during a 14-h day, at 75% efficiency, what is the average power output? Equation Transcription: Text Transcription: 1.00 x 10^5 kg/s

Water is stored in an artificial lake created by a dam (Fig. 15–27). The water depth is 48 m at the dam, and a steady flow rate of \(32~ \mathrm {m^3/s}\) is maintained through hydroelectric turbines installed near the base of the dam. How much electrical power can be produced? Equation Transcription: Text Transcription: 32 m^3/s

An inventor claims to have built an engine that produces 2.00 MW of usable work while taking in 3.00 MW of thermal energy at 425 K, and rejecting 1.00 MW of thermal energy at 215 K. Is there anything fishy about his claim? Explain

When \(5.80 \times 10^5~ \mathrm J\) of heat is added to a gas enclosed in a cylinder fitted with a light frictionless piston maintained at atmospheric pressure, the volume is observed to increase from \(\mathrm {1.9~ m^3}\) to \(\mathrm {4.1~ m^3}\). Calculate \((a)\) the work done by the gas, and \((b)\) the change in internal energy of the gas. \((c)\) Graph this process on a \(PV\) diagram. Equation Transcription: Text Transcription: 5.80 x 10^5 J 1.9 m^3 4.1 m^3

A restaurant refrigerator has a coefficient of performance of 4.6. If the temperature in the kitchen outside the refrigerator is \(\mathrm {32^\circ C}\), what is the lowest temperature that could be obtained inside the refrigerator if it were ideal? Equation Transcription: Text Transcription: 32^oC

A particular car does work at the rate of about 7.0 kJ/s when traveling at a steady 21.8 m/s along a level road. This is the work done against friction. The car can travel 17 km on 1.0 L of gasoline at this speed (about 40 mi/gal). What is the minimum value for \(T_\mathrm H\) if \(T_\mathrm L\) is \(25^\circ \mathrm C\)? The energy available from 1.0 L of gas is \(3.2 \times 10^7~ \mathrm J\). Equation Transcription: Text Transcription: T_H T_L 25^oC 3.2 x 10^7 J

A “Carnot” refrigerator (the reverse of a Carnot engine) absorbs heat from the freezer compartment at a temperature of \(\mathrm {-17^\circ C}\) and exhausts it into the room at \(\mathrm {25^\circ C}\). (a) How much work would the refrigerator do to change 0.65 kg of water at \(\mathrm {25^\circ C}\) into ice at \(\mathrm {-17^\circ C}\)? (b) If the compressor output is 105 W and runs 25% of the time, how long will this take? Equation Transcription: Text Transcription: -17^oC 25^oC 25^oC -17^oC

It has been suggested that a heat engine could be developed that made use of the temperature difference between water at the surface of the ocean and water several hundred meters deep. In the tropics, the temperatures may be \(\mathrm {27^\circ C}\) and \(\mathrm {4^\circ C}\), respectively. \((a)\) What is the maximum efficiency such an engine could have? \((b)\) Why might such an engine be feasible in spite of the low efficiency? \((c)\) Can you imagine any adverse environmental effects that might occur? Equation Transcription: Text Transcription: 27^oC 4^oC

A cooling unit for a new freezer has an inner surface area of \(\mathrm {8.0~ m^2}\) and is bounded by walls 12 cm thick with a thermal conductivity of \(\mathrm {0.050~ W/m \cdot K}\). The inside must be kept at \(\mathrm {-15^\circ C}\) in a room that is at \(\mathrm {22^\circ C}\). The motor for the cooling unit must run no more than 15% of the time. What is the minimum power requirement of the cooling motor? Equation Transcription: Text Transcription: 8.0 m^2 0.050 W/m cdot K -15^oC 22^oC

Refrigeration units can be rated in “tons.” A 1-ton air conditioning system can remove sufficient energy to freeze 1 ton (2000 pounds = 909 kg) of \(\mathrm {0^\circ C}\) water into \(\mathrm {0^\circ C}\) ice in one 24-h day. If, on a \(\mathrm {35^\circ C}\) day, the interior of a house is maintained at \(\mathrm {22^\circ C}\) by the continuous operation of a 5-ton air conditioning system, how much does this cooling cost the homeowner per hour? Assume the work done by the refrigeration unit is powered by electricity that costs $0.10 per kWh and that the unit’s coefficient of performance is 18% that of an ideal refrigerator. \(\mathrm {1~ kWh=3.60 \times 10^6~ J}\) ________________ Equation Transcription: Text Transcription: 0^oC 0^oC 35^oC 22^oC 1 kWh=3.60 x 10^6 J

Two 1100-kg cars are traveling 85 km/h in opposite directions when they collide and are brought to rest. Estimate the change in entropy of the universe as a result of this collision. Assume \(T=20^\circ \mathrm C\).

A 110-g insulated aluminum cup at \(35^{\circ}C\) is filled with 150 g of water at \(45^{\circ}C\). After a few minutes, equilibrium is reached. (a) Determine the final temperature, and (b) estimate the total change in entropy.

The burning of gasoline in a car releases about \(\mathrm {3.0 \times 10^4~ kcal/gal}\). If a car averages 41 km/gal when driving 110 km/h which requires 25 hp, what is the efficiency of the engine under those conditions? Equation Transcription: Text Transcription: 3.0 x 10^4 kcal/gal

A Carnot engine operates with \(T_\mathrm {L}=20^\circ \mathrm C\) and has an efficiency of 25%. By how many kelvins should the high operating temperature \(T_\mathrm {H}\) be increased to achieve an efficiency of 35%? Equation Transcription: Text Transcription: T_L=20^oC T_H

Calculate the work done by an ideal gas in going from state A to state C in Fig. 15–28 for each of the following processes: \((a)\) ADC, \((b)\) ABC, and \((c)\) AC directly. ________________ Equation Transcription: Text Transcription: P_C P_A V_A V_C

A 38% efficient power plant puts out 850 MW of electrical power. Cooling towers take away the exhaust heat. (a) If the air temperature is allowed to rise \(7.0 C^{\circ}\), estimate what volume of air \((km^3)\) is heated per day. Will the local climate be heated significantly? (b) If the heated air were to form a layer 180 m thick, estimate how large an area it would cover for 24 h of operation. Assume the air has density \(1.3 kg/m^3\) and has specific heat of about \(1.0 kJ/kg \cdot C^{\circ}\) at constant pressure.

