Download Some general information about thermodynamics

Some general information about thermodynamics
(gas laws, the laws of thermodynamics, pressure-volume graphs, and heat engines)
Alan D. Gleue
Lawrence High School
This discussion of thermodynamics was put together as a summary for information teachers might find
helpful at the high school level. For additional information, I have included a list of resources at the end.
Thermodynamics is the study of the flow of heat and our ability to harness this flow to create useable
work. A thermodynamic system is a definite amount of matter enclosed by surfaces or boundaries.
These surfaces may be real or imaginary. A thermodynamic system can be made such as a piston and
cylinder or a natural-occurring substance. Biological, chemical, and physical systems can be described
thermodynamically. The material around the thermodynamic system is called the surroundings.
We often like to discuss heat energy entering or leaving a system from the surroundings. There are
other times we can describe the system as insulated from heat flow. This is called a thermally-isolated
system. Even though the system is constructed of atoms and molecules, it is impossible to use Newton’s
Laws to analyze the motion of each particle. Even though the particles of system behave
microscopically, we look at macroscopic variables that can be measured such a pressure, temperature,
and volume.
See the figure below (http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node11.html)
It is common to describe our system as a gas in a piston. This is done historically as much of the
research of thermodynamics involved heat engines. Consequently, many concepts and equations about
gases are used. The properties of how gases act can be summarized by the kinetic theory of gases.
Experimental relationships among gas pressure, temperature, and volume can be expressed with the gas
laws.
Here are the gas laws (http://en.wikipedia.org/wiki/Gas_laws) typically discussed in high school
chemistry and physics courses:
-
Boyle’s Law: Pressure*Volume = constant, if Temperature is held constant [P1V1 = P2V2]
Charles’ Law: Volume/Temperature = constant, if Pressure is held constant [V1/T1 = V2/T2]
Gay-Lussac’s Law: Pressure/Temperature = constant, if Volume is held constant [P1/T1 = P2/T2]
Combined Gas Law: This incorporates the above three laws into one statement: [(P1 V1)/T1 = (P2
V2)/T2]. Essentially, the combine gas law states that PV/T = constant.
Avogadro’s Law: Volume/number of moles of the gas = constant [V α n]
Ideal Gas Law: PV = nRT. The constant is symbolized by ‘R’ and is called the gas law constant.
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It is important to note that all the prior gas laws can be derived from the Ideal Gas Law. For example if
we hold Temperature at a constant, then PV = nRT, and the right side becomes a combined constant and
then we have Boyle’s Law. It is important to note that these laws work for ideal gases. A real gas
behaves to a good approximation like an ideal gas in that in most cases we can ignore particle size and
particle attraction. In cases that we have high gas density or extremely cold temperatures, then our gas
laws become less accurate, and we have to add other variables to our equations. However, these
exceptions are beyond the nature of a high school course.
Many of the principles of thermodynamics and heat engines can be visually displayed on a pressurevolume (pV) graph. The pressure is displayed on the y axis and the volume on the x axis. Common units
for pressure would be psi or kilopascals. Volume is often displayed in mL or m3. The pressure, number
of moles, and temperature of a gas are called the state variables (there are others, too). The pressurevolume graph displays a point on the graph for a gas with specific values of these variables. When these
variables change for the gas, then the point moves along the graph to a different point.
For example, in the graph displayed below, the gas starts at state 1 with a volume of two volume units
and a pressure of 4 pressure units and changes to 6 volume units and 8 pressure units at state 2.
When a gas changes states, we say that it undergoes a thermodynamic process.
Temperature is indirectly shown on this graph. Even though the gas has the same volume in states 1
and 3, state 3 occurs at a higher temperature. States 2 and 3 exist at the same pressure but state 2
exists at a greater temperature. When a gas returns to its original state (State 1 to State 2 to State 3 and
then back to State 1, it undergoes a thermodynamic cycle.
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The gas laws discussed earlier can be portrayed on the pressure-volume graph as certain processes. We
can display Boyle’s Law as the following graph:
The curves generated are called isotherms as the points along the curve occur at the same temperature.
For example, if a gas goes from state 1 to state 2, the product of its volume and pressure represents a
constant. For this isotherm, the constant is equal to 8. When a gas goes from state 1 to state 2, the gas
is expanding; we can call this an isothermal expansion. If a gas starts at state 3 and goes to state 4, a
higher-temperature isotherm is generated, as we get a larger constant (in this case, 16). For this
isotherm, the gas is being compressed as it is losing volume, and this isotherm is called an isothermal
compression.
