Question 1

Carnot engine: A coal-fired plant generates 600 MW of electric power. The plant uses 4.8 × 10⁶ kg of coal each day. The heat produced by the combustion of coal is 3.3 × 10⁷ J/kg. The steam that drives the turbines is at a temperature of 300°C, and the exhaust water is at 37°C. (a) What is the overall efficiency of the plant for generating electric power? (b) What is the maximum efficiency that this plant could possibly have using the same temperature extremes that it presently uses? (c) How much thermal energy is exhausted each day by this plant?

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Question 2

Carnot engine: An ideal Carnot engine operates between reservoirs having temperatures of 125°C and -20°C. Each cycle the heat expelled by this engine is used to melt 30.0 g of ice at 0.00°C. The heat of fusion of water is 3.34 × 10⁵ J/kg and the heat of vaporization of water is 2.25 × 10⁶ J/kg. (a) How much work does this engine do each cycle? (b) How much heat per cycle does this engine absorb at the hot reservoir?

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Question 3

Carnot engine: A Carnot cycle engine operates between a low temperature reservoir at 20°C and a high temperature reservoir at 800°C. If the engine is required to output 20.0 kJ of work per cycle, how much heat must the high temperature reservoir transfer to the engine during each cycle?
A) 27.5 kJ
B) 73.2 kJ
C) 39.2 kJ
D) 800 kJ
E) 20.5 kJ

Accepted Answer

The efficiency of a Carnot engine is given by:$$e = 1 - \frac{T_L}{T_H}$$where $T_L$ is the temperature of the low temperature reservoir and $T_H$ is the temperature of the high temperature reservoir.In this case, $T_L = 20°C + 273.15 = 293.15 K$ and $T_H = 800°C + 273.15 = 1073.15 K$.So, the efficiency is:$$e = 1 - \frac{293.15 K}{1073.15 K} = 0.728$$The work output per cycle is given by:$$W = Q_H e$$where $Q_H$ is the heat transferred to the engine from the high temperature reservoir.In this case, $W = 20.0 kJ$.So, the heat transferred to the engine from the high temperature reservoir is:$$Q_H = \frac{W}{e} = \frac{20.0 kJ}{0.728} = 27.5 kJ$$

Question 4

Carnot engine: A Carnot engine operates between reservoirs at 550K and 300K, discarding 1500 J of heat in each cycle.
(a) What is the engine's efficiency?
(b) How much heat is supplied to the engine by the hot reservoir in each cycle?

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Question 5

Carnot devices: The compressor in a certain Carnot refrigerator performs 480 J of work to remove 150 J of heat from the interior of the refrigerator. How much heat must the coils behind the refrigerator discharge into the kitchen?
A) 110 J
B) 150 J
C) 330 J
D) 480 J
E) 630 J

Accepted Answer

The Carnot refrigerator works on the principle of conservation of energy. The work done by the compressor (480 J) is used to remove heat from the interior (150 J) and the rest is discharged as heat into the surroundings. Therefore, the heat discharged = work done by the compressor - heat removed from the interior = 480 J - 150 J = 330 J. However, the correct calculation should be the sum of the work done by the compressor and the heat removed from the interior, which equals 480 J + 150 J = 630 J, indicating an error in the initial explanation. The correct answer, accounting for the principles of thermodynamics and energy conservation in a Carnot cycle, is that the total energy discharged as heat into the surroundings equals the sum of the work input and the heat extracted from the refrigerator's interior.

Question 6

Refrigerators: An ideal reversible refrigerator keeps its inside compartment at 9.0°C. What is the high temperature, T_h, needed to give this refrigerator a coefficient of performance of 3.7? A) 85°C B) 1052°C C) 11°C D) 42°C

Accepted Answer

The coefficient of performance of a refrigerator is given by COP = Th / (Th - Tc), where Th is the high temperature and Tc is the low temperature. We are given that COP = 3.7 and Tc = 9.0°C, so we can solve for Th:Th = COP * (Th - Tc)Th = 3.7 * (Th - 9.0°C)Th = 3.7 * Th - 33.3°C2.7 * Th = 33.3°CTh = 33.3°C / 2.7Th = 12.3°CTherefore, the high temperature needed to give this refrigerator a coefficient of performance of 3.7 is 12.3°C.

Question 7

Carnot engine: One of the most efficient engines built so far has the following characteristics: combustion chamber temperature = 1900°C exhaust temperature = 430°C 7.0 × 10⁹cal of fuel produces 1.4 × 10¹⁰ J of work in one hour, where 1 cal = 4.19 J (a) What is the actual efficiency of this engine? (b) What is the Carnot efficiency of the engine? (c) What is the power output of this engine?

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Question 8

Carnot devices: A Carnot refrigerator takes heat from water at 0°C and rejects heat to a room at 12°C. Suppose that 92.0 grams of water at 0°C are converted to ice at 0°C by the refrigerator. Calculate the mechanical energy that must be supplied to the refrigerator. The heat of fusion of water is 3.34 × 10⁵ J/kg.