Suppose a power plant delivers energy at 880 MW using steam turbines. The steam goes into the turbines superheated at 625 K and deposits its unused heat in river water at 285 K. Assume that the turbine operates as an ideal Carnot engine. \((a)\) If the river flow rate is \(\mathrm {37~ m^3/s}\), estimate the average temperature increase of the river water immediately downstream from the power plant. \((b)\) What is the entropy increase per kilogram of the downstream river water in \(\mathrm {J/kg \cdot K}\)? Equation Transcription: Text Transcription: 37 m^3/s J/kg cdot K

A car engine whose output power is 135 hp operates at about 15% efficiency. Assume the engine’s water temperature of \(\mathrm {85^\circ C}\) is its cold-temperature (exhaust) reservoir and \(\mathrm {495^\circ C}\) is its thermal “intake” temperature (the temperature of the exploding gas–air mixture). \((a)\) What is the ratio of its efficiency relative to its maximum possible (Carnot) efficiency? \((b)\) Estimate how much power (in watts) goes into moving the car, and how much heat, in joules and in kcal, is exhausted to the air in 1.0 h. ________________ Equation Transcription: Text Transcription: 85^oC 495^oC

An ideal monatomic gas is contained in a tall cylindrical jar of cross-sectional area \(\mathrm {0.080~ m^2}\) fitted with an airtight frictionless 0.15-kg movable piston. When the gas is heated (at constant pressure) from \(\mathrm {25^\circ C}\) to \(\mathrm {55^\circ C}\), the piston rises 1.0 cm. How much heat was required for this process? Assume atmospheric pressure outside. [\(Hint\): See Section 14–2.] Equation Transcription: Text Transcription: 0.080 m^2 25^oC 55^oC

Metabolizing 1.0 kg of fat results in about \(\mathrm {3.7 \times 10^7~ J}\) of internal energy in the body. \((a)\) In one day, how much fat does the body burn to maintain the body temperature of a person staying in bed and metabolizing at an average rate of 95 W? \((b)\) How long would it take to burn 1.0 kg of fat this way assuming there is no food intake? Equation Transcription: Text Transcription: 3.7 x 10^7 J

\((a)\) At a steam power plant, steam engines work in pairs, the heat output of the first one being the approximate heat input of the second. The operating temperatures of the first are \(\mathrm {750^\circ C}\) and \(\mathrm {440^\circ C}\), and of the second \(\mathrm {415^\circ C}\)and 270°C. If the heat of combustion of coal is \(\mathrm {2.8 \times 10^7~J/kg}\), at what rate must coal be burned if the plant is to put out 950 MW of power? Assume the efficiency of the engines is 65% of the ideal (Carnot) efficiency. \((b)\) Water is used to cool the power plant. If the water temperature is allowed to increase by no more than \(\mathrm {4.5^\circ C}\), estimate how much water must pass through the plant per hour. ________________ Equation Transcription: Text Transcription: 750^oC 440^oC 415^oC 270^oC 2.8 x 10^7J/kg 4.5^oC

Suppose a heat pump has a stationary bicycle attachment that allows \(you\) to provide the work instead of using an electrical wall outlet. If your heat pump has a coefficient of performance of 2.0 and you can cycle at a racing pace (Table 15–2) for a half hour, how much heat can you provide?

An ideal air conditioner keeps the temperature inside a room at \(\mathrm {21^\circ C}\) when the outside temperature is \(\mathrm {32^\circ C}\). If 4.8 kW of power enters a room through the windows in the form of direct radiation from the Sun, how much electrical power would be saved if the windows were shaded so only 500 W enters? Equation Transcription: Text Transcription: 21^oC 32^oC

An ideal heat pump is used to maintain the inside temperature of a house at \(T_\mathrm {in}=22^\circ \mathrm C\) when the outside temperature is \(T_\mathrm {out}\). Assume that when it is operating, the heat pump does work at a rate of 1500 W. Also assume that the house loses heat via conduction through its walls and other surfaces at a rate given by \((650~\mathrm {W/C^\circ})(T_\mathrm {in}-T_\mathrm {out})\). \((a)\) For what outside temperature would the heat pump have to operate all the time in order to maintain the house at an inside temperature of \(\mathrm {22^\circ C}\)? \((b)\) If the outside temperature is \(\mathrm {8^\circ C}\), what percentage of the time does the heat pump have to operate in order to maintain the house at an inside temperature of \(\mathrm {22^\circ C}\)? ________________ Equation Transcription: Text Transcription: T_in=22^oC T_out (650 W/C^o)(T_in-T_out) 22^oC 8^oC 22^oC

Problem 1COQ Fossil-fuel electric generating plants produce “thermal pollution.” Part of the heat produced by the burning fuel is not converted to electric energy. The reason for this waste is (a) the efficiency is higher if some heat is allowed to escape. (b) engineering technology has not yet reached the point where 100% waste heat recovery is possible. (c) some waste heat must be produced: this is a fundamental property of nature when converting heat to useful work. (d) the plants rely on fossil fuels, not nuclear fuel. (e) None of the above.

In an isobaric compression of an ideal gas, (a) no heat flows into the gas. (b) the internal energy of the gas remains constant. (c) no work is done on the gas. (d) work is done on the gas. (e) work is done by the gas.

Problem 1P (I) An ideal gas expands isothermally, performing 4.30 X 103 J of work in the process. Calculate (a) the change in internal energy of the gas, and (b) the heat absorbed during this expansion.

Problem 1Q In an isothermal process, 3700 J of work is done by an ideal gas. Is this enough information to tell how much heat has been added to the system? If so, how much? If not, why not?

Problem 1SL What happens to the internal energy of water vapor in the air that condenses on the outside of a cold glass of water? Is work done or heat exchanged? Explain in detail.

Problem 2MCQ Which is possible: converting (i) 100 J of work entirely into 100 J of heat, (ii) 100 J of heat entirely into 100 J of work? (a) Only (i) is possible. (b) Only (ii) is possible. (c) Both (i) and (ii) are possible. (d) Neither (i) nor (ii) is possible.

(I) One liter of air is cooled at constant pressure until its volume is halved, and then it is allowed to expand isothermally back to its original volume. Draw the process on a PV diagram.

Problem 2Q Can mechanical energy ever be transformed completely into heat or internal energy? Can the reverse happen? In each case, if your answer is no, explain why not; if yes, give one or two examples.

Problem 2SL Draw a PV diagram for an ideal gas which undergoes a three-step cyclic thermodynamic process in which the first step has =0 and W > 0, the second step has W =0, and the third step has Q = 0 and W < 0.

Problem 3MCQ An ideal gas undergoes an isobaric compression and then an isovolumetric process that brings it back to its initial temperature. Had the gas undergone one isothermal process instead, (a) the work done on the gas would be the same. (b) the work done on the gas would be less. (c) the work done on the gas would be greater. (d) Need to know the temperature of the isothermal process

Problem 3P (II) Sketch a PV diagram of the following process: 2.5 L of ideal gas at atmospheric pressure is cooled at constant pressure to a volume of 1.0 L, and then expanded isothermally back to 2.5 L, whereupon the pressure is increased at constant volume until the original pressure is reached.

Problem 3Q Can the temperature of a system remain constant even though heat flows into or out of it? If so, give examples.

Problem 3SL What exactly is a Carnot engine and why is it important? How practical is it?

An ideal gas undergoes an isothermal expansion from state A to state B. In this process (use sign conventions, page 413) (a) Q = 0, \(\Delta U=0\), W > 0. (b) Q > 0, \(\Delta U=0\), W < 0. (c) Q = 0, \(\Delta U>0\), W > 0. (d) Q > 0, \(\Delta U=0\), W > 0. (e) Q = 0, \(\Delta U<0\), W < 0. Equation Transcription: Text Transcription: Delta U=0 Delta U>0 Delta U<0

Problem 4P (II) A gas is enclosed in a cylinder fitted with a light frictionless piston and maintained at atmospheric pressure. When 254 kcal of heat is added to the gas, the volume is observed to increase slowly from to 12.0 m3 to 16.2 m3. Calculate (a) the work done by the gas and (b) the change in internal energy of the gas.

Problem 4Q Explain why the temperature of a gas increases when it is compressed adiabatically.

Problem 4SL (a) Make up an advertisement for a refrigerator or air conditioner that violates the first law of thermodynamics (see Section 15–6). (b) Make up an ad for a car engine that violates the second law of thermodynamics (see Section 15–5).

