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Robert Fludd's 1618 "water screw" perpetual motion machine from a 1660 wood engraving. This device is widely credited as the first recorded attempt to describe such a device in order to produce useful work, that of driving millstones.[1]

The term perpetual motion, taken literally, refers to movement that goes on forever. However, the term more commonly refers to any device or system that perpetually (indefinitely) produces more energy than it consumes, resulting in a net output of energy for indefinite time. The law of conservation of energy, which states that energy cannot be created or destroyed, implies that such a perpetual motion machine cannot exist.

The most commonly contemplated type of perpetual motion machine is a mechanical system which (supposedly) sustains motion indefinitely, despite losing energy to friction and air resistance. A second type of impossible "perpetual motion machine" is one which does not violate conservation of energy, but produces work by spontaneously extracting heat from its surroundings, thereby cooling them down, and converting the heat energy into mechanical work. Such machines are forbidden by the second law of thermodynamics.


Basic principles

Perpetual motion violates either the first law of thermodynamics, the second law of thermodynamics, or both. The first law of thermodynamics is essentially a statement of conservation of energy. The second law can be phrased in several different ways, the most intuitive of which is that heat flows spontaneously from hotter to colder places; the most well known statement is that entropy tends to increase, or at the least stay the same; another statement is that no heat engine (an engine which produces work while moving heat between two separate places) can be more efficient than a Carnot heat engine. As a special case of this, any machine operating in a closed cycle cannot only transform thermal energy to work in a region of constant temperature.

Machines which are claimed not to violate either of the two laws of thermodynamics but rather to generate energy from unconventional sources are sometimes referred to as perpetual motion machines, although they are generally considered not to meet the standard criteria for the name. By way of example, it is possible to design a clock or other low-power machine, such as Cox's timepiece, which runs on the differences in barometric pressure or temperature between night and day. Such a machine has a source of energy, albeit one from which it is impractical to produce power in quantity.


It is customary to classify supposed perpetual motion machines according to which law of thermodynamics they purport to violate:

  1. A perpetual motion machine of the first kind produces energy from nothing, giving the user unlimited 'free' energy. It thus violates the law of conservation of energy.
  2. A perpetual motion machine of the second kind is a machine which spontaneously converts thermal energy into mechanical work. When the thermal energy is equivalent to the work done, this does not violate the law of conservation of energy. However it does violate the more subtle second law of thermodynamics (see also entropy). Such a machine is different from real heat engines (such as car engines), which always involve a transfer of heat from a hotter reservoir to a colder one, the latter being warmed up in the process. The signature of a perpetual motion machine of the second kind is that there is only one heat reservoir involved, which is being spontaneously cooled without involving a transfer of heat to a cooler reservoir. This conversion of heat into useful work, without any side effect, is impossible, according to the second law of thermodynamics. In contrast, for example, an automobile engine is not a perpetual motion machine because it works on the basis of the temperature difference between the fuel burning in the cylinder and the cooler environment outside. In order for it to function, some of the chemical energy released as heat when the fuel burns must not be converted to work, but must be exhausted to the cooler reservoir of the environment by the exhaust gas and the radiator. As explained by statistical mechanics, there are far more states in which heat distribution is close to thermodynamic equilibrium than states in which heat is concentrated in small regions, so temperatures will tend to even out over time, reducing the amount of free energy available for conversion to mechanical energy.
  3. A more obscure category is a perpetual motion machine of the third kind, usually (but not always)[2] defined as one that completely eliminates friction and other dissipative forces, to maintain motion forever (due to its mass inertia). Third in this case refers solely to the position in the above classification scheme, not the third law of thermodynamics. Although it is impossible to make such a machine,[3][4] as dissipation can never be 100% eliminated in a mechanical system, it is nevertheless possible to get very close to this ideal (see examples in the Low Friction section). Even if such a machine could be built, it would not serve as a source of energy but merely a perpetual energy storage device. A frictionless flywheel, for example, would eventually slow down and stop if its kinetic energy were tapped for useful work, and we would get no more energy out than the amount that was initially put in to spin up the flywheel.

Use of the term "impossible" and perpetual motion

October 1920 issue of Popular Science magazine, on perpetual motion. Although considered impossible by scientists, perpetual motion continues to capture the imagination of inventors. The device shown is a "mass leverage" device, where the spherical weights on our right have more leverage than those on the left, supposedly creating a perpetual rotation, but there are a greater number of weights to our left, balancing the device.

