Question 1

What is the efficiency of an ideal Carnot 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 atm for both reservoirs.

Accepted Answer

The efficiency of an ideal Carnot engine is given by the formula:

Efficiency = 1 - (T_low / T_high)

where T_low and T_high are the temperatures of the two reservoirs in Kelvin.

In this case, the low-temperature reservoir is the one in which ice and water coexist, which is at the melting point of water (273.15 K), and the high-temperature reservoir is the one in which water and steam coexist, which is at the boiling point of water (373.15 K).

Therefore, the efficiency of the Carnot engine is:

Efficiency = 1 - (273.15 K / 373.15 K) = 0.27 = 27% (rounded to two decimal places)

So, the best choice is A) 27%.

Question 2

The "hot shot" heat engine operating between 40°C and 380°C has an efficiency that is 60% of that of an ideal Carnot engine operating between the same temperatures. If the "hot shot" engine absorbs heat at a rate of 60 kW, at what rate does it exhaust heat?

Accepted Answer

The efficiency of the "hot shot" engine can be calculated using:
efficiency = (1 - Tc/Th) = 0.6*(1 - Tc/Th)_Carnot
where Th is the hot reservoir temperature (380°C), Tc is the cold reservoir temperature (40°C), and (1 - Tc/Th)_Carnot is the efficiency of an ideal Carnot engine operating between the same temperatures. Solving for (1 - Tc/Th)_Carnot yields:
(1 - Tc/Th)_Carnot = efficiency/0.6 = (1 - Tc/Th)*0.6/0.6 = 0.4*(1 - Tc/Th)
Substituting values, we get:
0.4*(1 - 313/653) = 0.299
So the ideal efficiency of the Carnot engine is 29.9%.

The "hot shot" engine absorbs heat at a rate of 60 kW, so the power output or work done by the engine is:
P_out = efficiency*P_in = 0.299*60 kW = 17.94 kW

The rate at which the engine exhausts heat is equal to the rate at which it absorbs heat minus the power output or work done by the engine:
P_exhaust = P_in - P_out = 60 kW - 17.94 kW = 42.06 kW

Therefore, the rate at which the "hot shot" engine exhausts heat is 42.06 kW, which is closest to choice B (41 kW).

Question 3

An ideal Carnot engine operating between a warm reservoir of unknown temperature and a cold reservoir at $1.76 \mathrm {~K}$ has an efficiency of $40.0 \%$ What is the temperature of the warm reservoir?

Accepted Answer

The efficiency of a Carnot engine is given by $\eta =1-\frac{T_C}{T_H}$, where $T_C$ is the temperature of the cold reservoir and $T_H$ is the temperature of the hot reservoir. We can rearrange this equation to solve for $T_H$: $T_H=\frac{T_C}{1-\eta}$.

Substituting the given values, we have $\eta = 0.40$ and $T_C = 1.76 	ext{ K}$. Therefore, $T_H = \frac{1.76}{1-0.40}= 2.93 	ext{ K}$.

So the answer is (A).

Question 4

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

Accepted Answer

The answer of An ideal Carnot air conditioner operates between...

Question 5

Suppose that the Department of Energy develops a new reversible engine that has a coefficient of performance (COP) of 4.0 when operated as a refrigerator and a COP of 5.0 when operated as a heat pump. What is its thermal efficiency when operated as a heat engine doing work?

Accepted Answer

The answer of Suppose that the Department of Energy develops...

Question 6

During each cycle, a refrigerator removes 20.0 kJ of heat from the freezing compartment and ejects 24.0 kJ into a room.
(a) How much work per cycle is required each cycle to run this refrigerator?
(b) What is the coefficient of performance of this refrigerator?

Accepted Answer

The answer of During each cycle, a refrigerator removes 20.0...

Question 7

A reversible engine takes in heat at a rate of 555 W and exhausts heat at 482 W.
(a) How much power does it produce?
(b) What is its efficiency?
(c) If operated in reverse as a refrigerator, what would be its performance coefficient (COP)?

Accepted Answer

The answer of A reversible engine takes in heat at...

