2054:), the gas is in thermal contact with the hot temperature reservoir, and is thermally isolated from the cold temperature reservoir. The gas is allowed to expand, doing work on the surroundings by pushing up the piston (Stage One figure, right). Although the pressure drops from points 1 to 2 (figure 1) the temperature of the gas does not change during the process because the heat transferred from the hot temperature reservoir to the gas is exactly used to do work on the surroundings by the gas. There is no change in the gas internal energy, and no change in the gas temperature if it is an ideal gas. Heat
2359:
2202:
2145:
2009:
2693:
2394:
1978:
42:
1612:
2726:
2777:, is the amount of work energy exchanged by the system with its surroundings. The amount of heat exchanged with the hot reservoir is the sum of the two. If the system is behaving as an engine, the process moves clockwise around the loop, and moves counter-clockwise if it is behaving as a refrigerator. The efficiency to the cycle is the ratio of the white area (work) divided by the sum of the white and red areas (heat absorbed from the hot reservoir).
2380:) Once again the gas in the engine is thermally insulated from the hot and cold reservoirs, and the engine is assumed to be frictionless and the process is slow enough, hence reversible. During this step, the surroundings do work on the gas, pushing the piston down further (Stage Four figure, right), increasing its internal energy, compressing it, and causing its temperature to rise back to the temperature infinitesimally less than
4317:
3104:
3134:
2673:), the isothermal stages follow the isotherm lines for the working fluid, the adiabatic stages move between isotherms, and the area bounded by the complete cycle path represents the total work that can be done during one cycle. From point 1 to 2 and point 3 to 4 the temperature is constant (isothermal process). Heat transfer from point 4 to 1 and point 2 to 3 are equal to zero (adiabatic process).
2177:. The gas continues to expand with reduction of its pressure, doing work on the surroundings (raising the piston; Stage Two figure, right), and losing an amount of internal energy equal to the work done. The loss of internal energy causes the gas to cool. In this step it is cooled to a temperature that is infinitesimally higher than the cold reservoir temperature
4360:
Looking at this formula an interesting fact becomes apparent: Lowering the temperature of the cold reservoir will have more effect on the ceiling efficiency of a heat engine than raising the temperature of the hot reservoir by the same amount. In the real world, this may be difficult to achieve since the cold reservoir is often an existing ambient temperature.
4359:
Rearranging the right side of the equation gives what may be a more easily understood form of the equation, namely that the theoretical maximum efficiency of a heat engine equals the difference in temperature between the hot and cold reservoir divided by the absolute temperature of the hot reservoir.
4395:
In mesoscopic heat engines, work per cycle of operation in general fluctuates due to thermal noise. If the cycle is performed quasi-statically, the fluctuations vanish even on the mesoscale. However, if the cycle is performed faster than the relaxation time of the working medium, the fluctuations of
2910:
which is the amount of heat transferred in the process. If the process moves the system to greater entropy, the area under the curve is the amount of heat absorbed by the system in that process; otherwise, it is the amount of heat removed from or leaving the system. For any cyclic process, there is
2919:
diagrams for a clockwise cycle, the area under the upper portion will be the energy absorbed by the system during the cycle, while the area under the lower portion will be the energy removed from the system during the cycle. The area inside the cycle is then the difference between the two (the
4237:. This time, the cycle remains exactly the same except that the directions of any heat and work interactions are reversed. Heat is absorbed from the low-temperature reservoir, heat is rejected to a high-temperature reservoir, and a work input is required to accomplish all this. The
3099:
diagram is (a) equal to the total work performed by the system on the surroundings if the loop is traversed in a clockwise direction, and (b) is equal to the total work done on the system by the surroundings as the loop is traversed in a counterclockwise direction.
4697:
can improve the thermal efficiency of steam power plants and why the thermal efficiency of combined-cycle power plants (which incorporate gas turbines operating at even higher temperatures) exceeds that of conventional steam plants. The first prototype of the
5182:
2817:). For a simple closed system (control mass analysis), any point on the graph represents a particular state of the system. A thermodynamic process is represented by a curve connecting an initial state (A) and a final state (B). The area under the curve is:
4987:...since the Carnot heat engine, setting an upper bound on the efficiency of a heat engine is an ideal, reversible engine of which a single cycle must be performed in infinite time which is impractical and so the Carnot engine has zero power.
3669:
4224:
This is the Carnot heat engine working efficiency definition as the fraction of the work done by the system to the thermal energy received by the system from the hot reservoir per cycle. This thermal energy is the cycle initiator.