It is widely understood that the laws of physics are incomplete. Outside of pure mathematics, stating that things are absolutely impossible is considered un-scientific by many. Nevertheless, the term is commonly used to describe those things which absolutely cannot occur within the context of our current formulation of physical laws.[5]

The conservation laws are particularly robust. Noether's theorem states that any conservation law can be derived from a corresponding continuous symmetry.[6] In other words, so long as the laws of physics (not simply the current understanding of them, but the actual laws, which may still be undiscovered) and the various physical constants remain invariant over time — so long as the laws of the universe are fixed — then the conservation laws must be true, in the sense that they follow from the presupposition using mathematical logic. To put it the other way around: if perpetual motion or "overunity" machines were possible, then most of what we believe to be true about physics, mathematics, or both would have to be false.[7]

We can investigate whether the laws of physics are invariant over time: using telescopes we can examine the universe in the distant past; the fact that stars even exist and are, to the limits of our measurements, identical to stars today, is a direct visual demonstration that physics was similar in the past. Combining different measurements such as spectroscopy, direct measurement of the speed of light in the past and similar measurements demonstrates that physics appears to have remained substantially the same, if not identical, for all of observable history spanning billions of years.[8]

The principles of thermodynamics are so well established, both theoretically and experimentally, that proposals for perpetual motion machines are universally met with disbelief on the part of physicists. Any proposed perpetual motion design offers a potentially instructive challenge to physicists: one is almost completely certain that it can't work, so one must explain how it fails to work. The difficulty (and the value) of such an exercise depends on the subtlety of the proposal; the best ones tend to arise from physicists' own thought experiments and often shed light upon certain aspects of physics.

The law that entropy always increases, holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations — then so much the worse for Maxwell's equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation. — Sir Arthur Stanley Eddington, The Nature of the Physical World (1927)

Thought experiments

Serious work in theoretical physics often involves thought experiments that test the boundaries of understanding of physical laws. Some such thought experiments involve apparent perpetual motion machines, and insight may be had from understanding why they either don't work or work in a way that does not violate the laws of physics.

  • Maxwell's demon: A thought experiment which led to physicists considering the interaction between entropy and information.
  • Feynman's "Brownian ratchet": A "perpetual motion" machine which extracts work from thermal fluctuations and appears to run forever but really only runs as long as the environment is warmer than the ratchet.
  • Self-perpetuating cosmic inflation: Andrei Linde has proposed that during the theoretical period of cosmic inflation in the early universe, quantum fluctuations in energy could be magnified by the very inflationary process, preventing the global cooling trend from ever being fully consummated. This would violate both the first and second laws of thermodynamics; indeed, it may constitute the origin of a low-entropy past that gets the second law going in the first place. However, a machine to harness this principle would have several serious flaws. It would need to use unimaginable amounts of energy (on at least a Planck scale); it might have consequences cataclysmic to the area around it for an unknown distance (there is no prior natural limit to the scale of the damage); and at least the majority of and possibly all the energy it generated would be in a newly-created universe which might be inaccessibly far away along a wormhole.


One day man will connect his apparatus to the very wheelwork of the universe [...] and the very forces that motivate the planets in their orbits and cause them to rotate will rotate his own machinery.

Nikola Tesla

Some common ideas recur repeatedly in perpetual motion machine designs. Many ideas that continue to appear today were stated as early as 1670 by John Wilkins, Bishop of Chester and an official of the Royal Society. He outlined three potential sources of power for a perpetual motion machine, "Chymical Extractions", "Magnetical Virtues" and "the Natural Affection of Gravity".[1]

The seemingly mysterious ability of magnets to influence motion at a distance without any apparent energy source has long appealed to inventors. One of the earliest examples of a system using magnets was proposed by Wilkins and has been widely copied since: it consists of a ramp with a magnet at the top, which pulled a metal ball up the ramp. Near the magnet was a small hole that was supposed to allow the ball to drop under the ramp and return to the bottom, where a flap allowed it to return to the top again. The device simply could not work: any magnet strong enough to pull the ball up the ramp would necessarily be too powerful to allow it to drop through the hole. Faced with this problem, more modern versions typically use a series of ramps and magnets, positioned so the ball is to be handed off from one magnet to another as it moves. The problem remains the same.