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For Charles’ Law, we hold the pressure constant and manipulate the temperature and volume of the
gas. A pV diagram might look like the following graph below.
Processes where the pressure stays constant are called isobaric processes. Going from state 1 to state
2, we have an expansion as our volume is increasing. Our temperature increases. Going the other
direction, we have an isobaric compression as our volume decreases. Isobaric processes represent
horizontal lines on a pV graph.
Vertical processes on a pV graph
reflect Gay-Lussac’s relationship.
The volume stays constant while
the pressure of the gas changes.
See the following graph. These
processes are called isometric or
isovolumetric or isochoric. A gas
going from state 1 to state 2 has a
pressure increase with no change
in volume. Its temperature
increases.
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Another common process is an adiabatic process. An adiabatic process occurs when no heat flows in or
out of the system. The system is isolated thermally from its surroundings. It is impossible to completely
isolate the system from the surroundings but approximate adiabatic processes can occur with good
insulation and if the process occurs fast enough so there is little time for heat to flow in our out of the
system. An adiabatic process is curved similarly to an isothermal (Boyle’s Law) but it is more
complicated:
P1V1γ = P2V2γ
Gamma (γ) is a dimensionless quantity that is the ratio of the specific heats (c) of the gas at constant
pressure and volume, i.e. γ = cp /cv. For a monatomic gas at 20 degrees Celsius (helium, for example)
the gamma is about 1.67. For a diatomic gas (hydrogen, for example) at the same temperature, the
gamma is approximately 1.4. Dry air at 20 degrees Celsius has a gamma of 1.4, too.
(http://en.wikipedia.org/wiki/Specific_heat_ratio)
The following is the graph of the two isotherms displayed earlier as compared with an adiabatic process,
called an adiabat:
If we begin at state 1, there is a volume of 2 units and a pressure of 8 units. At state 2 for an isothermal
expansion, we end with a volume of 8 units and a pressure of 2 units. However, with an adiabat, for a
volume of 8 units, the pressure would be 1.15 instead of 2. With this adiabat, there is also a cooling
effect which would not be seen with the isothermal expansion.
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These four processes: isothermal, isobaric, isometric, and adiabatic are all statements of the ideal gas
law. For example, during an isothermal we are holding the temperature constant. Additionally, we are
not changing the mass or molar amount of the gas. When we state the ideal gas law, PV = nRT, all the
variables on the right side are constant values and can be combined to form one constant, PV =
constant. We are back to Boyles’ Law which diagrammatically graph as isotherms.
In addition to this, we can summarize the nature of pressure and volume of a gas as a polytropic
process. In this case, we say that P*Vn = constant. Our value of the exponent will change depending
upon the process. For n = 1, we have Boyles’s Law and our process is isothermal. For n = 0, we have an
isobaric process and Charles’ Law is demonstrated. If n = ∞, we have Gay-Lussac’s relationship and an
isometric process. Finally, as discussed earlier, if n = γ (gamma), we have our adiabatic process. This
diagram and summary statements are helpful
(http://www.taftan.com/thermodynamics/POLYTROP.HTM):
Point 0 is the initial state of the system
•
•
•
•
•
•
•
•
0 to 1= constant pressure heating (isobaric expansion)
0 to 2= constant volume heating (isometric)
0 to 3= adiabatic compression
0 to 4= isothermal compression
0 to 5= constant pressure cooling (isobaric compression)
0 to 6= constant volume cooling (isometric)
0 to 7= adiabatic expansion
0 to 8= isothermal expansion
These are the four basic thermodynamic processes and how they are displayed on a pressure-volume
graph. We often take these pV processes and link them together to form a thermodynamic cycle. The
following picture shows an example of a thermodynamic cycle from the 2010 Advanced Placement
Physics B exam (http://apcentral.collegeboard.com/apc/public/repository/ap10_frq_physics_b.pdf):
In this case, we have four processes linked together to form a cycle. Two of them (AB and CD) are
isometric processes where the volume stays constant and two (BC and DA) are isobaric, where the
pressure stays constant.