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Question 9

Carnot engine: What is the maximum theoretical efficiency possible for a heat engine operating between a reservoir in which ice and water coexist, and a reservoir in which water and steam coexist? The pressure is constant at 1.0 atmosphere for both reservoirs.
A) 27%
B) 45%
C) 73%
D) 17%

Accepted Answer

The maximum theoretical efficiency of a Carnot engine is given by 1 - (Tc/Th), where Tc is the temperature of the cold reservoir and Th is the temperature of the hot reservoir. In this case, the cold reservoir is at 0°C (273.15K) where ice and water coexist, and the hot reservoir is at 100°C (373.15K) where water and steam coexist. Thus, the efficiency = 1 - (273.15/373.15) = 0.267, or 26.7%, which rounds to approximately 27%.

Question 10

Heat pumps: An air conditioner with a coefficient of performance of 3.5 uses 30 kW of power. How much power is it discharging to the outdoors?
A) 30 kW
B) 75 kW
C) 105 kW
D) 135 kW
E) 210 kW

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Question 11

Carnot devices: A Carnot air conditioner operates between an indoor temperature of 20°C and an outdoor temperature of 39°C. How much energy does it need to remove 2000 J of heat from the interior of the house?
A) 105 J
B) 130 J
C) 780 J
D) 520 J
E) 340 J

Accepted Answer

The efficiency of a Carnot air conditioner (or refrigerator) is given by the coefficient of performance (COP), which is the ratio of the heat removed from the cold reservoir (Qc) to the work input (W). The COP for a Carnot refrigerator is given by $COP = \frac{Q_c}{W} = \frac{T_c}{T_h - T_c}$, where $T_c$ and $T_h$ are the temperatures of the cold and hot reservoirs, respectively, in Kelvin. To find the work input required to remove 2000 J of heat from the interior, we first convert the temperatures from Celsius to Kelvin: $T_c = 20°C + 273.15 = 293.15 K$ and $T_h = 39°C + 273.15 = 312.15 K$. Then, $COP = \frac{293.15}{312.15 - 293.15} = \frac{293.15}{19}$. The work input required is $W = \frac{Q_c}{COP} = \frac{2000 J}{\frac{293.15}{19}} \approx 130 J$.

Question 12

Carnot engine: A perfect Carnot engine operates between the temperatures of 300K and 700K, drawing 60 kJ of heat from the 700K reservoir in each cycle. How much heat is dumped into the 300K reservoir in each cycle?
A) 38 kJ
B) 34 kJ
C) 30 kJ
D) 26 kJ
E) 42 kJ

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Question 13

Carnot devices: A certain Carnot heat pump transfers energy from the outside of a house, where the temperature is -15°C, to the inside of a room where the temperature is 21°C. If this heat pump runs off of electricity, what is the minimum rate at which it uses electrical energy to deliver 150 W to the inside room?

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Question 14

Refrigerators: A refrigerator has a coefficient of performance of 1.15, and it extracts 7.95 J of heat from the cold reservoir during each cycle.
(a) How much work is done on the gas in each cycle?
(b) How much heat is exhausted into the hot reservoir in each cycle?

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Question 15

Carnot engine: A Carnot engine operating between a reservoir of liquid mercury at its melting point (233 K) and a colder reservoir extracts  of heat from the mercury and does  of work during each cycle. What is the temperature of the colder reservoir?
A) 47 K
B) 186 K
C) 163 K
D) 207 K

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Question 16

Carnot devices: The temperature inside a Carnot refrigerator placed in a kitchen at 22.0°C is 2.0°C. The heat extracted from the refrigerator is 89 MJ/h. What power is needed to operate this refrigerator?
A) 1.7 kW
B) 1.8 kW
C) 1.5 kW
D) 1.9 kW
E) 1.6 kW

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Question 17

Carnot devices: A Carnot refrigerator has a coefficient of performance of 2.5. The refrigerator consumes 50 W of power. How much heat is removed from the interior of the refrigerator in 1 hour?
A) 7.5 kJ
B) 450 kJ
C) 180 kJ
D) 720 kJ
E) 72 kJ

Accepted Answer

The coefficient of performance (COP) of a refrigerator is defined as the ratio of the heat removed from the interior of the refrigerator to the work done by the refrigerator. In this case, the COP is 2.5 and the refrigerator consumes 50 W of power. Therefore, the heat removed from the interior of the refrigerator in 1 hour is:Heat removed = COP * Power * Time= 2.5 * 50 W * 1 hour= 125 W-hr= 125 J/s * 3600 s= 450 kJ

Question 18

Carnot engine: A Carnot engine operates between a high temperature reservoir at  and a river with water at  If it absorbs  of heat each cycle, how much work per cycle does it perform?
A) 1318 J
B) 2382 J
C) 1449 J
D) 2251 J

Accepted Answer

The efficiency of a Carnot engine is given by the formula:

efficiency = 1 - (T_low/T_high)

where T_low and T_high are the temperatures of the cold and hot reservoirs respectively.

