Question 1

An ideal Carnot engine operates between a warm reservoir at $233 \mathrm {~K}$ 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

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, $T_h$ is the temperature of the hot reservoir, and $\eta$ is the efficiency. The work done $W$ is related to the heat extracted $Q_h$ and the efficiency by $W = \eta Q_h$. Given $W = 4.0 \, \mathrm{J}$ and $Q_h = 15.0 \, \mathrm{J}$, we find $\eta = \frac{4.0}{15.0} = \frac{4}{15}$. Substituting the given $T_h = 233 \, \mathrm{K}$ and solving for $T_c$, we get $T_c = T_h(1 - \eta) = 233 \times (1 - \frac{4}{15}) = 233 \times \frac{11}{15} = 171 \, \mathrm{K}$.

Question 2

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

Accepted Answer

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Question 3

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 ($\eta$) of an ideal Carnot 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, in kelvins. For water and ice coexisting, $T_C = 273 K$, and for water and steam coexisting, $T_H = 373 K$. Thus, $\eta = 1 - \frac{273}{373} = 0.268$ or $26.8 \%$, which rounds to $27 \%$.

Question 4

A heat engine absorbs 85.6 kJ of heat each cycle and exhausts 61.8 kJ.
(a) What is the efficiency of the engine?
(b) How much work does it do each cycle?

Accepted Answer

The answer of A heat engine absorbs 85.6 kJ of...

Question 5

An ideal Carnot engine is operated between a hot and a cold reservoir. The temperature difference between the two reservoirs is $284 ^ { \circ } \mathrm { C }$. If the efficiency of this ideal engine is $24.0 \%$, find the temperature of the cold reservoir in degrees Celsius.

Accepted Answer

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Question 6

A gas follows the $p V$ trajectory shown in Figure 16.2. How much work is done per cycle by the gas if The gas in a heat engine follows the cycle shown in the $p V$ diagram. How much work does this engine do each cycle if $p _ { 0 } = 5.4$ atm?

Accepted Answer

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Question 7

For a certain ideal Carnot engine, the hot reservoir is $35 \mathrm { C } ^ { \circ }$ higher than the cold reservoir. If this engine is to have an efficiency of $20 \%$, what must be the temperature of the hot reservoir?

Accepted Answer

The answer of For a certain ideal Carnot engine, the...

Question 8

An ideal Carnot heat engine operates between reservoirs at 1740 K and 150 K. In each
cycle, 260 J of heat energy is rejected to the low temperature reservoir. In each cycle, how
much mechanical work W is performed by the engine?

Accepted Answer

The answer of An ideal Carnot heat engine operates between...

Question 9

A real (non-Carnot) heat engine, operating between heat reservoirs at temperatures of 450 $\mathrm { K }$ and $270 \mathrm {~K}$, performs $3.3 \mathrm {~kJ}$ of net work, and rejects $8.2 \mathrm {~kJ}$ 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 $40.0 \%$. What is the temperature of the warm reservoir?

Accepted Answer

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Question 11

The ocean thermal energy conversion project uses the surface water near tropical islands with a temperature of $20 ^ { \circ } \mathrm { C }$ as the hot temperature reservoir, and the water at some depth, with a temperature of $5.0 ^ { \circ } \mathrm { C }$, as the cold temperature reservoir for a heat engine. What is the maximum possible efficiency of an engine running between those two temperatures?

Accepted Answer

The maximum possible efficiency of a heat engine operating between two temperatures is given by the Carnot efficiency formula: $\eta = 1 - \frac{T_{c}}{T_{h}}$, where $\eta$ is the efficiency, $T_{c}$ is the cold reservoir temperature in Kelvin, and $T_{h}$ is the hot reservoir temperature in Kelvin. To use this formula, first convert the given temperatures from Celsius to Kelvin: $20 ^{\circ}C = 293K$ and $5.0 ^{\circ}C = 278K$. Then, $\eta = 1 - \frac{278}{293} = 1 - 0.949$, which gives $\eta \approx 0.051$ or $5.1\%$.