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There areimportant ideas in thermodynamics called the laws of thermodynamics
(http://en.wikipedia.org/wiki/Laws_of_thermodynamics). The zeroth law of thermodynamics states
that if two thermodynamic systems are in thermal equilibrium with a third system, then the first two
systems are in thermal equilibrium.
http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/thereq.html#c2
The first law of thermodynamics is a conservation of energy equation. It states that the change of
internal energy of a system (ΔU) is equal to the heat added to the system (Q) minus any work (W) that
the system does on the surroundings. In an equation format we have: ΔU = Q - W.
Internal energy (U) is a measure of the total energy (kinetic, chemical, and potential energies) of the
particles of the system. Internal energy is related to the temperature of the system. When a system
gains internal energy from a process, this change of internal energy is a positive quantity (+ΔU).
Heat is symbolized by Q. When Q is positive, we add heat energy to the system. Physical work is
symbolized by W. When work is added by the system on the surroundings, W is positive. So the first
law of thermodynamics tells us that we can change the internal energy, U, of a system in two ways: we
can add or subtract heat, Q, and/or we can do work on the system or have the system do work on the
surroundings.
We can analyze our basic thermodynamic processes using a pressure-volume diagram and the first law.
For an isothermal process, we saw the following graph:
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
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Earlier, we mentioned that a system undergoing an isothermal process doesn’t change its temperature.
Because the temperature doesn’t change, the system will have no change to its internal energy (ΔU = 0).
Work can be visualized as the area under the process curve. Using the first law of thermodynamics, ΔU
= Q – W, we get 0 = Q – W or Q = W. This implies that the heat added to the system goes to work done
by the system on its surroundings. Numerically, we can derive the work (for an ideal gas) by using the
expression above. It is obtained by integrating the isothermal function.
Earlier, we saw that a constant pressure
process gives us a horizontal line on the
pressure-volume diagram. This constant
pressure process was called an isobaric
process and the curve is called an isobar.
The work, as before, is the area under the
isobar.
We have a gas expanding from an initial
volume to a final volume. Using the first law
of thermodynamics, ΔU = Q – W, we see that
the internal energy does increase when the
heat added to the system is greater than any
work done by the system on the
surroundings.
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Vertical processes displayed on a pressure-volume graph are called isometric (isovolumetric or isochoric)
processes. Pressure of the system changes but volume remains the same value. Because there is no
area under the process, the work done by or on the system equals zero.
http://www.ux1.eiu.edu/~cfadd/1150/14Thermo/work.html
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Using the First Law of Thermodynamics, ΔU = Q – W, W = 0, and ΔU = Q. To obtain an isometric process,
we could add heat (+Q) to a gas in a closed container; in this case, we would see the pressure of the gas
rise. Alternatively, we could cool a gas (-Q) in a closed container and its pressure would decrease.
Added heat would increase the internal energy (+ΔU) and subtracting heat would decrease the internal
energy (-ΔU). (picture: http://www.wikipremed.com/image_archive.php?code=010303)
Our last common process is the adiabatic compression or expansion. An adiabatic process occurs when
no heat is added or subtracted from the system. Using our First Law of Thermodynamics, ΔU = Q – W, Q
= 0. Consequently, ΔU = – W.
http://en.wikipedia.org/wiki/Adiabatic_process
When we have an adiabatic expansion, the system gains volume, cools, and work is done by the system
on the surroundings. Consequently, the system loses internal energy (-ΔU). When the system is
compressed adiabatically (see picture, http://www.wikipremed.com/image_archive.php?code=010303),
work is done on the system and its internal energy increases. No heat is added or subtracted. The work
either done by or on the system changes the internal energy.
I have found that this page from our textbook has a very helpful summary.
(Physics, Wilson,
Buffa, and Lou, 6th
edition, p. 409)
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This may also help students with pressure-volume graphs and interpreting how the quantities of the
First Law of Thermodynamics change during different processes.
(Physics, Wilson, Buffa, and Lou, 6th edition, p. 409)
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In terms of a pressure-volume graph, we looked at various processes. We said that we could put
processes together and come up with what is called a thermodynamic cycle. In this case the system’s
pressure, temperature, volume, and internal energy end up back where they started.
http://www.wikipremed.com/image_archive.php?code=010304
Important thermodynamic cycles are created by heat engines. A heat engine is a device that takes heat
(Q) and converts some of this heat into mechanical work (W) that can be used to power machines or
biological processes. A heat engine takes heat from a high temperature reservoir, converts some of this
heat to useable work, and then transfers the remaining heat to a low temperature reservoir.