Problem 5MCQ An ideal gas undergoes an isothermal process. Which of the following statements are true? (i) No heat is added to or removed from the gas. (ii) The internal energy of the gas does not change. (iii) The average kinetic energy of the molecules does not change. (a) (i) only. (b) (i) and (ii) only. (c) (i) and (iii) only. (d) (ii) and (iii) only. (e) (i), (ii), and (iii). (f) None of the above.

Problem 5P (II) A 1.0-L volume of air initially at 3.5 atm of (gauge) pressure is allowed to expand isothermally until the pressure is 1.0 atm. It is then compressed at constant pressure to its initial volume, and lastly is brought back to its original pressure by heating at constant volume. Draw the process on a PV diagram, including numbers and labels for the axes.

Problem 5Q An ideal monatomic gas expands slowly to twice its volume (1) isothermally; (2) adiabatically; (3) isobarically. Plot each on a PV diagram. In which process is ?U the greatest, and in which is ?U the least? In which is W the greatest and the least? In which is Q the greatest and the least?

One day a person sleeps for 7.0 h, goes running for an hour, gets dressed and eats breakfast for an hour, sits at work for the next 9.0 h, does household chores for a couple of hours, eats dinner for an hour, surfs the Internet and watches TV for a couple of hours, and finally takes an hour for a bath and getting ready for bed. If all of the energy associated with these activities is considered as heat that the body outputs to the environment, estimate the change in entropy the person has provided. (See Sections 15–3 and 15–7.)

Problem 6MCQ An ideal gas undergoes an adiabatic expansion, a process in which no heat flows into or out of the gas. As a result, (a) the temperature of the gas remains constant and the pressure decreases. (b) both the temperature and pressure of the gas decrease. (c) the temperature of the gas decreases and the pressure increases. (d) both the temperature and volume of the gas increase. (e) both the temperature and pressure of the gas increase.

Problem 6P (II) The pressure in an ideal gas is cut in half slowly, while being kept in a container with rigid walls. In the process, 465 kJ of heat left the gas. (a) How much work was done during this process? (b) What was the change in internal energy of the gas during this process?

Problem 6Q (a) What happens if you remove the lid of a bottle containing chlorine gas? (b) Does the reverse process ever happen? Why or why not? (c) Can you think of two other examples of irreversibility?

Problem 6SL A particular 1.5 m2 photovoltaic panel operating in direct sunlight produces electricity at 20% efficiency. The resulting electricity is used to operate an electric stove that can be used to heat water. A second system uses a 1.5 m2 curved mirror to concentrate the Sun’s energy directly onto a container of water. Estimate how long it takes each system to heat 1.0 kg of water from 25°C to 95°C. (See also Chapter 14.)

A heat engine operates between a high temperature of about \(600^{\circ}C\) and a low temperature of about \(300^{\circ}C\). What is the maximum theoretical efficiency for this engine? (a) = 100%. (b) \(\approx\ 66%\) (c) \(\approx\ 50%\) (d) \(\approx\ 34%\) (e) Cannot be determined from the given information.

Problem 7P (II) In an engine, an almost ideal gas is compressed adiabatically to half its volume. In doing so, 2630 J of work is done on the gas. (a) How much heat flows into or out of the gas? (b) What is the change in internal energy of the gas? (c) Does its temperature rise or fall?

Would a definition of heat engine efficiency as \(e=W / Q_{L}\) be useful? Explain. Equation Transcription: Text Transcription: e = W/Q_L

Problem 7SL A dehumidifier removes water vapor from air and has been referred to as a “refrigerator with an open door.” The humid air is pulled in by a fan and passes over a cold coil, whose temperature is less than the dew point, and some of the air’s water condenses. After this water is extracted, the air is warmed back to its original temperature and sent into the room. In a well-designed dehumidifier, the heat that is removed by the cooling coil mostly comes from the condensation of water vapor to liquid, and this heat is used to re-warm the air. Estimate how much water is removed in 1.0 h by an ideal dehumidifier, if the temperature of the room is 25°C, the water condenses at 8°C, and the dehumidifier does work at the rate of 600 W of electrical power. (See Sections 15–6, 13–12, and 14–5.)

Problem 8MCQ On a very hot day, could you cool your kitchen by leaving the refrigerator door open? (a) Yes, but it would be very expensive. (b) Yes, but only if the humidity is below 50%. (c) No, the refrigerator would exhaust the same amount of heat into the room as it takes out of the room. (d) No, the heat exhausted by the refrigerator into the room is more than the heat the refrigerator takes out of the room.

Problem 8P (II) An ideal gas expands at a constant total pressure of 3.0 atm from 410 mL to 690 mL. Heat then flows out of the gas at constant volume, and the pressure and temperature are allowed to drop until the temperature reaches its original value. Calculate (a) the total work done by the gas in the process, and (b) the total heat flow into the gas.

What are the high-temperature and the low-temperature areas for (a) an internal combustion engine, and (b) a steam engine? Are they, strictly speaking, heat reservoirs?

Which of the following possibilities could increase the efficiency of a heat engine or an internal combustion engine? (a) Increase the temperature of the hot part of the system and reduce the temperature of the exhaust. (b) Increase the temperatures of both the hot part and the exhaust part of the system by the same amount. (c) Decrease the temperatures of both the hot part and the exhaust part of the system by the same amount. (d) Decrease the temperature of the hot part and increase the temperature of the exhaust part by the same amount. (e) None of the above; only redesigning the engine or using better gas could improve the engine’s efficiency.

Problem 9P (II) 8.5 moles of an ideal monatomic gas expand adiabatically, performing 8300 J of work in the process. What is the change in temperature of the gas during this expansion?

Problem 9Q The oceans contain a tremendous amount of thermal (internal) energy. Why, in general, is it not possible to put this energy to useful work?

Problem 10MCQ About what percentage of the heat produced by burning gasoline is turned into useful work by a typical automobile? (a) 20%. (b) 50%. (c) 80%. (d) 90%. (e) Nearly 100%.

(II) Consider the following two-step process. Heat is allowed to flow out of an ideal gas at constant volume so that its pressure drops from 2.2 atm to 1.4 atm. Then the gas expands at constant pressure, from a volume of 5.9 L to 9.3 L, where the temperature reaches its original value. See Fig.15–22. Calculate (a) the total work done by the gas in the process, (b) the change in internal energy of the gas in the process, and (c) the total heat flow into or out of the gas.

Problem 10Q Can you warm a kitchen in winter by leaving the oven door open? Can you cool the kitchen on a hot summer day by leaving the refrigerator door open? Explain.

Problem 11MCQ Which statement is true regarding the entropy change of an ice cube that melts? (a) Since melting occurs at the melting point temperature, there is no temperature change so there is no entropy change. (b) Entropy increases. (c) Entropy decreases.

Problem 11Q The COPs are defined differently for heat pumps and air conditioners. Explain why.

(III) The PV diagram in Fig. 15–23 shows two possible states of a system containing 1.75 moles of a monatomic ideal gas. \(\left(P_{1}=P_{2}=425 \mathrm{~N} / \mathrm{m}^{2}, V_{1}=2.00 \mathrm{~m}^{3}, V_{2}=8.00 \mathrm{~m}^{3}\right)\) (a) Draw the process which depicts an isobaric expansion from state 1 to state 2, and label this process A. (b) Find the work done by the gas and the change in internal energy of the gas in process A. (c) Draw the two-step process which depicts an isothermal expansion from state 1 to the volume \(V_{2}\) followed by an isovolumetric increase in temperature to state 2, and label this process B. (d) Find the change in internal energy of the gas for the two-step process B. Equation transcription: Text transcription: (P{1}=P{2}=425{~N} /{m}^{2}, V{1}=2.00{~m}^{3}, V{2}=8.00{~m}^{3}) V_{2}

Problem 12Q You are asked to test a machine that the inventor calls an “in-room air conditioner”: a big box, standing in the middle of the room, with a cable that plugs into a power outlet. When the machine is switched on, you feel a stream of cold air coming out of it. How do you know that this machine cannot cool the room?