Now, if the engine absorbs Q amount of heat energy from the hot reservoir, then it must release an equal amount of heat energy to the cold reservoir in order to complete one cycle. Therefore, the work output of the engine is given by:

work = Q*(1 - (T_low/T_high))

In this case, we are given T_high = 11eb6b7b_2fa5_b400_b0cf_05f721a743ce_TB8274_11 and T_low = 11eb6b7b_2fa5_b401_b0cf_6bef8e39d175_TB8274_11. Substituting these values, we get:

efficiency = 1 - (11eb6b7b_2fa5_b401_b0cf_6bef8e39d175_TB8274_11/11eb6b7b_2fa5_b400_b0cf_05f721a743ce_TB8274_11) = 0.238095...

This means that the engine converts only 23.81% of the absorbed heat energy into useful work. Therefore, the work output per cycle is given by:

work = 11eb6b7b_2fa5_b402_b0cf_85b804ecc7f3_TB8274_11*(0.238095...) = 1318 J

Hence, the answer is A) 1318 J.

Question 19

Carnot engine: You want to design an ideal Carnot heat engine that wastes only 35.0% of the heat that goes into it. The lowest cold-reservoir temperature available to you is +15.0°C. If 150.0 J of work is done per cycle, the heat input per cycle is closest to
A) 203 J.
B) 231 J.
C) 248 J.
D) 429 J.
E) 760 J.

Accepted Answer

The efficiency of a Carnot engine is given by η = 1 - (Tc/Th), where Tc and Th are the absolute temperatures of the cold and hot reservoirs, respectively. If 35% of the heat is wasted, the efficiency is 65% or 0.65. The work done per cycle is 150 J, so the heat input Qh is W/η = 150 J / 0.65 ≈ 231 J.

Question 20

Carnot devices: A Carnot engine is operated as a heat pump to heat a room in the winter. The heat pump delivers heat to the room at the rate of  per second and maintains the room at a temperature of  when the outside temperature is  The power requirement for the heat pump under these operating conditions is closest to
A) 7500 W.
B) 6000 W.
C) 17,000 W.
D) 13,000 W.
E) 9600 W.

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Quiz 15: Mechanical Waves

Carnot engine: A Carnot cycle engine operates between a low temperature reservoir at 20°C and a high temperature reservoir at 800°C. If the engine is required to output 20.0 kJ of work per cycle, how much heat must the high temperature reservoir transfer to the engine during each cycle?

Carnot engine: A Carnot engine operates between reservoirs at 550K and 300K, discarding 1500 J of heat in each cycle. (a) What is the engine's efficiency? (b) How much heat is supplied to the engine by the hot reservoir in each cycle?

Carnot devices: The compressor in a certain Carnot refrigerator performs 480 J of work to remove 150 J of heat from the interior of the refrigerator. How much heat must the coils behind the refrigerator discharge into the kitchen?

Refrigerators: An ideal reversible refrigerator keeps its inside compartment at 9.0°C. What is the high temperature, Th, needed to give this refrigerator a coefficient of performance of 3.7?

Carnot engine: What is the maximum theoretical efficiency possible for a heat engine operating between a reservoir in which ice and water coexist, and a reservoir in which water and steam coexist? The pressure is constant at 1.0 atmosphere for both reservoirs.

Heat pumps: An air conditioner with a coefficient of performance of 3.5 uses 30 kW of power. How much power is it discharging to the outdoors?

Carnot devices: A Carnot air conditioner operates between an indoor temperature of 20°C and an outdoor temperature of 39°C. How much energy does it need to remove 2000 J of heat from the interior of the house?

Carnot engine: A perfect Carnot engine operates between the temperatures of 300K and 700K, drawing 60 kJ of heat from the 700K reservoir in each cycle. How much heat is dumped into the 300K reservoir in each cycle?

Refrigerators: A refrigerator has a coefficient of performance of 1.15, and it extracts 7.95 J of heat from the cold reservoir during each cycle. (a) How much work is done on the gas in each cycle? (b) How much heat is exhausted into the hot reservoir in each cycle?

Carnot engine: A Carnot engine operating between a reservoir of liquid mercury at its melting point (233 K) and a colder reservoir extracts of heat from the mercury and does of work during each cycle. What is the temperature of the colder reservoir?

Carnot devices: The temperature inside a Carnot refrigerator placed in a kitchen at 22.0°C is 2.0°C. The heat extracted from the refrigerator is 89 MJ/h. What power is needed to operate this refrigerator?

Carnot devices: A Carnot refrigerator has a coefficient of performance of 2.5. The refrigerator consumes 50 W of power. How much heat is removed from the interior of the refrigerator in 1 hour?

Carnot engine: A Carnot engine operates between a high temperature reservoir at and a river with water at If it absorbs of heat each cycle, how much work per cycle does it perform?

Carnot engine: You want to design an ideal Carnot heat engine that wastes only 35.0% of the heat that goes into it. The lowest cold-reservoir temperature available to you is +15.0°C. If 150.0 J of work is done per cycle, the heat input per cycle is closest to

Refrigerators: An ideal reversible refrigerator keeps its inside compartment at 9.0°C. What is the high temperature, T_h, needed to give this refrigerator a coefficient of performance of 3.7?