Question 12

Two ideal Carnot heat engines have the same efficiency. One operates between $5.0 \times 10 ^ { 2 } \mathrm {~K}$ and $3.0$ $\times 10 ^ { 2 } \mathrm {~K}$, and the other between $4.0 \times 10 ^ { 2 } \mathrm {~K}$ and some lower temperature. What is the lower temperature?

Accepted Answer

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Question 13

A coal-fired plant generates $600 \mathrm { MW }$ of electric power. The plant uses $4.8 \times 10 ^ { 6 } \mathrm {~kg}$ of coal each day, and the heat of combustion of coal is $3.3 \times 10 ^ { 7 } \mathrm {~J} / \mathrm { kg }$. The steam that drives the turbines is at a temperature of $300 ^ { \circ } \mathrm { C }$, and the exhaust water is at $37 ^ { \circ } \mathrm { C }$.
(a) What is the overall efficiency of the plant for generating electric power?
(b) How much thermal energy is exhausted each day?
(c) Using the same heat reservoirs, what is the maximum possible efficiency for a heat engine?

Accepted Answer

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Question 14

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} $. Substituting $ T_h = 492 \mathrm{~K} $ and $ T_c = 290 \mathrm{~K} $, we find $ \eta = 1 - \frac{290}{492} \approx 0.41 $. The work done per cycle is $ W = \eta \times Q_h = 0.41 \times 4000 \mathrm{~J} \approx 1642 \mathrm{~J} $.

Question 15

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, $T _ { d } T _ { h }$ ?

Accepted Answer

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

Question 16

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 17

One of the most efficient engines built so far has the following characteristics: The combustion chamber temperature is $1900 ^ { \circ } \mathrm { C }$, the exhaust temperature $= 430 ^ { \circ } \mathrm { C } , 7.0 \times 10 ^ { 9 }$ cal of fuel produces $1.4 \times 1010 \mathrm {~J}$ of work in one hour. ( $1 \mathrm { cal } = 4.186 \mathrm {~J}$ )
(a) What is the actual efficiency of this engine?
(b) What is the power output of this engine?
(c) What would be the maximum possible efficiency for an engine using the same temperature extremes?

Accepted Answer

The answer of One of the most efficient engines built...

Question 18

The figure shows a $p V$ diagram for a cycle of a heat engine for which $Q \mathrm { H } = 59 \mathrm {~J}$. What is the thermal efficiency of the engine?

Accepted Answer

The answer of The figure shows a $p V$ diagram...

Question 19

An ideal Carnot heat engine operates between $285 \mathrm {~K}$ and $460 \mathrm {~K}$. What is its efficiency?

Accepted Answer

The efficiency ($\eta$) of an ideal Carnot engine is given by $\eta = 1 - \frac{T_C}{T_H}$, where $T_C$ is the cold reservoir temperature and $T_H$ is the hot reservoir temperature, both in Kelvin. Thus, $\eta = 1 - \frac{285}{460} = 1 - 0.6196 = 0.3804$, which rounds to $0.38$.

Question 20

A heat engine has an efficiency of 31.4% and receives 8.72 kJ of heat per cycle.
(a) How much work does it perform in each cycle?
(b) How much heat does it exhaust in each cycle?

Accepted Answer

The answer of A heat engine has an efficiency of...

Quiz 15: The Laws of Thermodynamics

An ideal Carnot engine operates between a warm reservoir at $233 \mathrm {~K}$ 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?

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.

A heat engine absorbs 85.6 kJ of heat each cycle and exhausts 61.8 kJ. (a) What is the efficiency of the engine? (b) How much work does it do each cycle?

An ideal Carnot engine is operated between a hot and a cold reservoir. The temperature difference between the two reservoirs is $284 ^ { \circ } \mathrm { C }$ . If the efficiency of this ideal engine is $24.0 \%$ , find the temperature of the cold reservoir in degrees Celsius.