Problem 13Q Think up several processes (other than those already mentioned) that would obey the first law of thermodynamics, but, if they actually occurred, would violate the second law.

Problem 14P (I) How much energy would the person of Example 15–7 transform if instead of working 11.0 h she took a noontime break and ran at 15 km/h for 1.0 h?

Problem 14Q Suppose a lot of papers are strewn all over the floor; then you stack them neatly. Does this violate the second law of thermodynamics? Explain.

What would be the internal energy change in Example 15–1 if 2500 J of heat is added to the system and 1800 J of work is done by the system (i.e., as output)?

Is the work done by the gas in process ADB of Fig. 15–6 greater than, less than, or equal to the work done in the isothermal process AB?

In Example 15–4, if the heat lost from the gas in the process BD is \(8.4 x 10^{3} J\), what is the change in internal energy of the gas during process BD? Equation transcription: Text transcription: 8.4 x 10^{3} J

A motor is running with an intake temperature \(T_{H}=400 K\) and an exhaust temperature Which of the following are not possible efficiencies for the engine? (a) 0.10; (b) 0.16; (c) 0.24; (d) 0.30; (e) 0.33. Equation transcription: Text transcription: T_{H}=400 K

Problem 15EE Return to the Chapter-Opening Question, page 412, and answer it again now. Try to explain why you may have answered differently the first time.

In the Table above, what is the probability that there will be at least two heads? (a) \(\frac{1}{2}\); (b) \(\frac{1}{16}\); (c) \(\frac{1}{8}\); (d) \(\frac{3}{8}\); (e) \(\frac{11}{16}\) Equation transcription: Text transcription: frac{1}{2} frac{1}{16} frac{1}{8} frac{3}{8} frac{11}{16}

Problem 15P (I) Calculate the average metabolic rate of a 65-kg person who sleeps 8.0 h, sits at a desk 6.0 h, engages in light activity 6.0 h, watches TV 2.0 h, plays tennis 1.5 h, and runs 0.50 h daily

Problem 15Q The first law of thermodynamics is sometimes whimsically stated as, “You can’t get something for nothing,” and the second law as, “You can’t even break even.” Explain how these statements could be equivalent to the formal statements.

Problem 16P (II) A 65-kg person decides to lose weight by sleeping one hour less per day, using the time for light activity. How much weight (or mass) can this person expect to lose in 1 year, assuming no change in food intake? Assume that 1 kg of fat stores about 40,000 kJ of energy.

Problem 16Q A gas is allowed to expand (a) adiabatically and (b) isothermally. In each process, does the entropy increase, decrease, or stay the same? Explain.

Problem 17P (II) (a) How much energy is transformed by a typical 65-kg person who runs at 15 km/h for 30 min/day in one week (Table 15–2)? (b) How many food calories would the person have to eat to make up for this energy loss?

Problem 17Q Which do you think has the greater entropy, 1 kg of solid iron or 1 kg of liquid iron? Why?

Problem 18P (I) A heat engine exhausts 8200 J of heat while performing 2600 J of useful work. What is the efficiency of this engine?

Problem 18Q Give three examples, other than those mentioned in this Chapter, of naturally occurring processes in which order goes to disorder. Discuss the observability of the reverse process.

Problem 19P (I) What is the maximum efficiency of a heat engine whose operating temperatures are 560°C and 345°C?

Entropy is often called “time’s arrow” because it tells us in which direction natural processes occur. If a movie were run backward, name some processes that you might see that would tell you that time was “running backward.”

Problem 20P (I) The exhaust temperature of a heat engine is 230°C.What is the high temperature if the Carnot efficiency is 34%?

Living organisms, as they grow, convert relatively simple food molecules into a complex structure. Is this a violation of the second law of thermodynamics? Explain your answer.

Problem 21P (I) A heat engine does 9200 J of work per cycle while absorbing 25.0 kcal of heat from a high-temperature reservoir. What is the efficiency of this engine?

(I) A heat engine’s high temperature \(T_{H}\) could be ambient temperature, because liquid nitrogen at 77 K could be \(T_{L}\) and is cheap. What would be the efficiency of a Carnot engine that made use of heat transferred from air at room temperature (293 K) to the liquid nitrogen “fuel”(Fig.15–25)?

Problem 23P (II) Which will improve the efficiency of a Carnot engine more: a 10 C° increase in the high-temperature reservoir, or a 10 C° decrease in the low-temperature reservoir? Give detailed results. Can you state a generalization?

(II) A certain power plant puts out 580 MW of electric power. Estimate the heat discharged per second, assuming that the plant has an efficiency of 32%.

Problem 25P (II) A nuclear power plant operates at 65% of its maximum theoretical (Carnot) efficiency between temperatures of 660°C and 330°C. If the plant produces electric energy at the rate of 1.4GW, how much exhaust heat is discharged per hour?

Problem 26P (II) A heat engine exhausts its heat at 340°C and has a Carnot efficiency of 36%.What exhaust temperature would enable it to achieve a Carnot efficiency of 42%?

Problem 27P (II) A Carnot engine operating temperatures are 210°C and 45°C. The engine’s power output is 910W. Calculate the rate of heat output.

Problem 28P (II) A four-cylinder gasoline engine has an efficiency of 0.22 and delivers 180 J of work per cycle per cylinder. If the engine runs at 25 cycles per second (1500 rpm), determine (a) the work done per second, and (b) the total heat input per second from the gasoline. (c) If the energy content of gasoline is 130 MJ per gallon, how long does one gallon last?

Problem 29P (II) A Carnot engine performs work at the rate of 520 kW with an input of 950 kcal of heat per second. If the temperature of the heat source is 520°C, at what temperature is the waste heat exhausted?

Problem 30P (II) A heat engine uses a heat source at 580°C and has an ideal (Carnot) efficiency of 22%. To increase the ideal efficiency to 42%, what must be the temperature of the heat source?

(I) If an ideal refrigerator keeps its contents at \(2.5^{\circ}C\) when the house temperature is \(22^{\circ}C\), what is its COP?

Problem 33P (I) The low temperature of a freezer cooling coil is -8°C and the discharge temperature is 33°C. What is the maximum theoretical coefficient of performance?

Problem 34P (II) What is the temperature inside an ideal refrigerator-freezer that operates with a in a 22°C room?

Problem 35P (II) A heat pump is used to keep a house warm at 22°C. How much work is required of the pump to deliver 3100 J of heat into the house if the outdoor temperature is (a) 0°C, (b) -15°C (b) Assume a COP of 3.0. (c) Redo for both temperatures, assuming an ideal (Carnot) coefficient of performance COP =TH(TH-TL)

(II) (a) What is the coefficient of performance of an ideal heat pump that extracts heat from \(6^{0} \mathrm{C}\) air outside and deposits heat inside a house at \(24^{0} \mathrm{C}\)? (b) If this heat pump operates on 1200 W of electrical power, what is the maximum heat it can deliver into the house each hour? See Problem 35. Equation transcription: Text transcription: 6^{0}{C} 24^{0}{C}

Problem 37P (II) What volume of water at 0°C can a freezer make into ice cubes in 1.0 h, if the coefficient of performance of the cooling unit is 6.0 and the power input is 1.2 kilowatt?