So far we have stated the Zeroth and First Laws of Thermodynamics. Our discussion of heat engines can
lead us to a Second Law of Thermodynamics. The Second Law of Thermodynamics puts a limit on our
heat engine: one statement of the second law says that a one cannot create a heat engine that converts
heat energy completely into mechanical work –no machine is 100% efficient. In essence, one cannot
create perpetual motion machines which run forever with no additional energy or heat spontaneously
added from the surroundings. A concept called entropy is a measure of a thermodynamic system’s
ability to convert heat to useable work.
The Second Law of Thermodynamics can be stated in different ways depending upon the application and
which resources are consulted. Additional statements of this law as well as a further discussion of
entropy can be found at the following website: http://hyperphysics.phyastr.gsu.edu/hbase/thermo/seclaw.html#c3.
Within the high school setting, further
discussion about heat engines is helpful. As
mentioned above, a heat engine is a device
that produces some useable work from
heat energy. Heat engines often work in
repeating cycles that can be visually seen
on a pressure-volume graph.
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In terms of a pressure-volume graph, the net work can be
calculated per cycle. In the following pV diagram, we
have two isobaric processes and two isometric processes.
The net work is found by taking the area bounded by the
processes; in this case, we can take the area of the
rectangle.
There are similar questions found on advanced
placement exams. Here is one question taken from the
2006 AP Physics B exam:
These problems are often constructed to ask the students to find net work, system temperatures,
pressure and volume values using gas laws and the First and Second Laws of Thermodynamics.
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In the case of heat engines, one would like to find a series of processes that can be formed into a
thermodynamic cycle that maximizes the amount of useable work obtained. The engine can rely on
external combustion or internal combustion and can use different fuels and energy sources. Almost all
power planets, automobiles, and aircraft rely on internal or external combustion engines and high
school students would find studying several engines design and pV diagrams obtained by the engine a
useful activity.
This diagram shows the Rankine thermodynamic cycle. The Rankine thermodynamic cycle describes an
external combustion heat engine using steam and liquid water and can be found in many power plants.
The combustion fuels are often coal, oil, and natural gas. Its pressure –volume diagram includes two
isobaric processes and two adiabats. This cycle was first described by the Scottish engineer, William
J.M. Rankine in 1859.
Fundamentals of Classical Thermodynamics (Wylen, Sonntag, and Borgnakke), pp. 342-3
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Around the same period of time, European inventors and engineers were investigating engines that
would become known as internal combustion engines. Internal combustion engines have combustion
chambers inside the engine itself. In 1860, Jean Joseph Etienne Lenoir developed an engine that was
gas-fired instead of driven by steam. In 1861, the first 4-cycle engine was patented by Alphonse Beau
de Rochas. A year later, the German inventor, Nikolaus Otto, was the first individual to build and then
sell a modified version of this engine. This engine operates on the ‘Otto’ cycle and its pressure-volume
graph is included below. There are adiabatic, isobaric, and isometric processes involved with this cycle.
There are other types of internal combustion engine cycles. Another, used widely is the Diesel cycle,
which was developed by Rudolph Diesel in 1897. A Diesel engine operates differently than an Otto
engine in that it can operate at higher compression ratios. Fuel is compressed and ignited by the
compression instead of the use of spark plugs as in the Otto engine.
Another type of heat engine uses a turbine. Common uses include aircraft and ship propulsion.
Turbojets and turbofans are common types of aircraft engines. It was developed by George Brayton, an
American engineer in the 19th century. A picture of the Brayton cycle is found on the next page.
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The following is a table of different external and internal engine cycles and the thermodynamic
processes involved.
Cycle
Ericsson (1833), Brayton (external
combustion)
Bell Coleman (external combustion,
reverse Brayton)
Stoddard (external combustion)
Stirling (external combustion)
Ericsson (1853)
Otto
Diesel
Brayton (jet)
Lenoir (pulse jet)
Compression Heat
Process
Addition
Process
Adiabatic
Isobaric
Expansion
Process
Adiabatic
Heat
Rejection
Process
Isobaric
Adiabatic
Isobaric
Adiabatic
Isobaric
Adiabatic
Isothermal
Isothermal
Adiabatic
Adiabatic
Adiabatic
Isobaric
Isometric
Isometric
Isobaric
Isometric
Isobaric
Isobaric
Isometric
Adiabatic
Isothermal
Isothermal
Adiabatic
Adiabatic
Adiabatic
Adiabatic
Isometric
Isometric
Isobaric
Isometric
Isometric
Isobaric
Isobaric
http://en.wikipedia.org/wiki/Thermodynamic_cycle
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It was noted earlier that heat engines cannot convert all of the inputted heat energy to useable work.