Question 8

An ideal Carnot engine operates between a high temperature reservoir at $492 \mathrm {~K}$ and a river with water at $290 \mathrm {~K}$ If it absorbs $4000 \mathrm {~J}$ of heat each cycle, how much work per cycle does it perform?

Accepted Answer

The efficiency of a Carnot engine is given by:$$\eta = 1 - \frac{T_c}{T_h}$$where $T_c$ is the temperature of the cold reservoir and $T_h$ is the temperature of the hot reservoir.In this case, $T_c = 290 \mathrm {~K}$ and $T_h = 492 \mathrm {~K}$, so:$$\eta = 1 - \frac{290 \mathrm {~K}}{492 \mathrm {~K}} = 0.411$$The work done per cycle is given by:$$W = Q_h \eta$$where $Q_h$ is the heat absorbed per cycle.In this case, $Q_h = 4000 \mathrm {~J}$, so:$$W = 4000 \mathrm {~J} \times 0.411 = 1642 \mathrm {~J}$$Therefore, the correct answer is A.

Question 9

A real (non-Carnot) heat engine, operating between heat reservoirs at temperatures of 450 K and 270 K, performs 3.3 kJ of net work, and rejects 8.2 kJ of heat in a single cycle.
(a) What is the thermal efficiency of this heat engine?
(b) What is the maximum efficiency it could possibly have?

Accepted Answer

The answer of A real (non-Carnot) heat engine, operating between...

Question 10

An ideal Carnot engine operating between a warm reservoir of unknown temperature and a cold reservoir at $1.76 \mathrm {~K}$ has an efficiency of $11.0 \%$ What is the temperature of the warm reservoir?

Accepted Answer

The answer of An ideal Carnot engine operating between a...

Question 11

During each cycle, the compressor in a certain ideal Carnot refrigerator performs 480 J of work to remove 150 J of heat from the interior of the refrigerator. How much heat do the coils behind the refrigerator discharge into the kitchen each cycle?

Accepted Answer

The answer of During each cycle, the compressor in a...

Question 12

A Carnot engine operates between two reservoirs with unknown temperatures. If the Carnot engine operates at $67 \%$ efficiency, what is the ratio of the absolute temperatures of the reservoirs, T_c/T_h?

Accepted Answer

The efficiency of a Carnot engine is given by $1 - \frac{T_c}{T_h}$. If the efficiency is 67%, then $1 - \frac{T_c}{T_h} = 0.67$. Solving for $\frac{T_c}{T_h}$, we get $\frac{T_c}{T_h} = 0.33$.

Question 13

A refrigerator has a coefficient of performance equal to 4.2. How much work must be done on the operating gas in the refrigerator in order to remove 250 J of heat from the interior compartment?

Accepted Answer

The coefficient of performance (COP) for a refrigerator is given by COP = Qc/W, where Qc is the heat removed from the cold reservoir (interior compartment) and W is the work done on the system. Given COP = 4.2 and Qc = 250 J, we can rearrange the formula to solve for W: W = Qc / COP = 250 J / 4.2 ≈ 60 J.

Question 14

An air conditioner with a coefficient of performance of 3.50 uses 30.0 kW of power to operate. What power is it discharging to the outdoors?

Accepted Answer

The answer of An air conditioner with a coefficient of...

Question 15

An ideal Carnot engine operates between a warm reservoir at 233 K and a colder reservoir. During each cycle, this engine extracts $15.0 \mathrm {~J}$ of heat from the warm reservoir and does $4.0 \mathrm {~J}$ of work. What is the temperature of the colder reservoir?

Accepted Answer

From the Carnot cycle, we know that the efficiency of the engine is given by 
$$\eta = 1 - \frac{T_c}{T_h}$$
where $T_c$ and $T_h$ are the temperatures of the cold and hot reservoirs, respectively. We are given that the engine extracts $15.0 \mathrm{~J}$ of heat from the warm reservoir and does $4.0 \mathrm{~J}$ of work during each cycle. Thus, the amount of heat rejected to the cold reservoir is 
$$Q_c = Q_h - W = 15.0 \mathrm{~J} - 4.0 \mathrm{~J} = 11.0 \mathrm{~J}$$
Since the engine is ideal and operates in a Carnot cycle, we know that it is a reversible process and the amount of heat rejected to the cold reservoir is related to the amount of heat absorbed from the hot reservoir by 
$$\frac{Q_h}{T_h} = \frac{Q_c}{T_c}$$
Substituting our values and manipulating the equation we have 
$$T_c = \frac{Q_c}{Q_h}\cdot T_h = \frac{11.0 \mathrm{~J}}{15.0\mathrm{~J}}\cdot 233 \mathrm{~K} \approx 171 \mathrm{~K}$$
Therefore, the temperature of the colder reservoir is approximately $171 \mathrm{~K}$. Thus, the best choice is (A).

