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# 19.4: Heat Engines

Difficulty Level: At Grade Created by: CK-12

Heat engines transform input heat into work in accordance with the laws of thermodynamics. For instance, as we learned in the previous chapter, increasing the temperature of a gas at constant volume will increase its pressure. This pressure can be transformed into a force that moves a piston.

The mechanics of various heat engines differ but their fundamentals are quite similar and involve the following steps:

1. Heat is supplied to the engine from some source at a higher temperature $(T_h)$.
2. Some of this heat is transferred into mechanical energy through work done $(W)$.
3. The rest of the input heat is transferred to some source at a lower temperature $(T_c)$ until the system is in its original state.

A single cycle of such an engine can be illustrated as follows:

In effect, such an engine allows us to 'siphon off' part of the heat flow between the heat source and the heat sink. The efficiency of such an engine is define as the ratio of net work performed to input heat; this is the fraction of heat energy converted to mechanical energy by the engine:

$e = \frac{W}{Q_i} && \text{[5] Efficiency of a heat engine}$

If the engine does not lose energy to its surroundings (of course, all real engines do), then this efficiency can be rewritten as

$e = \frac{Q_i-Q_o}{Q_i} && \text{[6] Efficiency of a lossless heat engine}$

A Carnot Engine, the most efficient heat engine possible, has an efficiency equal to

$e_c = 1-\frac{T_c}{T_h} && \text{[7] Efficiency of a Carnot (ideal) heat engine}$

where $T_c$ and $T_h$ are the temperatures of the hot and cold reservoirs, respectively.

## Some Important Points

• In a practical heat engine, the change in internal energy must be zero over a complete cycle. Therefore, over a complete cycle $W = \Delta Q$.
• The work done by a gas during a portion of a cycle = $P\Delta\!V$, note $\Delta\!V$can be positive or negative.

## Gas Heat Engines

• When gas pressure-forces are used to move an object then work is done on the object by the expanding gas. Work can be done on the gas in order to compress it.
• If you plot pressure on the vertical axis and volume on the horizontal axis (see $P-V$ diagrams in the last chapter), the work done in any complete cycle is the area enclosed by the graph. For a partial process, work is the area underneath the curve, or$P\Delta\!V$.

Question: A heat engine operates at a temperature of $650\mathrm{K}$. The work output is used to drive a pile driver, which is a machine that picks things up and drops them. Heat is then exhausted into the atmosphere, which has a temperature of $300\mathrm{K}$.

a) What is the ideal efficiency of this engine?

b.) The engine drives a $1200\mathrm{kg}$ weight by lifting it $50m$ in $2.5 sec$. What is the engine’s power output?

c) If the engine is operating at $50\%$ of ideal efficiency, how much power is being consumed?

d) The fuel the engine uses is rated at $2.7\times 10^6\mathrm{J/kg}$. How many kg of fuel are used in one hour?

a) We will plug the known values into the formula to get the ideal efficiency.

$\eta = 1 -\frac{T_{\mathrm{cold}}}{T_\mathrm{hot}}=1-\frac{300\mathrm{K}}{650\mathrm{K}}=54\%$

b) To find the power of the engine, we will use the power equation and plug in the known values.

$P=\frac{W}{t}=\frac{Fd}{t}=\frac{mad}{t}=\frac{1200\mathrm{kg}\times 9.8\mathrm{m/s^2}\times 50\mathrm{m}}{2.5\mathrm{sec}}=240\mathrm{kW}$

c) First, we know that it is operating at $50\%$ of ideal efficiency. We also know that the max efficiency of this engine is $54\%$. So the engine is actually operating at

$.5\times 54\%=27\%$

of $100\%$ efficiency. So $240\mathrm{kW}$ is $27\%$ of what?