Perpetuum Mobile of Villard de Honnecourt (about 1230).

Gravity also acts at a distance, without an apparent energy source. But to get energy out of a gravitational field (for instance, by dropping a heavy object, producing kinetic energy as it falls) you have to put energy in (for instance, by lifting the object up), and some energy is always dissipated in the process. A typical application of gravity in a perpetual motion machine is Bhaskara's wheel in the 12th century, whose key idea is itself a recurring theme, often called the overbalanced wheel: Moving weights are attached to a wheel in such a way that they fall to a position further from the wheel's center for one half of the wheel's rotation, and closer to the center for the other half. Since weights further from the center apply a greater torque, the result is (or would be, if such a device worked) that the wheel rotates forever. The moving weights may be hammers on pivoted arms, or rolling balls, or mercury in tubes; the principle is the same.

Yet another theoretical machine involves a frictionless environment for motion. This involves the use of diamagnetic or electromagnet levitation to float an object. This is done in a vacuum to eliminate air friction and friction from an axle. The levitated object is then free to rotate around its center of gravity without interference. However, this machine has no practical purpose because the rotated object cannot do any work as work requires the levitated object to cause motion in other objects, bringing friction into the problem.

To extract work from heat, thus producing a perpetual motion machine of the second kind, the most common approach (dating back at least to Maxwell's demon) is unidirectionality. Only molecules moving fast enough and in the right direction are allowed through the demon's trap door. In a Brownian ratchet, forces tending to turn the ratchet one way are able to do so while forces in the other direction aren't. A diode in a heat bath allows through currents in one direction and not the other. These schemes typically fail in two ways: either maintaining the unidirectionality costs energy (Maxwell's demon needs light to look at all those particles and see what they're doing), or the unidirectionality is an illusion and occasional big violations make up for the frequent small non-violations (the Brownian ratchet will be subject to internal Brownian forces and therefore will sometimes turn the wrong way).

Invention history

Orffyreus Wheel. The device was designed by Johann Bessler.

The earliest references to perpetual motion machines, by an Indian mathematician-astronomer, Bhāskara II, date back to 1150. He described a wheel that he claimed would run forever.[9]

Villard de Honnecourt in 1235 described, in a 33 page manuscript, a perpetual motion machine of the first kind. His idea was based on the changing torque of a series of weights attached with hinges to the rim of a wheel. While ascending they would hang close to the wheel and have little torque, but they would topple after reaching the top and drag the wheel down on descent due to their greater torque during the swing. His device spawned a variety of imitators that continued to refine the basic design.

In 1607 Cornelius Drebbel in "Wonder-vondt van de eeuwighe bewegingh" dedicated a Perpetuum motion machine to James I of England. It was described by Heinrich Hiesserle von Chodaw in 1621.

Robert Boyle's self-flowing flask appears to fill itself through siphon action. This is not possible in reality: a siphon requires its "output" to be lower than the "input".

Blaise Pascal introduced a primitive form of roulette and the roulette wheel in the 17th century in his search for a perpetual motion machine.[10]

In 1775 the Royal Academy of Sciences in Paris issued the statement that the Academy "will no longer accept or deal with proposals concerning perpetual motion". Johann Bessler (also known as Orffyreus) created a series of claimed perpetual motion machines in the 18th century. In the 19th century, the invention of perpetual motion machines became an obsession for many scientists. Many machines were designed based on electricity, but none of them lived up to their promises. Another early prospector in this field was John Gamgee. Gamgee developed the Zeromotor, a perpetual motion machine of the second kind.

Devising these machines is a favourite pastime of many eccentrics, who often come up with elaborate machines in the style of Rube Goldberg or Heath Robinson. These designs may appear to work on paper at first glance. Usually, though, various flaws or obfuscated external power sources have been incorporated into the machine. Such activity has made them useless in the practice of "invention".