Problem 38P (II) How much less per year would it cost a family to operate a heat pump that has a coefficient of performance of 2.9 than an electric heater that costs $2000 to heat their home for a year? If the conversion to the heat pump costs $15,000, how long would it take the family to break even on heating costs? How much would the family save in 20 years?

Problem 39P (I) What is the change in entropy of 320 g of steam at 100°C when it is condensed to water at 100°C?

Problem 40P (I) 1.0 kg of water is heated from 0°C to 100°C. Estimate the change in entropy of the water.

Problem 41P (I) What is the change in entropy of of water at 1.00 m3 0°C when it is frozen to ice at 0°C?

Problem 42P (II) A 5.8-kg box having an initial speed of 4.0 m/s slides along a rough table and comes to rest. Estimate the total change in entropy of the universe. Assume all objects are at room temperature (293 K).

Problem 43P (II) If 1.00 m3 of water at 0°C is frozen and cooled to -8.0°C by being in contact with a great deal of ice at -8.0°C estimate the total change in entropy of the process.

Problem 44P (II) An aluminum rod conducts 8.40 cal/s from a heat source maintained at 225°C to a large body of water at 22°C. Calculate the rate at which entropy increases in this process.

Problem 45P (II) A 2.8-kg piece of aluminum at 28.5°C is placed in 1.0 kg of water in a Styrofoam container at room temperature (20.0°C). Estimate the net change in entropy of the system.

Problem 46P (II) A falling rock has kinetic energy KE just before striking the ground and coming to rest. What is the total change in entropy of rock plus environment as a result of this collision?

Problem 47P (II) 1.0 kg of water at 35°C is mixed with 1.0 kg of water at 45°C in a well-insulated container. Estimate the net change in entropy of the system.

Problem 48P (III) A real heat engine working between heat reservoirs at 970 K and 650 K produces 550 J of work per cycle for a heat input of 2500 J. (a) Compare the efficiency of this real engine to that of an ideal (Carnot) engine. (b) Calculate the total entropy change of the universe per cycle of the real engine, and (c) also if the engine is ideal (Carnot).

Problem 49P (II) Calculate the probabilities, when you throw two dice, of obtaining (a) a 4, and (b) a 10.

Problem 50P (II) Suppose that you repeatedly shake six coins in your hand and drop them on the floor. Construct a table showing the number of microstates that correspond to each macrostate. What is the probability of obtaining (a) three heads and three tails, and (b) six heads?

Problem 51P (III) A bowl contains many red, orange, and green jelly beans, in equal numbers. You are to make a line of 3 jelly beans by randomly taking 3 beans from the bowl. (a) Construct a table showing the number of microstates that correspond to each macrostate. Then determine the probability of (b) all 3 beans red, and (c) 2 greens, 1 orange.

Problem 52P (III) Rank the following five-card hands in order of increasing probability: (a) four aces and a king; (b) six of hearts, eight of diamonds, queen of clubs, three of hearts, jack of spades; (c) two jacks, two queens, and an ace; and (d) any hand having no two equal-value cards (no pairs, etc.). Discuss your ranking in terms of microstates and macrostates.

(I) Solar cells (Fig. 15–26) can produce about 40 W of electricity per square meter of surface area if directly facing the Sun. How large an area is required to supply the needs of a house that requires 24 kWh/day? Would this fit on the roof of an average house? (Assume the Sun shines about 9 h/day.)

Problem 54P (II) Energy may be stored by pumping water to a high reservoir when demand is low and then releasing it to drive turbines during peak demand. Suppose water is pumped to a lake 115 m above the turbines at a rate of 1.00 X 105 kg/s for 10.0 h at night. (a) How much energy (kWh) is needed to do this each night? (b) If all this energy is released during a 14-h day, at 75% efficiency, what is the average power output?

(II) Water is stored in an artificial lake created by a dam (Fig. 15–27). The water depth is 48 m at the dam, and a steady flow rate of \(32 m^{3} / s\) is maintained through hydroelectric turbines installed near the base of the dam. How much electrical power can be produced? Equation transcription: Text transcription: 32 m^{3} / s

Problem 56GP An inventor claims to have built an engine that produces 2.00 MW of usable work while taking in 3.00 MW of thermal energy at 425 K, and rejecting 1.00MW of thermal energy at 215 K. Is there anything fishy about his claim? Explain.

When \(5.80 \times 10^{5}\) of heat is added to a gas enclosed in a cylinder fitted with a light frictionless piston maintained at atmospheric pressure, the volume is observed to increase from \(1.9 m^{3}\) to \(4.1 m^{3}\). Calculate (a) the work done by the gas, and (b) the change in internal energy of the gas. (c) Graph this process on a PV diagram. Equation transcription: Text transcription: 5.80 times 10^{5} 1.9 m^{3} 4.1 m^{3}

Problem 58GP A restaurant refrigerator has a coefficient of performance of 4.6. If the temperature in the kitchen outside the refrigerator is 32°C, what is the lowest temperature that could be obtained inside the refrigerator if it were ideal?

Problem 59GP A particular car does work at the rate of about 7.0 kJ/s when traveling at a steady 21.8 m/s along a level road. This is the work done against friction. The car can travel 17 km on 1.0 L of gasoline at this speed (about 40 mi/gal ). What is the minimum value for TH if TL is 25°C? The energy available from 1.0 L of gas is 3.2*107 J.

Problem 60GP A “Carnot” refrigerator (the reverse of a Carnot engine) absorbs heat from the freezer compartment at a temperature of -17o C and exhausts it into the room at 25°C. (a) How much work would the refrigerator do to change 0.65 kg of water at 25°C into ice at -17o C? (b) If the compressor output is 105 W and runs 25% of the time, how long will this take?

Problem 61GP It has been suggested that a heat engine could be developed that made use of the temperature difference between water at the surface of the ocean and water several hundred meters deep. In the tropics, the temperatures may be 27°C and 4°C, respectively. (a) What is the maximum efficiency such an engine could have? (b) Why might such an engine be feasible in spite of the low efficiency? (c) Can you imagine any adverse environmental effects that might occur?

Problem 62GP A cooling unit for a new freezer has an inner surface area of 8.0 m2 and is bounded by walls 12 cm thick with a thermal conductivity of 0.050 W/m.K. The inside must be kept at -15o C in a room that is at 22°C. The motor for the cooling unit must run no more than 15% of the time. What is the minimum power requirement of the cooling motor?

Refrigeration units can be rated in “tons.” A 1-ton air conditioning system can remove sufficient energy to freeze 1 ton of \(0^{0} \mathrm{C}\) water into \(0^{0} \mathrm{C}\) ice in one 24-h day. If, on a \(35^{0} \mathrm{C}\) day, the interior of a house is maintained at \(22^{0} \mathrm{C}\) by the continuous operation of a 5-ton air conditioning system, how much does this cooling cost the homeowner per hour? Assume the work done by the refrigeration unit is powered by electricity that costs $0.10 per kWh and that the unit’s coefficient of performance is 18% that of an ideal refrigerator. \(1 k W h=3.60 x 10^{6} J\). Equation transcription: Text transcription: 0^{0}{C} 35^{0}{C} 22^{0}{C} 1 k W h=3.60 x 10^{6} J

Problem 64GP Two 1100-kg cars are traveling 85 km/h in opposite directions when they collide and are brought to rest. Estimate the change in entropy of the universe as a result of this collision. Assume T=20o C

Problem 65GP A 110-g insulated aluminum cup at 35°C is filled with 150 g of water at 45°C. After a few minutes, equilibrium is reached. (a) Determine the final temperature, and (b) estimate the total change in entropy.