$.27x=240\mathrm{kW} \Rightarrow x=\frac{240\mathrm{kW}}{.27}=890\mathrm{kW}$

## Thermodynamics and Heat Engines Problem Set

1. Consider a molecule in a closed box. If the molecule collides with the side of the box, how is the force exerted by the molecule on the box related to the momentum of the molecule? Explain conceptually, in words rather than with equations.
2. If the number of molecules is increased, how is the pressure on a particular area of the box affected? Explain conceptually, in words rather than with equations.
3. The temperature of the box is related to the average speed of the molecules. Use momentum principles to relate temperature to pressure. Explain conceptually, in words rather than with equations.
4. What would happen to the number of collisions if temperature and the number of molecules remained fixed, but the volume of the box increased? Explain conceptually, in words rather than with equations.
5. Use the reasoning in the previous four questions to qualitatively derive the ideal gas law.
6. Typical room temperature is about $300 \;\mathrm{K}$. As you know, the air in the room contains both $O_2$ and $N_2$ gases, with nitrogen the lower mass of the two. If the average kinetic energies of the oxygen and nitrogen gases are the same (since they are at the same temperature), which gas has a higher average speed?
7. Use the formula $P = F / A$ to argue why it is easier to pop a balloon with a needle than with a finger (pretend you don’t have long fingernails).
8. Take an empty plastic water bottle and suck all the air out of it with your mouth. The bottle crumples. Why, exactly, does it do this?
9. You will notice that if you buy a large drink in a plastic cup, there will often be a small hole in the top of the cup, in addition to the hole that your straw fits through. Why is this small hole necessary for drinking?
10. Suppose you were swimming in a lake of liquid water on a planet with a lower gravitational constant $g$ than Earth. Would the pressure $10$ meters under the surface be the same, higher, or lower, than for the equivalent depth under water on Earth? (You may assume that the density of the water is the same as for Earth.)
11. Why is it a good idea for Noreen to open her bag of chips before she drives to the top of a high mountain?
12. Explain, using basic physics conservation laws, why the following conditions would cause the ideal gas law to be violated:
1. There are strong intermolecular forces in the gas.
2. The collisions between molecules in the gas are inelastic.
3. The molecules are not spherical and can spin about their axes.
4. The molecules have non-zero volume.

To the right is a graph of the pressure and volume of a gas in a container that has an adjustable volume. The lid of the container can be raised or lowered, and various manipulations of the container change the properties of the gas within. The points $a, b,$ and $c$ represent different stages of the gas as the container undergoes changes (for instance, the lid is raised or lowered, heat is added or taken away, etc.) The arrows represent the flow of time. Use the graph to answer the following questions.