Proposals for such inoperable machines have become so common that the United States Patent and Trademark Office (USPTO) has made an official policy of refusing to grant patents for perpetual motion machines without a working model. The USPTO Manual of Patent Examining Practice states:

With the exception of cases involving perpetual motion, a model is not ordinarily required by the Office to demonstrate the operability of a device. If operability of a device is questioned, the applicant must establish it to the satisfaction of the examiner, but he or she may choose his or her own way of so doing.[11]

And, further, that:

A rejection [of a patent application] on the ground of lack of utility includes the more specific grounds of inoperativeness, involving perpetual motion. A rejection under 35 U.S.C. 101 for lack of utility should not be based on grounds that the invention is frivolous, fraudulent or against public policy.[12]

The filing of a patent is a clerical task, and the USPTO won't refuse filings for perpetual motion machines; the patent will be filed and then most probably rejected by the patent examiner, after he has done a formal examination.[13] Even if the patent is granted, it doesn't mean that the invention actually works, it just means that the examiner thinks that it works, or that he couldn't figure out why it wouldn't work.[13]

The USPTO maintains a collection of Perpetual Motion Gimmicks as Digest 9 in Class 74

The USPTO has granted a few patents for motors that are claimed to run without net energy input. Some of these are:

Howard R. Johnson, U.S. Patent 4,151,431
  • Johnson, Howard R., U.S. Patent 4,151,431 "Permanent magnet motor", April 24, 1979[13]
  • Baker, Daniel, U.S. Patent 4,074,153 "Magnetic propulsion device", February 14, 1978
  • Hartman; Emil T., U.S. Patent 4,215,330 "Permanent magnet propulsion system", December 20, 1977 (this device is related to the Simple Magnetic Overunity Toy (SMOT)),
  • Flynn; Charles J., U.S. Patent 6,246,561 "Methods for controlling the path of magnetic flux from a permanent magnet and devices incorporating the same", July 31, 1998
  • Patrick, et al., U.S. Patent 6,362,718 "Motionless electromagnetic generator" , March 26, 2002
  • Green, Willie A., U.S. Patent 6,526,925 "Piston Driven Rotary Engine", March 4, 2003 "Fluid driven device utilizing a leveraged system with minimal displacement"
  • Goldenblum, Halm, U.S. Patent 6,962,052 "Energy generation mechanism, device and system", November 8, 2005 "A chamber with a partition which lets gas molecules flow one way and not the other. The pressure which builds up on one side of the partition is used to drive a generator."
  • Flynn, Joe, U.S. Patent 6,246,561 "Methods for controlling the path of magnetic flux from a permanent magnet and devices incorporating the same", June 12, 2001
  • Gates; Glenn A., U.S. Patent 6,523,646 "Spring driven apparatus", February 23, 2003 "Energy is stored in the springs and power is generated by way of the various forces which cause the springs to wind and unwind."
  • McQueen; Jesse, U.S. Patent 7,095,126 "Internal energy generating power source", August 22, 2006 "An external power source such as a battery is used to initially supply power to start an alternator and generator. Once the system has started it is not necessary for the battery to supply power to the system. The battery can then be disconnected. The alternator and electric motor work in combination to generate electrical power." Examiners: Schuberg, Darren ; Mohandesi, Iraj A.
  • Haisch, et al. U.S. Patent 7,379,286 "Quantum vacuum energy extraction", May 27, 2008 "[...] converting energy from the electromagnetic quantum vacuum available at any point in the universe to usable energy in the form of heat, electricity, mechanical energy or other forms of power. [...] When atoms enter into suitable micro Casimir cavities a decrease in the orbital energies of electrons in atoms will thus occur. Such energy will be captured in the claimed devices. Upon emergence form such micro Casimir cavities the atoms will be re-energized by the ambient electromagnetic quantum vacuum. [...] process is also consistent with the conservation of energy in that all usable energy does come at the expense of the energy content of the electromagnetic quantum vacuum."

In 1979, Joseph Newman filed a US Patent application for his "energy machine" which unambiguously claimed over-unity operation, where power output exceeded power input; the source of energy was claimed to be the atoms of the machine's copper conductor.[14] The Patent Office rejected the application after the National Bureau of Standards measured the electrical input to be greater than the electrical output. Newman challenged the decision in court and lost.[15]

Other patent offices around the world, such as the United Kingdom Patent Office, have similar practices. Section 4.05 of the UKPO Manual of Patent Practice states:

Processes or articles alleged to operate in a manner which is clearly contrary to well-established physical laws, such as perpetual motion machines, are regarded as not having industrial application.[16]

Examples of decisions by the UK Patent Office to refuse patent applications for perpetual motion machines include:[17]