Question 16

An inventor tries to sell you his new heat engine that takes in 40 J of heat at 87°C on each cycle, expels 30 J at 27°C, and does 10 J of work. Would it be wise to invest in this engine? Back up your conclusion with numerical calculations.

Accepted Answer

The answer of An inventor tries to sell you his...

Question 17

A refrigerator has a performance coefficient (COP) 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?

Accepted Answer

The answer of A refrigerator has a performance coefficient (COP)...

Question 18

An ideal reversible heat pump is taking heat from the outside air at -10.0°C and discharging it into the house at 18.0°C. What is the coefficient of performance of this heat pump?

Accepted Answer

The answer of An ideal reversible heat pump is taking...

Question 19

If the efficiency of a reversible engine is 28%, what is its COP (performance coefficient) operated (a) as a refrigerator, and (b) as a heat pump?

Accepted Answer

The answer of If the efficiency of a reversible engine...

Question 20

During each cycle of operation, a refrigerator absorbs 230 J of heat from the freezer and expels 356 J of heat to the room. How much work input is required in each cycle?

Accepted Answer

The work input required for each cycle of a refrigerator can be calculated using the first law of thermodynamics, which states that the change in the internal energy of a system is equal to the heat added to the system minus the work done by the system. In the context of a refrigerator, the work input is equal to the heat expelled to the room minus the heat absorbed from the freezer. Therefore, the work input is 356 J - 230 J = 126 J.

Quiz 11: Using Energy

What is the efficiency of an ideal Carnot 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 atm for both reservoirs.

The "hot shot" heat engine operating between 40°C and 380°C has an efficiency that is 60% of that of an ideal Carnot engine operating between the same temperatures. If the "hot shot" engine absorbs heat at a rate of 60 kW, at what rate does it exhaust heat?

An ideal Carnot engine operating between a warm reservoir of unknown temperature and a cold reservoir at 1.76 K1.76 \mathrm {~K}1.76 K has an efficiency of 40.0%40.0 \%40.0% What is the temperature of the warm reservoir?

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

Suppose that the Department of Energy develops a new reversible engine that has a coefficient of performance (COP) of 4.0 when operated as a refrigerator and a COP of 5.0 when operated as a heat pump. What is its thermal efficiency when operated as a heat engine doing work?

During each cycle, a refrigerator removes 20.0 kJ of heat from the freezing compartment and ejects 24.0 kJ into a room. (a) How much work per cycle is required each cycle to run this refrigerator? (b) What is the coefficient of performance of this refrigerator?

A reversible engine takes in heat at a rate of 555 W and exhausts heat at 482 W. (a) How much power does it produce? (b) What is its efficiency? (c) If operated in reverse as a refrigerator, what would be its performance coefficient (COP)?

An ideal Carnot engine operates between a high temperature reservoir at 492 K492 \mathrm {~K}492 K and a river with water at 290 K290 \mathrm {~K}290 K If it absorbs 4000 J4000 \mathrm {~J}4000 J of heat each cycle, how much work per cycle does it perform?

A real (non-Carnot) heat engine, operating between heat reservoirs at temperatures of 450 K and 270 K, performs 3.3 kJ of net work, and rejects 8.2 kJ of heat in a single cycle. (a) What is the thermal efficiency of this heat engine? (b) What is the maximum efficiency it could possibly have?

An ideal Carnot engine operating between a warm reservoir of unknown temperature and a cold reservoir at 1.76 K1.76 \mathrm {~K}1.76 K has an efficiency of 11.0%11.0 \%11.0% What is the temperature of the warm reservoir?