13. Consider the change the gas undergoes as it transitions from point $b$ to point $c$. What type of process is this?
2. isothermal
3. isobaric
4. isochoric
5. entropic
14. Consider the change the gas undergoes as it transitions from point $c$ to point $a$. What type of process is this?
2. isothermal
3. isobaric
4. isochoric
5. none of the above
15. Consider the change the gas undergoes as it transitions from point $a$ to point $b$. Which of the following best describes the type of process shown?
1. isothermal
2. isobaric
3. isochoric
16. How would an isothermal process be graphed on $a$ $P-V$ diagram?
17. Write a scenario for what you would do to the container to make the gas within undergo the cycle described above.
18. Why is it so cold when you get out of the shower wet, but not as cold if you dry off first before getting out of the shower? _____________________________________________________________
19. Antonio is heating water on the stove to boil eggs for a picnic. How much heat is required to raise the temperature of his 10.0-kg vat of water from $20^\circ C$ to $100^\circ C$?
20. Amy wishes to measure the specific heat capacity of a piece of metal. She places the 75-g piece of metal in a pan of boiling water, then drops it into a styrofoam cup holding 50 g of water at $22^\circ C$. The metal and water come to an equilibrium temperature of $25^\circ C$. Calculate:
1. The heat gained by the water
2. The heat lost by the metal
3. The specific heat of the metal
21. John wishes to heat a cup of water to make some ramen for lunch. His insulated cup holds 200 g of water at $20^\circ C$. He has an immersion heater rated at 1000 W (1000 J/s) to heat the water.
1. How many JOULES of heat are required to heat the water to $100^\circ C$?
2. How long will it take to do this with a 1000-W heater?
22. You put a 20g cylinder of aluminum $(c=0.2 \ cal/g/^\circ C)$ in the freezer $(T=-10^\circ C)$. You then drop the aluminum cylinder into a cup of water at $20^\circ C$. After some time they come to a common temperature of $12^\circ C$. How much water was in the cup?
23. Emily is testing her baby’s bath water and finds that it is too cold, so she adds some hot water from a kettle on the stove. If Emily adds 2.00 kg of water at $80.0^\circ C$ to 20.0 kg of bath water at $27.0^\circ C$, what is the final temperature of the bath water?
24. You are trying to find the specific heat of a metal. You heated a metal in an oven to $250^\circ C$. Then you dropped the hot metal immediately into a cup of cold water. To the right is a graph of the temperature of the water versus time that you took in the lab. The mass of the metal is 10g and the mass of the water is 100g. Recall that water has a specific heat of $1 \ cal/g^\circ C$.
25. How much heat is required to melt a 20 g cube of ice if
1. the ice cube is initially at $0^\circ C$
2. the ice cube is initially at $-20^\circ C$ (be sure to use the specific heat of ice)
26. A certain alcohol has a specific heat of $0.57 \ cal/g^\circ C$ and a melting point of $-114^\circ C$. You have a 150 g cup of liquid alcohol at $22^\circ C$ and then you drop a 10 g frozen piece of alcohol at $-114^\circ C$ into it. After some time the alcohol cube has melted and the cup has come to a common temperature of $7^\circ C$. a. What is the latent heat of fusion (i.e. the ‘$L$’ in the $Q = mL$ equation) for this alcohol? b. Make a sketch of the graph of the alcohol’s temperature vs. time c. Make a sketch of the graph of the water’s temperature vs. time
27. Calculate the average speed of $N_2$ molecules at room temperature $(300 \;\mathrm{K})$. (You remember from your chemistry class how to calculate the mass (in $kg$) of an $N_2$ molecule, right?)
28. How high would the temperature of a sample of $O_2$ gas molecules have to be so that the average speed of the molecules would be $10$% the speed of light?
29. How much pressure are you exerting on the floor when you stand on one foot? (You will need to estimate the area of your foot in square meters.)
30. Calculate the amount of force exerted on a $2 \;\mathrm{cm} \times 2 \;\mathrm{cm}$ patch of your skin due to atmospheric pressure $(P_0 = 101,000 \;\mathrm{Pa})$. Why doesn’t your skin burst under this force?
31. Use the ideal gas law to estimate the number of gas molecules that fit in a typical classroom.
32. Assuming that the pressure of the atmosphere decreases exponentially as you rise in elevation according to the formula $P = P_0 e^\frac{-h} {a}$, where $P_0$ is the atmospheric pressure at sea level $(101,000 \;\mathrm{Pa})$, $h$ is the altitude in km and a is the scale height of the atmosphere $(a \approx 8.4 \;\mathrm{km})$.
1. Use this formula to determine the change in pressure as you go from San Francisco to Lake Tahoe, which is at an elevation approximately $2 \;\mathrm{km}$ above sea level.
2. If you rise to half the scale height of Earth’s atmosphere, by how much does the pressure decrease?
3. If the pressure is half as much as on sea level, what is your elevation?
33. At Noah’s Ark University the following experiment was conducted by a professor of Intelligent Design (formerly Creation Science). A rock was dropped from the roof of the Creation Science lab and, with expensive equipment, was observed to gain $100 \;\mathrm{J}$ of internal energy. Dr. Dumb explained to his students that the law of conservation of energy required that if he put $100 \;\mathrm{J}$ of heat into the rock, the rock would then rise to the top of the building. When this did not occur, the professor declared the law of conservation of energy invalid.
1. Was the law of conservation of energy violated in this experiment, as was suggested? Explain.
2. If the law wasn’t violated, then why didn’t the rock rise?
34. An instructor has an ideal monatomic helium gas sample in a closed container with a volume of $0.01\;\mathrm{m}^3$, a temperature of $412 \;\mathrm{K}$, and a pressure of $474 \;\mathrm{kPa}$.
1. Approximately how many gas atoms are there in the container?
2. Calculate the mass of the individual gas atoms.
3. Calculate the speed of a typical gas atom in the container.
4. The container is heated to $647 \;\mathrm{K}$. What is the new gas pressure?
5. While keeping the sample at constant temperature, enough gas is allowed to escape to decrease the pressure by half. How many gas atoms are there now?
6. Is this number half the number from part (a)? Why or why not?
7. The closed container is now compressed isothermally so that the pressure rises to its original pressure. What is the new volume of the container?
8. Sketch this process on a P-V diagram.
9. Sketch cubes with volumes corresponding to the old and new volumes.
35. A famous and picturesque dam, $80 \;\mathrm{m}$ high, releases $24,000 \;\mathrm{kg}$ of water a second. The water turns a turbine that generates electricity.
1. What is the dam’s maximum power output? Assume that all the gravitational potential energy of the water is converted into electrical energy.
2. If the turbine only operates at $30$% efficiency, what is the power output?
3. How many Joules of heat are exhausted into the atmosphere due to the plant’s inefficiency?