For a certain ideal Carnot engine, the hot reservoir is $35 \mathrm { C } ^ { \circ }$ higher than the cold reservoir. If this engine is to have an efficiency of $20 \%$ , what must be the temperature of the hot reservoir?

An ideal Carnot heat engine operates between reservoirs at 1740 K and 150 K. In each cycle, 260 J of heat energy is rejected to the low temperature reservoir. In each cycle, how much mechanical work W is performed by the engine?

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?

Two ideal Carnot heat engines have the same efficiency. One operates between $5.0 \times 10 ^ { 2 } \mathrm {~K}$ and $3.0$ $\times 10 ^ { 2 } \mathrm {~K}$ , and the other between $4.0 \times 10 ^ { 2 } \mathrm {~K}$ and some lower temperature. What is the lower temperature?

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?

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, $T _ { d } T _ { h }$ ?

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?

An ideal Carnot heat engine operates between $285 \mathrm {~K}$ and $460 \mathrm {~K}$ . What is its efficiency?

A heat engine has an efficiency of 31.4% and receives 8.72 kJ of heat per cycle. (a) How much work does it perform in each cycle? (b) How much heat does it exhaust in each cycle?

Quiz 15: The Laws of Thermodynamics

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 1.01.01.0 atm for both reservoirs.

A heat engine absorbs 85.6 kJ of heat each cycle and exhausts 61.8 kJ. (a) What is the efficiency of the engine? (b) How much work does it do each cycle?

An ideal Carnot engine is operated between a hot and a cold reservoir. The temperature difference between the two reservoirs is 284∘C284 ^ { \circ } \mathrm { C }284∘C . If the efficiency of this ideal engine is 24.0%24.0 \%24.0% , find the temperature of the cold reservoir in degrees Celsius.

A gas follows the pVp VpV trajectory shown in Figure 16.2. How much work is done per cycle by the gas if The gas in a heat engine follows the cycle shown in the pVp VpV diagram. How much work does this engine do each cycle if p0=5.4p _ { 0 } = 5.4p0​=5.4 atm?

For a certain ideal Carnot engine, the hot reservoir is 35C∘35 \mathrm { C } ^ { \circ }35C∘ higher than the cold reservoir. If this engine is to have an efficiency of 20%20 \%20% , what must be the temperature of the hot reservoir?

An ideal Carnot heat engine operates between reservoirs at 1740 K and 150 K. In each cycle, 260 J of heat energy is rejected to the low temperature reservoir. In each cycle, how much mechanical work W is performed by the engine?

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 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 Carnot engine operates between two reservoirs with unknown temperatures. If the Carnot engine operates at 67%67 \%67% efficiency, what is the ratio of the absolute temperatures of the reservoirs, TdThT _ { d } T _ { h }Td​Th​ ?

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?

The figure shows a pVp VpV diagram for a cycle of a heat engine for which QH=59 JQ \mathrm { H } = 59 \mathrm {~J}QH=59 J . What is the thermal efficiency of the engine?

An ideal Carnot heat engine operates between 285 K285 \mathrm {~K}285 K and 460 K460 \mathrm {~K}460 K . What is its efficiency?

A heat engine has an efficiency of 31.4% and receives 8.72 kJ of heat per cycle. (a) How much work does it perform in each cycle? (b) How much heat does it exhaust in each cycle?

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.

An ideal Carnot engine is operated between a hot and a cold reservoir. The temperature difference between the two reservoirs is $284 ^ { \circ } \mathrm { C }$ . If the efficiency of this ideal engine is $24.0 \%$ , find the temperature of the cold reservoir in degrees Celsius.

For a certain ideal Carnot engine, the hot reservoir is $35 \mathrm { C } ^ { \circ }$ higher than the cold reservoir. If this engine is to have an efficiency of $20 \%$ , what must be the temperature of the hot reservoir?

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?

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?

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, $T _ { d } T _ { h }$ ?

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?

An ideal Carnot heat engine operates between $285 \mathrm {~K}$ and $460 \mathrm {~K}$ . What is its efficiency?