Problem 66GP The burning of gasoline in a car releases about 3.0*104 kcal/gal. If a car averages 41 km/gal when driving 110 km/h, which requires 25 hp, what is the efficiency of the engine under those conditions?

Problem 67GP A Carnot engine operates with TL =20o C and has an efficiency of 25%. By how many kelvins should the high operating temperature TH be increased to achieve an efficiency of 35%?

Calculate the work done by an ideal gas in going from state A to state C in Fig. 15–28 for each of the following processes: (a) ADC, (b) ABC, and (c) AC directly.

Problem 69GP A 38% efficient power plant puts out 850 MW of electrical power. Cooling towers take away the exhaust heat. (a) If the air temperature is allowed to rise 7.0 C°, estimate what volume of air is (km3) heated per day. Will the local climate be heated significantly? (b) If the heated air were to form a layer 180 m thick, estimate how large an area it would cover for 24 h of operation. Assume the air has density 1.3kg/m3 and has specific heat of about 1.0 kJ/kg.Co at constant pressure.

Problem 70GP Suppose a power plant delivers energy at 880 MW using steam turbines. The steam goes into the turbines superheated at 625 K and deposits its unused heat in river water at 285 K. Assume that the turbine operates as an ideal Carnot engine. (a) If the river flow rate is 37 m3/s, estimate the average temperature increase of the river water immediately downstream from the power plant. (b) What is the entropy increase per kilogram of the downstream river water in J/kg.K?

Problem 71GP A car engine whose output power is 135 hp operates at about 15% efficiency. Assume the engine’s water temperature of 85°C is its cold-temperature (exhaust) reservoir and 495°C is its thermal “intake” temperature (the temperature of the exploding gas–air mixture). (a) What is the ratio of its efficiency relative to its maximum possible (Carnot) efficiency? (b) Estimate how much power (in watts) goes into moving the car, and how much heat, in joules and in kcal, is exhausted to the air in 1.0 h.

Problem 72GP An ideal monatomic gas is contained in a tall cylindrical jar of cross-sectional area 0.080 m2 fitted with an airtight frictionless 0.15-kg movable piston. When the gas is heated (at constant pressure) from 25°C to 55°C, the piston rises 1.0 cm. How much heat was required for this process? Assume atmospheric pressure outside. [Hint: See Section 14–2.]

Problem 73GP Metabolizing 1.0 kg of fat results in about 3.7*107 J of internal energy in the body. (a) In one day, how much fat does the body burn to maintain the body temperature of a person staying in bed and metabolizing at an average rate of 95 W? (b) How long would it take to burn 1.0 kg of fat this way assuming there is no food intake?

Problem 74GP (a) At a steam power plant, steam engines work in pairs, the heat output of the first one being the approximate heat input of the second. The operating temperatures of the first are 750°C and 440°C, and of the second 415°C and 270°C. If the heat of combustion of coal is 2.8*107 J/kg, at what rate must coal be burned if the plant is to put out 950 MW of power? Assume the efficiency of the engines is 65% of the ideal (Carnot) efficiency. (b) Water is used to cool the power plant. If the water temperature is allowed to increase by no more than 4.5 C°, estimate how much water must pass through the plant per hour.

Suppose a heat pump has a stationary bicycle attachment that allows you to provide the work instead of using an electrical wall outlet. If your heat pump has a coefficient of performance of 2.0 and you can cycle at a racing pace (Table 15–2) for a half hour, how much heat can you provide?

Problem 76GP An ideal air conditioner keeps the temperature inside a room at 21°C when the outside temperature is 32°C. If 4.8 kW of power enters a room through the windows in the form of direct radiation from the Sun, how much electrical power would be saved if the windows were shaded so only 500W enters?

An ideal heat pump is used to maintain the inside temperature of a house at \(T_{i n}\)= \(22^{0} \mathrm{C}\) when the outside temperature is \(T_{\text {out }}\). Assume that when it is operating, the heat pump does work at a rate of 1500 W. Also assume that the house loses heat via conduction through its walls and other surfaces at a rate given by \(\left.650 \mathrm{~W} /{ }^{0} \mathrm{C}\right)\left(T_{\text {in }}-T_{\text {out }}\right)\). (a) For what outside temperature would the heat pump have to operate all the time in order to maintain the house at an inside temperature of \(22^{0} \mathrm{C}\)? (b) If the outside temperature is \(8^{0} C\), what percentage of the time does the heat pump have to operate in order to maintain the house at an inside temperature of \(22^{0} \mathrm{C}\)? Equation transcription: Text transcription: T{i n} 22^{0}{C} T{{out }} 650{~W} /{ }^{0}{C})(T{{in }}-T{ {out }}) 8^{0} C

An ideal gas expands isothermally, performing of work in the process. Calculate (a) the change in internal energy of the gas, and (b) the heat absorbed during this expansion.

One liter of air is cooled at constant pressure until its volume is halved, and then it is allowed to expand isothermally back to its original volume. Draw the process on a PV diagram

Sketch a PV diagram of the following process: 2.5 L of ideal gas at atmospheric pressure is cooled at constant pressure to a volume of 1.0 L, and then expanded isothermally back to 2.5 L, whereupon the pressure is increased at constant volume until the original pressure is reached.

A gas is enclosed in a cylinder fitted with a light frictionless piston and maintained at atmospheric pressure. When 254 kcal of heat is added to the gas, the volume is observed to increase slowly from to Calculate (a) the work done by the gas and (b) the change in internal energy of the gas

(II) A 1.0-L volume of air initially at 3.5 atm of (gauge) pressure is allowed to expand isothermally until the pressure is 1.0 atm. It is then compressed at constant pressure to its initial volume, and lastly is brought back to its original pressure by heating at constant volume. Draw the process on a PV diagram, including numbers and labels for the axes.

The pressure in an ideal gas is cut in half slowly, while being kept in a container with rigid walls. In the process, 465 kJ of heat left the gas. (a) How much work was done during this process? (b) What was the change in internal energy of the gas during this process?

In an engine, an almost ideal gas is compressed adiabatically to half its volume. In doing so, 2630 J of work is done on the gas. (a) How much heat flows into or out of the gas? (b) What is the change in internal energy of the gas? (c) Does its temperature rise or fall?

An ideal gas expands at a constant total pressure of 3.0 atm from 410 mL to 690 mL. Heat then flows out of the gas at constant volume, and the pressure and temperature are allowed to drop until the temperature reaches its original value. Calculate (a) the total work done by the gas in the process, and (b) the total heat flow into the gas

8.5 moles of an ideal monatomic gas expand adiabatically, performing 8300 J of work in the process. What is the change in temperature of the gas during this expansion?

Consider the following two-step process. Heat is allowed to flow out of an ideal gas at constant volume so that its pressure drops from 2.2 atm to 1.4 atm. Then the gas expands at constant pressure, from a volume of 5.9 L to 9.3 L, where the temperature reaches its original value. See Fig.1522. Calculate (a) the total work done by the gas in the process, (b) the change in internal energy of the gas in the process, and (c) the total heat flow into or out of the gas.