Decision BL O/044/06, John Frederick Willmott's application no. 0502841[18]
Decision BL O/150/06, Ezra Shimshi's application no. 0417271[19]

The European Patent Classification (ECLA) has classes including patent applications on perpetual motion systems: ECLA classes "F03B17/04: Alleged perpetua mobilia ..." and "F03B17/00B: [... machines or engines] (with closed loop circulation or similar : ...Installations wherein the liquid circulates in a closed loop; Alleged perpetua mobilia of this or similar kind ...".[20]

Recent examples

As the term "perpetual energy" increasingly became associated with fraud in the late 19th century, inventors have generally come to avoid using it. One common alternative term used is "over-unity," even though it has essentially the same meaning. Today devices described as perpetual motion devices claim to operate by extracting "zero point energy" or some other source of external energy. Modern demonstrations of such machines sometimes involve using a small battery (called an exciter) to spin a flywheel which then turns a generator to (briefly) illuminate a common mains-powered type of lightbulb; this is a straightforward conversion between potential and kinetic energy.[citation needed]

Apparent perpetual motion machines

Even though they fully respect the laws of thermodynamics, there are a few conceptual or real devices that appear to be in "perpetual motion." Closer analysis reveals that they actually "consume" some sort of natural resource or latent energy, such as the phase changes of water or other fluids or small natural temperature gradients. In general, extracting large amounts of work using these devices is difficult to impossible.

Resource consuming

Some examples of such devices include:

  • The drinking bird toy functions using small ambient temperature gradients and evaporation.
  • A capillarity based water pump functions using small ambient temperature gradients and vapour pressure differences.
  • A Crookes radiometer consists of a partial vacuum glass container with a lightweight propeller moved by (light-induced) temperature gradients.
  • Any device picking up minimal amounts of energy from the natural electromagnetic radiation around it, such as a solar powered motor.
  • The Atmos clock uses changes in the vapor pressure of ethyl chloride with temperature to wind the clock spring.
  • A device powered by radioactive decay from an isotope with a relatively long half-life; such a device could plausibly operate for hundreds or thousands of years.

Low friction

  • In flywheel energy storage, "modern flywheels can have a zero-load rundown time measurable in years."[21]
  • Once spun up, objects in the vacuum of space—stars, black holes, planets, moons, spin-stabilized satellites, etc.—continue spinning almost indefinitely with no further energy input.
  • In certain quantum-mechanical systems (such as superfluidity and superconductivity), dissipation-free "motion" is possible.

Free energy suppression

Because perpetual motion claims have been around for some time, conspiracy theories are often invoked to explain the lack of acceptance and/or availability of such technology.


This is a gallery of some of the perpetual motion machine plans.