36. A heat engine operates at a temperature of $650 \;\mathrm{K}$. The work output is used to drive a pile driver, which is a machine that picks things up and drops them. Heat is then exhausted into the atmosphere, which has a temperature of $300 \;\mathrm{K}$.
1. What is the ideal efficiency of this engine?
2. The engine drives a $1200 \;\mathrm{kg}$ weight by lifting it $50\;\mathrm{m}$ in $2.5 \;\mathrm{sec}$. What is the engine’s power output?
3. If the engine is operating at $50$% of ideal efficiency, how much power is being consumed?
4. How much power is exhausted?
5. The fuel the engine uses is rated at $2.7\times10^6 \;\mathrm{J/kg}$. How many kg of fuel are used in one hour?
37. Calculate the ideal efficiencies of the following sci-fi heat engines:
1. A nuclear power plant on the moon. The ambient temperature on the moon is $15 \;\mathrm{K}$. Heat input from radioactive decay heats the working steam to a temperature of $975 \;\mathrm{K}$.
2. A heat exchanger in a secret underground lake. The exchanger operates between the bottom of a lake, where the temperature is $4 \;\mathrm{C}$, and the top, where the temperature is $13 \;\mathrm{C}$.
3. A refrigerator in your dorm room at Mars University. The interior temperature is $282 \;\mathrm{K}$; the back of the fridge heats up to $320 \;\mathrm{K}$.
38. How much external work can be done by a gas when it expands from $0.003 \;\mathrm{m}^3$ to $0.04 \;\mathrm{m}^3$ in volume under a constant pressure of $400 \;\mathrm{kPa}$? Can you give a practical example of such work?
39. In the above problem, recalculate the work done if the pressure linearly decreases from $400 \;\mathrm{kPa}$ to $250\;\mathrm{kPa}$ under the same expansion. Hint: use a $PV$ diagram and find the area under the line.
40. One mole $(N = 6.02\times10^{23})$ of an ideal gas is moved through the following states as part of a heat engine. The engine moves from state A to state B to state C, and then back again. Use the Table (below) to answer the following questions:
1. Draw a P-V diagram.
2. Determine the temperatures in states A, B, and C and then fill out the table.
3. Determine the type of process the system undergoes when transitioning from A to B and from B to C. (That is, decide for each if it is isobaric, isochoric, isothermal, or adiabatic.)
4. During which transitions, if any, is the gas doing work on the outside world? During which transitions, if any, is work being done on the gas?
5. What is the amount of net work being done by this gas?
State Volume $(m3)$ Pressure $(atm)$ Temperature $(K)$
A $0.01$ $0.60$
B $0.01$ $0.25$
C $0.02$ $0.25$
1. A sample of gas is used to drive a piston and do work. Here’s how it works:
• The gas starts out at standard atmospheric pressure and temperature. The lid of the gas container is locked by a pin.
• The gas pressure is increased isochorically through a spigot to twice that of atmospheric pressure.
• The locking pin is removed and the gas is allowed to expand isobarically to twice its volume, lifting up a weight. The spigot continues to add gas to the cylinder during this process to keep the pressure constant.
• Once the expansion has finished, the spigot is released, the high-pressure gas is allowed to escape, and the sample settles back to $1 \;\mathrm{atm}$.
• Finally, the lid of the container is pushed back down. As the volume decreases, gas is allowed to escape through the spigot, maintaining a pressure of $1 \;\mathrm{atm}$. At the end, the pin is locked again and the process restarts.
1. Draw the above steps on a $P-V$ diagram.
2. Calculate the highest and lowest temperatures of the gas.
2. A heat engine operates through $4$ cycles according to the $PV$ diagram sketched below. Starting at the top left vertex they are labeled clockwise as follows: a, b, c, and d.
1. From $a-b$ the work is $75 \;\mathrm{J}$ and the change in internal energy is $100 \;\mathrm{J}$; find the net heat.
2. From the a-c the change in internal energy is $-20 \;\mathrm{J}$. Find the net heat from b-c.
3. From c-d the work is $-40 \;\mathrm{J}$. Find the net heat from c-d-a.
4. Find the net work over the complete $4$ cycles.
5. The change in internal energy from b-c-d is $-180 \;\mathrm{J}$. Find:
1. the net heat from c-d
2. the change in internal energy from d-a
3. the net heat from d-a