Use the conservation of energy to explain why the temperature of a well-insulated gas increases when it is compressedsay, by pushing down on a pistonwhereas the temperature decreases when the gas expands. Show your reasoning.

The PV diagram in Fig. 1523 shows two possible states of a system containing 1.75 moles of a monatomic ideal gas. (a) Draw the process which depicts an isobaric expansion from state 1 to state 2, and label this process A. (b) Find the work done by the gas and the change in internal energy of the gas in process A. (c) Draw the two-step process which depicts an isothermal expansion from state 1 to the volume followed by an isovolumetric increase in temperature to state 2, and label this process B. (d) Find the change in internal energy of the gas for the two-step process B

When a gas is taken from a to c along the curved path in Fig. 1524, the work done by the gas is and the heat added to the gas is Along path abc, the work done by the gas is (That is, 56 J of work is done on the gas.) (a) What is Q for path abc? (b) If what is W for path cda? (c) What is Q for path cda? (d) What is (e) If what is Q for path da?

How much energy would the person of Example 157 transform if instead of working 11.0 h she took a noontime break and ran at for 1.0 h?

(I) Calculate the average metabolic rate of a 65-kg person who sleeps 8.0 h, sits at a desk 6.0 h, engages in light activity 6.0 h, watches TV 2.0 h, plays tennis 1.5 h, and runs 0.50 h daily.

A 65-kg person decides to lose weight by sleeping one hour less per day, using the time for light activity. How much weight (or mass) can this person expect to lose in 1 year, assuming no change in food intake? Assume that 1 kg of fat stores about 40,000 kJ of energy.

(a) How much energy is transformed by a typical 65-kg person who runs at for in one week (Table 152)? (b) How many food calories would the person have to eat to make up for this energy loss?

A heat engine exhausts 8200 J of heat while performing 2600 J of useful work. What is the efficiency of this engine?

What is the maximum efficiency of a heat engine whose operating temperatures are 560C and 345C?

The exhaust temperature of a heat engine is 230C. What is the high temperature if the Carnot efficiency is 34%?

A heat engine does 9200 J of work per cycle while absorbing 25.0 kcal of heat from a high-temperature reservoir. What is the efficiency of this engine?

A heat engines high temperature could be ambient temperature, because liquid nitrogen at 77 K could be and is cheap. What would be the efficiency of a Carnot engine that made use of heat transferred from air at room temperature (293 K) to the liquid nitrogen fuel(Fig.1525)?

Which will improve the efficiency of a Carnot engine more: a 10 C increase in the high-temperature reservoir, or a 10 C decrease in the low-temperature reservoir? Give detailed results. Can you state a generalization?

A certain power plant puts out 580 MW of electric power. Estimate the heat discharged per second, assuming that the plant has an efficiency of 32%

A nuclear power plant operates at 65% of its maximum theoretical (Carnot) efficiency between temperatures of 660C and 330C. If the plant produces electric energy at the rate of 1.4 GW, how much exhaust heat is discharged per hour?

A heat engine exhausts its heat at 340C and has a Carnot efficiency of 36%. What exhaust temperature would enable it to achieve a Carnot efficiency of 42%

A Carnot engines operating temperatures are 210C and 45C. The engines power output is 910 W. Calculate the rate of heat output

A four-cylinder gasoline engine has an efficiency of 0.22 and delivers 180 J of work per cycle per cylinder. If the engine runs at 25 cycles per second (1500 rpm), determine (a) the work done per second, and (b) the total heat input per second from the gasoline. (c) If the energy content of gasoline is 130 MJ per gallon, how long does one gallon last?

A Carnot engine performs work at the rate of 520 kW with an input of 950 kcal of heat per second. If the temperature of the heat source is 520C, at what temperature is the waste heat exhausted?

A heat engine uses a heat source at 580C and has an ideal (Carnot) efficiency of 22%. To increase the ideal efficiency to 42%, what must be the temperature of the heat source?

A typical compact car experiences a total drag force of about 350 N at If this car gets 32 miles per gallon of gasoline at this speed, and a liter of gasoline releases about when burned, what is the cars efficiency?

If an ideal refrigerator keeps its contents at 2.5C when the house temperature is 22C, what is its COP?

The low temperature of a freezer cooling coil is and the discharge temperature is 33C. What is the maximum theoretical coefficient of performance?

What is the temperature inside an ideal refrigeratorfreezer that operates with a in a 22C room?

A heat pump is used to keep a house warm at 22C. How much work is required of the pump to deliver 3100 J of heat into the house if the outdoor temperature is (a) 0C, (b) Assume a COP of 3.0. (c) Redo for both temperatures, assuming an ideal (Carnot) coefficient of performance

What is the coefficient of performance of an ideal heat pump that extracts heat from 6C air outside and deposits heat inside a house at 24C? (b) If this heat pump operates on 1200 W of electrical power, what is the maximum heat it can deliver into the house each hour? See Problem 35

What volume of water at 0C can a freezer make into ice cubes in 1.0 h, if the coefficient of performance of the cooling unit is 6.0 and the power input is 1.2 kilowatt?

How much less per year would it cost a family to operate a heat pump that has a coefficient of performance of 2.9 than an electric heater that costs $2000 to heat their home for a year? If the conversion to the heat pump costs $15,000, how long would it take the family to break even on heating costs? How much would the family save in 20 years?

What is the change in entropy of 320 g of steam at 100C when it is condensed to water at 100C?

1.0 kg of water is heated from 0C to 100C. Estimate the change in entropy of the water.

What is the change in entropy of of water at 0C when it is frozen to ice at 0C?

A 5.8-kg box having an initial speed of slides along a rough table and comes to rest. Estimate the total change in entropy of the universe. Assume all objects are at room temperature (293 K).

If of water at 0C is frozen and cooled to by being in contact with a great deal of ice at estimate the total change in entropy of the process.

An aluminum rod conducts from a heat source maintained at 225C to a large body of water at 22C. Calculate the rate at which entropy increases in this process.

A 2.8-kg piece of aluminum at 28.5C is placed in 1.0 kg of water in a Styrofoam container at room temperature (20.0C). Estimate the net change in entropy of the system

A falling rock has kinetic energy KE just before striking the ground and coming to rest. What is the total change in entropy of rock plus environment as a result of this collision?

1.0 kg of water at 35C is mixed with 1.0 kg of water at 45C in a well-insulated container. Estimate the net change in entropy of the system.

A real heat engine working between heat reservoirs at 970 K and 650 K produces 550 J of work per cycle for a heat input of 2500 J. (a) Compare the efficiency of this real engine to that of an ideal (Carnot) engine. (b) Calculate the total entropy change of the universe per cycle of the real engine, and (c) also if the engine is ideal (Carnot).

Calculate the probabilities, when you throw two dice, of obtaining (a) a 4, and (b) a 10

Suppose that you repeatedly shake six coins in your hand and drop them on the floor. Construct a table showing the number of microstates that correspond to each macrostate. What is the probability of obtaining (a) three heads and three tails, and (b) six heads?