See also


  1. ^ a b Angrist, Stanley (January 1968). "Perpetual Motion Machines". Scientific American 218 (1): 115–122. 
  2. ^ An alternative definition is given, for example, by (Schadewald 2008:55-56), who defines a "perpetual motion machine of the third kind" as a machine that violates the third law of thermodynamics.
  3. ^ Wong, Kau-Fui Vincent (2000), Thermodynamics for Engineers, CRC Press, p. 154, ISBN 978-0-84-930232-9, 
  4. ^ Akshoy, Ranjan Paul; Sanchayan, Mukherjee; Pijush, Roy (2005), Mechanical Sciences: Engineering Thermodynamics and Fluid Mechanics, Prentice-Hall India, p. 51, ISBN 978-8-12-032727-6, 
  5. ^ Barrow, John D.. "1". Impossibility:The limits of science and the science of limits. 
  6. ^ Goldstein, Herbert; Poole, Charles; Safko, John (2002), Classical Mechanics (3rd edition), San Francisco: Addison Wesley, pp. 589–598, ISBN 0-201-65702-3 
  7. ^ "The perpetual myth of free energy". BBC News. 9 July 2007. Retrieved 9 July 2007. 
  8. ^ "CE410: Are constants constant?", talkorigins
  9. ^ Lynn Townsend White, Jr. (April 1960). "Tibet, India, and Malaya as Sources of Western Medieval Technology", The American Historical Review 65 (3), p. 522-526.
  10. ^ MIT, "Inventor of the Week Archive: Pascal : Mechanical Calculator", May 2003. "Pascal worked on many versions of the devices, leading to his attempt to create a perpetual motion machine. He has been credited with introducing the roulette machine, which was a by-product of these experiments."
  11. ^ "600 Parts, Form, and Content of Application - 608.03 Models, Exhibits, Specimens", Manual of Patent Examining Procedure (8 ed.), August 2001, 
  12. ^ "700 Examination of Applications II. UTILITY - 706.03(a) Rejections Under 35 U.S.C. 101", Manual of Patent Examining Procedure (8 ed.), August 2001, 
  13. ^ a b c Nolo, ed (2008). Patent It Yourself (13, illustrated, revised ed.). p. 99. ISBN 1413308546.,151,431&client=opera&hl=es. 
  14. ^ Peterson, Ivars (1 June 1985). "A patent pursuit: Joe Newman’s ‘energy machine’ - inventor fights for patent on machine that generates more energy than it takes in from external sources". Science News. 
  15. ^ Peterson, Ivars (5 July 1986). "NBS Report Short Circuits Energy Machine". Science News. Retrieved 8 May 2009. 
  16. ^ Manual of Patent Practice, Section 4, United Kingdom Patent Office, 
  17. ^ See also, for more examples of refused patent applications at the United Kingdom Patent Office (UK-IPO), UK-IPO gets tougher on perpetual motion, IPKat, 12 June 2008. Consulted on June 12, 2008.
  18. ^ Patents Ex parte decision (O/044/06)
  19. ^
  20. ^ ECLA classes F03B17/04 and F03B17/00B. Consulted on June 12, 2008.
  21. ^ WO patent application 2008037004, "An energy storage device and method of use", published 2008-04-03  

External links

Simple English

[[File:|thumb|right|Robert Fludd's 1618 "water screw" perpetual motion machine from a 1660 woodcut. This device is widely credited as the first recorded attempt to describe such a device in order to produce useful work - driving millstones.[1]]]

Perpetual motion refers to a movement that goes on forever. A machine that could be set in motion once would continue to move forever. Such a device or system would be against the law of conservation of energy. This law states that energy cannot be created, or destroyed, but one form of energy can be transformed into a different one.

Perpetual motion machines have interested inventors for a long time, and even though it has been shown that such a machine cannot exist, people still try to build them.

Perpetual motion machines have been classified into different types. Each type breaks a different law of thermodynamics:

  1. A perpetual motion machine of the first kind produces energy from nothing. It gives its user unlimited free energy. It violates the law of the conservation of energy.[2]
  2. A perpetual motion machine of the second kind spontaneously converts thermal energy into mechanical work. In itself, this does not violate the law of the conservation of energy,because there is the same amount of mechanical work than there was of thermal energy. Such machines violate the Second law of thermodynamics. Real heat engines, as they are found in cars, also convert heat energy to work. Such engines have two reservoirs: the heat is transferred from the hot one to the cold one. Some of the heat is converted into mechanical work, the rest is used to heat the cold reservoir. Perpetual motion machines only have one reservoir for the heat. For this reason, they can convert all the heat into mechanical work.
  3. A perpetual motion machine of the third kind is one that can eliminate effects like friction and other effects that use energy. Because of the intertia of the mass, it can therefore maintain motion forever. This violates the third law of thermodynamics. Although it is impossible to make such a machine,[3][4] because the loss of energy cannot be completely eliminated in a mechanical system, it is still possible to reduce the loss a great deal. Even if such a machine could be built, it could only be used to store energy forever, and not to produce energy. A frictionless flywheel, for example, would eventually slow down and stop if its kinetic energy were tapped for useful work, and we would get no more energy out than the amount that was initially put in to spin up the flywheel.


  1. Stanley Angrist, "Perpetual Motion Machines", Scientific American", January 1968, Vol. 218, No. 1, pp. 115-122
  2. "" (htm). 1999. Retrieved 2010-08-02. 
  3. Wong, Kau-Fui Vincent (2000). Thermodynamics for Engineers. CRC Press. p. 154. ISBN 978-0-84-930232-9. 
  4. Akshoy, Ranjan Paul; Sanchayan, Mukherjee; Pijush, Roy (2005). Mechanical Sciences: Engineering Thermodynamics and Fluid Mechanics. Prentice-Hall India. p. 51. ISBN 978-8-12-032727-6. 

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