Doing so would violate the Second Law of Thermodynamics. The efficiency of a heat engine is a ratio of
work out to heat energy in.
When expressed as a percentage, efficiencies range between zero and 100 percent. Efficiency is lost to
factors such as friction and heat loss. A typical Otto gasoline engine operates around 25 to 30%
efficient. Large coal fueled externally combustion power plant heat engines might have an efficiency of
around 45 to 50%. Diesel engines may have efficiencies around 50%.
Heat engines are constrained by the limit imposed on it by the Carnot cycle efficiency. The Carnot cycle
efficiency is the maximum efficiency set up between two temperatures, TH and TC. TH is the temperature
of the high temperature reservoir and TC is the temperature of the low temperature reservoir. The
Carnot cycle efficiency was studied by Sadi Carnot, a French engineer.
Carnot wanted to answer the question, “What is the maximum possible efficiency of a heat engine?” He
answered this question by analyzing theoretical processes of heat engines. Carnot found that two
reversible isothermal processes connected by two reversible adiabatic processes (isentropic processes)
can make for the most idealized, efficient engine if these processes are reversible. A reversible process
is one that can be taken in the other direction with no change in either the system or the surroundings.
An isentropic process will not gain or lose entropy during the process.
However, real heat engines do not have reversible processes, with the result leading to an overall net
increase of entropy and lost efficiency. Friction exists and systems cannot be perfectly insulated.
Consequently, real engines are always less efficient for a given set of temperatures that the Carnot
engine working at the same two temperatures.
The Carnot efficiency of any heat engine operating between two temperatures can be calculated with
the equation below (http://en.wikipedia.org/wiki/Carnot_cycle). On the following page is a diagram of
the Carnot processes and cyle.
where
W is the work done by the system (energy exiting the system as work),
QH is the heat put into the system (heat energy entering the system),
TC is the absolute temperature of the cold reservoir, and
TH is the absolute temperature of the hot reservoir.
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http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
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Students enrolling into an Advanced Physics program can take the AP Physics B or AP Physics C program.
The AP Physics C program is a calculus-based physics course that covers two areas of physics: mechanics
and electricity and magnetism. The AP Physics B course covers mechanics, fluids, thermodynamics,
electricity and magnetism, optics, and modern physics. It is a pre-calculus based course.
Students spend the year studying the concepts and then take the Advanced Physics exam in May of
every year. The exam is split into two parts, a multiple choice part and a free response part. On the free
response part students are given an equation sheet
(http://www.collegeboard.com/prod_downloads/ap/students/physics/physics_equation_tables.pdf)
that can be used on the exam. The following is the fluid mechanics and thermodynamic equations. I
included it in this document to show what concepts and equations are pertinent for the AP teacher to
cover. The fluids/thermodynamics questions are approximately 15% of the entire AP exam.
http://www.collegeboard.com/prod_downloads/ap/students/physics/physics_equation_tables.pdf
Much more can be written about thermodynamics. In this document, I wanted to present the basics of
the gas laws, the pressure-volume diagram, the Laws of Thermodynamics, and heat engines for the high
school instructor. There is much more about heat transfer, entropy, refrigerators, heat pumps, and
biological and chemical thermodynamics not included. Much more detail can be found on the internet
or in other sources (see next page).
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TEXTBOOKS I USED FOR THIS DOCUMENT
Physics, Wilson, Buffa and Lou, 6th edition, Pearson/Prentice Hall, 2007.
Fundamentals of Classical Thermodynamics, Wylen, Sonntag, and Borgnakke, 4th edition, John Wiley and
Sons, Inc., 1994.
I also used some of Dr. Chris Depcik’s notes and powerpoint documents. Dr. Depcik is a professor at the
University of Kansas and was one our mentoring professors for our project.
OPEN COURSEWORK OVER THE INTERNET
Most college physics textbooks will have several chapters about heat transfer and thermodynamics.