3. A $0.1$ sample mole of an ideal gas is taken from state A by an isochoric process to state B then to state C by an isobaric process. It goes from state C to D by a process that is linear on a $PV$ diagram, and then it goes back to state A by an isobaric process. The volumes and pressures of the states are given in the Table (below); use this data to complete the following:
1. Find the temperature of the $4$ states
2. Draw a $PV$ diagram of the process
3. Find the work done in each of the four processes
4. Find the net work of the engine through a complete cycle
5. If $75 \;\mathrm{J}$ of heat is exhausted in D-A and A-B and C-D are adiabatic, how much heat is inputted in B-C?
6. What is the efficiency of the engine?
state Volume in $m^3 \times 10^{-3}$ Pressure in $N/m^2 \times 10^5$
A $1.04$ $2.50$
B $1.04$ $4.00$
C $1.25$ $4.00$
D $1.50$ $2.50$

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2. .
3. .
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12. .
13. .
14. .
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18. .
19. 800,000 cal or 3360 kJ
1. 150 cal (630 J)
2. same as a!
3. $0.027 \ cal/g^\circ C \ (0.11 \ J/g^\circ C)$
1. 67,000 J
2. 67.2 s
3. 1.1 min
20. 11.0 g
21. $31.8^\circ C$
22. $0.44 \ cal/g^\circ C$
1. 1600 cal (6720 J)
2. 1800 cal (7560 J)
23. 59.3 cal/g
24. $517 \;\mathrm{m/s}$
25. $1.15 \times 10^{12}\;\mathrm{K}$
26. .
27. $40 \;\mathrm{N}$
28. $\approx \ 10^{28}$ molecules
1. $21,000 \;\mathrm{Pa}$
2. Decreases to $61,000 \;\mathrm{Pa}$
3. $5.8 \;\mathrm{km}$ }}
1. No
2. allowed by highly improbable state. More likely states are more disordered.
29. a. $8.34 \times 10^{23}$ b. $6.64 \times 10^{-27}\;\mathrm{kg}$ c. $1600 \;\mathrm{m/s}$ d. $744 \;\mathrm{kPa}$ e. $4.2 \times 10^{20}$ or $0.0007 \;\mathrm{moles}$ g. $0.00785 \;\mathrm{m}^3$
1. $1.9 \;\mathrm{MW}$
2. $0.56 \;\mathrm{MW}$
3. $1.3 \;\mathrm{Mw}$
1. $54$%
2. $240 \;\mathrm{kW}$
3. $890 \;\mathrm{kW}$
4. $590 \;\mathrm{kW}$
5. $630 \;\mathrm{kg}$
1. $98$%
2. $4.0$%
3. $12$%
30. $14800 \;\mathrm{J}$
31. $12,000 \;\mathrm{J}$
32. b. $720 \;\mathrm{K}, 300 \;\mathrm{K}, 600 \;\mathrm{K}$ c. isochoric; isobaric d. $\;\mathrm{C}$ to $\;\mathrm{A}; \;\mathrm{B-C}$ e. $0.018 \;\mathrm{J}$
33. b. $300 \;\mathrm{K}, 1200 \;\mathrm{K}$
1. $1753 \;\mathrm{J}$
2. $-120 \;\mathrm{J}$
3. $80 \;\mathrm{J}$
4. $35 \;\mathrm{J}$
5. $-100 \;\mathrm{J}, 80 \;\mathrm{J}, 80 \;\mathrm{J}$

Feb 23, 2012

Aug 01, 2014