A bowl contains many red, orange, and green jelly beans, in equal numbers. You are to make a line of 3 jelly beans by randomly taking 3 beans from the bowl. (a) Construct a table showing the number of microstates that correspond to each macrostate. Then determine the probability of (b) all 3 beans red, and (c) 2 greens, 1 orange

Rank the following five-card hands in order of increasing probability: (a) four aces and a king; (b) six of hearts, eight of diamonds, queen of clubs, three of hearts, jack of spades; (c) two jacks, two queens, and an ace; and (d) any hand having no two equal-value cards (no pairs, etc.). Discuss your ranking in terms of microstates and macrostate

Solar cells (Fig. 1526) can produce about 40 W of electricity per square meter of surface area if directly facing the Sun. How large an area is required to supply the needs of a house that requires Would this fit on the roof of an average house? (Assume the Sun shines about 9 hday.

Energy may be stored by pumping water to a high reservoir when demand is low and then releasing it to drive turbines during peak demand. Suppose water is pumped to a lake 115 m above the turbines at a rate of for 10.0 h at night. (a) How much energy (kWh) is needed to do this each night? (b) If all this energy is released during a 14-h day, at 75% efficiency, what is the average power output

Water is stored in an artificial lake created by a dam (Fig. 1527). The water depth is 48 m at the dam, and a steady flow rate of is maintained through hydroelectric turbines installed near the base of the dam. How much electrical power can be produced

An inventor claims to have built an engine that produces 2.00 MW of usable work while taking in 3.00 MW of thermal energy at 425 K, and rejecting 1.00 MW of thermal energy at 215 K. Is there anything fishy about his claim? Explain

at 215 K. Is there anything fishy about his claim? Explain. 57. When of heat is added to a gas enclosed in a cylinder fitted with a light frictionless piston maintained at atmospheric pressure, the volume is observed to increase from to Calculate (a) the work done by the gas, and (b) the change in internal energy of the gas. (c) Graph this process on a PV diagram

A restaurant refrigerator has a coefficient of performance of 4.6. If the temperature in the kitchen outside the refrigerator is 32C, what is the lowest temperature that could be obtained inside the refrigerator if it were ideal?

A particular car does work at the rate of about when traveling at a steady along a level road. This is the work done against friction. The car can travel 17 km on 1.0 L of gasoline at this speed (about ). What is the minimum value for if is 25C? The energy available from 1.0 L of gas is 3.2 * 107 J.

A Carnot refrigerator (the reverse of a Carnot engine) absorbs heat from the freezer compartment at a temperature of and exhausts it into the room at 25C. (a) How much work would the refrigerator do to change 0.65 kg of water at 25C into ice at (b) If the compressor output is 105 W and runs 25% of the time, how long will this take?

It has been suggested that a heat engine could be developed that made use of the temperature difference between water at the surface of the ocean and water several hundred meters deep. In the tropics, the temperatures may be 27C and 4C, respectively. (a) What is the maximum efficiency such an engine could have? (b) Why might such an engine be feasible in spite of the low efficiency? (c) Can you imagine any adverse environmental effects that might occur?

A cooling unit for a new freezer has an inner surface area of and is bounded by walls 12 cm thick with a thermal conductivity of The inside must be kept at in a room that is at 22C. The motor for the cooling unit must run no more than 15% of the time. What is the minimum power requirement of the cooling motor?

Refrigeration units can be rated in tons. A 1-ton air conditioning system can remove sufficient energy to freeze 1 ton of 0C water into 0C ice in one 24-h day. If, on a 35C day, the interior of a house is maintained at 22C by the continuous operation of a 5-ton air conditioning system, how much does this cooling cost the homeowner per hour? Assume the work done by the refrigeration unit is powered by electricity that costs $0.10 per kWh and that the units coefficient of performance is 18% that of an ideal refrigerator.

Two 1100-kg cars are traveling in opposite directions when they collide and are brought to rest. Estimate the change in entropy of the universe as a result of this collision. Assume T = 20C.

A 110-g insulated aluminum cup at 35C is filled with 150 g of water at 45C. After a few minutes, equilibrium is reached. (a) Determine the final temperature, and (b) estimate the total change in entropy

The burning of gasoline in a car releases about If a car averages when driving which requires 25 hp, what is the efficiency of the engine under those conditions?

A Carnot engine operates with and has an efficiency of 25%. By how many kelvins should the high operating temperature be increased to achieve an efficiency of 35%?

Calculate the work done by an ideal gas in going from state A to state C in Fig. 1528 for each of the following processes: (a) ADC, (b) ABC, and (c) AC directly

A 38% efficient power plant puts out 850 MW of electrical power. Cooling towers take away the exhaust heat. (a) If the air temperature is allowed to rise 7.0 C, estimate what volume of air is heated per day. Will the local climate be heated significantly? (b) If the heated air were to form a layer 180 m thick, estimate how large an area it would cover for 24 h of operation. Assume the air has density and has specific heat of about at constant pressure.

Suppose a power plant delivers energy at 880 MW using steam turbines. The steam goes into the turbines superheated at 625 K and deposits its unused heat in river water at 285 K. Assume that the turbine operates as an ideal Carnot engine. (a) If the river flow rate is estimate the average temperature increase of the river water immediately downstream from the power plant. (b) What is the entropy increase per kilogram of the downstream river water in JkgK?

A car engine whose output power is 135 hp operates at about 15% efficiency. Assume the engines water temperature of 85C is its cold-temperature (exhaust) reservoir and 495C is its thermal intake temperature (the temperature of the exploding gasair mixture). (a) What is the ratio of its efficiency relative to its maximum possible (Carnot) efficiency? (b) Estimate how much power (in watts) goes into moving the car, and how much heat, in joules and in kcal, is exhausted to the air in 1.0 h

An ideal monatomic gas is contained in a tall cylindrical jar of cross-sectional area fitted with an airtight frictionless 0.15-kg movable piston. When the gas is heated (at constant pressure) from 25C to 55C, the piston rises 1.0 cm. How much heat was required for this process? Assume atmospheric pressure outside. [Hint: See Section 142.]

Metabolizing 1.0 kg of fat results in about \(3.7 \times 10^7 \ \mathrm J\) of internal energy in the body. (a) In one day, how much fat does the body burn to maintain the body temperature of a person staying in bed and metabolizing at an average rate of 95 W? (b) How long would it take to burn 1.0 kg of fat this way assuming there is no food intake?

(a) At a steam power plant, steam engines work in pairs, the heat output of the first one being the approximate heat input of the second. The operating temperatures of the first are 750C and 440C, and of the second 415C and 270C. If the heat of combustion of coal is at what rate must coal be burned if the plant is to put out 950 MW of power? Assume the efficiency of the engines is 65% of the ideal (Carnot) efficiency. (b) Water is used to cool the power plant. If the water temperature is allowed to increase by no more than 4.5 C, estimate how much water must pass through the plant per hour

Suppose a heat pump has a stationary bicycle attachment that allows you to provide the work instead of using an electrical wall outlet. If your heat pump has a coefficient of performance of 2.0 and you can cycle at a racing pace (Table 152) for a half hour, how much heat can you provide?

An ideal air conditioner keeps the temperature inside a room at 21C when the outside temperature is 32C. If 4.8 kW of power enters a room through the windows in the form of direct radiation from the Sun, how much electrical power would be saved if the windows were shaded so only 500 W enters?

An ideal heat pump is used to maintain the inside temperature of a house at when the outside temperature is Assume that when it is operating, the heat pump does work at a rate of 1500 W. Also assume that the house loses heat via conduction through its walls and other surfaces at a rate given by (a) For what outside temperature would the heat pump have to operate all the time in order to maintain the house at an inside temperature of 22C? (b) If the outside temperature is 8C, what percentage of the time does the heat pump have to operate in order to maintain the house at an inside temperature of 22C