There are many internet and web-based links that also have information about thermodynamics. One
helpful resource is the MIT open courseware. Here are several links to pdf lecture notes from several of
the open courses:
http://ocw.mit.edu/courses/chemistry/5-60-thermodynamics-kinetics-spring-2008/lecture-notes/
http://ocw.mit.edu/courses/physics/8-01t-physics-i-fall-2004/lecture-notes/w13d1_class_31.pdf
http://ocw.mit.edu/courses/physics/8-01t-physics-i-fall-2004/lecture-notes/w13d2_class_32.pdf
http://ocw.mit.edu/courses/physics/8-21-the-physics-of-energy-fall-2009/lecturenotes/MIT8_21s09_lec04.pdf
http://ocw.mit.edu/courses/physics/8-21-the-physics-of-energy-fall-2009/lecturenotes/MIT8_21s09_lec09.pdf
OTHER HELPFUL WEBSITES OVER GENERAL THERMODYNAMICS
Another helpful website is a flowchart that summarizes different thermodynamic concepts. This
website has some good topical information:
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
The physics hypertextbook offers valuable information:
http://physics.info/
NASA’s thermodynamics site:
http://www.grc.nasa.gov/WWW/K-12/airplane/thermo.html
http://www.scribd.com/doc/10050141/Thermodynamics-Notes-9: these are some additional lecture
notes over thermodynamics.
http://www.nuceng.ca/ep716/chap4.pdf: This website is a text-book style website.
http://www.physicsplanet.com/articles/three-laws-of-thermodynamics: This is a short article about the
laws of thermodynamics.
http://www.physics.uc.edu/~sitko/CollegePhysicsIII/12-Thermodynamics/Thermodynamics.htm : this
has some general information about thermodynamics.
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http://en.wikipedia.org/wiki/Thermodynamics: this is the standard Wikipedia article over
thermodynamics.
VIDEOS ON THE INTERNET
There are also many videos posted over thermodynamics.
Here is a general youtube link:
http://www.youtube.com/results?search_query=thermodynamics+lecture&aq=1
Khan Academy has many thermodyanamics videos. Here is a link to his channel:
http://www.youtube.com/user/khanacademy
Activate, “playlists” and then type in ‘thermodynamics’
HISTORY OF THERMODYNAMICS
http://www.nuc.berkeley.edu/courses/classes/E-115/Slides/A_Brief_History_of_Thermodynamics.pdf:
this website includes some history of thermodynamics.
http://www.eoht.info/page/Laws+of+thermodynamics: This website discusses the history of the laws of
thermodynamics.
GRAPHICS
These links have thermodynamics graphics that can be pasted into documents:
http://www.wikipremed.com/image_archive.php?code=010303
http://www.wikipremed.com/image_archive.php?code=010304%2522
WEBSITES CONCERNING PRESSURE-VOLUME GRAPHS
http://www.grc.nasa.gov/WWW/K-12/airplane/pvtsplot.html: This is a website from NASA with good
background information.
HEAT ENGINE INFORMATION
http://tinyrock.com/docs/Heat-Engines.pdf
http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/engcyc.html#c2
http://www.antonine-education.co.uk/physics_a2/options/module_7/topic_4/topic_4.htm
(this site above also includes some question and answers)
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ANIMATIONS AND SIMULATIONS OF HEAT ENGINES AND GAS LAWS
http://www.educypedia.be/education/carjava.htm
http://www.shermanlab.com/science/physics/index.php
http://www.animatedengines.com/
http://auto.howstuffworks.com/diesel.htm
http://auto.howstuffworks.com/engine.htm
http://phet.colorado.edu/en/simulations/category/physics
http://websites.kahoks.org/Richert_Gary/sciweb/thermo.htm
http://www.walter-fendt.de/ph14e/gaslaw.htm
http://physics.bu.edu/~duffy/classroom.html (scroll down to the thermodynamics section)
RESOURCES ABOUT SADI CARNOT AND THE CARNOT ENGINE
http://www.ias.ac.in/resonance/Nov2001/pdf/Nov2001p42-48.pdf
http://en.wikipedia.org/wiki/Thermal_efficiency
http://galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/carnot.htm (Carnot cycle simulation)
AP Physics Websites:
http://apphysicsresources.blogspot.com/2008/05/ap-physics-b-thermodynamics-equations.html (this
includes a concise summary of thermodynamics and some practice questions)
http://apcentral.collegeboard.com/apc/public/courses/teachers_corner/2262.html (This is the AP
Physics B home page and this link has lots of information for the instructor)
http://apcentral.collegeboard.com/apc/members/courses/teachers_corner/44428.html (article about
PV diagrams and the First Law of Thermodynamics)
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