3065:
4594:
4518:
2721:. The vertical axis is the system temperature, the horizontal axis is the system entropy. A-to-B (isothermal expansion), B-to-C (isentropic expansion), C-to-D (isothermal compression), D-to-A (isentropic compression).
1936:
4396:
work are inevitable. Nevertheless, when work and heat fluctuations are counted, an exact equality relates the exponential average of work performed by any heat engine to the heat transfer from the hotter heat bath.
5179:
2481:
3255:
4123:
3453:
2555:
4022:
3352:
4420:. In addition, real engines that operate along the Carnot cycle style (isothermal expansion / isentropic expansion / isothermal compression / isentropic compression) are rare. Nevertheless, Equation
2920:
absorbed net heat energy), but since the internal energy of the system must have returned to its initial value, this difference must be the amount of work done by the system per cycle. Referring to
4198:
1331:
2892:
3927:
1801:
done by the system or engine to the environment per Carnot cycle depends on the temperatures of the thermal reservoirs and the entropy transferred from the hot reservoir to the system
2315:
2137:
4596:
at which the first integral is over a part of a cycle where heat goes into the system and the second integral is over a cycle part where heat goes out from the system. Then, replace
2248:. The surroundings do work on the gas, pushing the piston down (Stage Three figure, right). An amount of energy earned by the gas from this work exactly transfers as a heat energy
2351:
3527:
4522:
2653:
1166:
1111:
1056:
4446:
863:
816:
731:
684:
596:
549:
1773:, merely transferred between the thermal reservoirs and the system without gain or loss. When work is applied to the system, heat moves from the cold to hot reservoir (
1822:
767:
635:
1001:
4311:
4284:
4233:
A Carnot heat-engine cycle described is a totally reversible cycle. That is, all the processes that compose it can be reversed, in which case it becomes the Carnot
4218:
3870:
3841:
3812:
3779:
3750:
3721:
3559:
3549:
2611:
2584:
1963:
1759:
1732:
4363:
In other words, the maximum efficiency is achieved if and only if entropy does not change per cycle. An entropy change per cycle is made, for example, if there is
4323:: A real engine (left) compared to the Carnot cycle (right). The entropy of a real material changes with temperature. This change is indicated by the curve on a
2783:(energy lost to the cold reservoir) can be seen as a direct subtraction, or expressed as the sum of a negative quantity, which can lead to different conventions.
500:
3475:
2079:
1799:
839:
792:
707:
660:
572:
525:
5131:
Kostic, M (2011). "Revisiting The Second Law of Energy
Degradation and Entropy Generation: From Sadi Carnot's Ingenious Reasoning to Holistic Generalization".
2404:
to illustrate the work done. 1-to-2 (isothermal expansion), 2-to-3 (isentropic expansion), 3-to-4 (isothermal compression), 4-to-1 (isentropic compression).
1761:(referred to as the hot and cold reservoirs, respectively), and a part of this transferred energy is converted to the work done by the system. The cycle is
2936:
2415:
4739:, the Carnot engine may be thought as the theoretical limit of macroscopic scale heat engines rather than any practical device that could ever be built.
2038:. (The infinitesimal temperature difference allows the heat to transfer into the gas without a significant change in the gas temperature. This is called
3149:
2387:
due solely to the work added to the system, but the entropy remains unchanged. At this point the gas is in the same state as at the start of step 1.
3357:
2486:
4338:). Irreversible systems and losses of energy (for example, work due to friction and heat losses) prevent the ideal from taking place at every step.
1641:
1827:
3262:
5234:
4731:. However, on a macroscopic scale limitations placed by the model's assumptions prove it impractical, and, ultimately, incapable of doing any
4245:
diagram of the reversed Carnot cycle is the same as for the Carnot heat-engine cycle except that the directions of the processes are reversed.
1342:
1230:
4379:, the required dumping of heat into the environment to dispose of excess entropy leads to a (minimal) reduction in efficiency. So Equation
464:
4355:
gives the maximum efficiency possible for any engine using the corresponding temperatures. A corollary to Carnot's theorem states that:
4027:
2173:) the gas in the engine is thermally insulated from both the hot and cold reservoirs, thus they neither gain nor lose heat. It is an
1320:
4347:
No engine operating between two heat reservoirs can be more efficient than a Carnot engine operating between those same reservoirs.
5396:
5032:
1353:
3932:
4426:
is extremely useful for determining the maximum efficiency that could ever be expected for a given set of thermal reservoirs.
5172:
5121:
5095:
5068:
923:
4128:
4736:
4254:
1669:
1634:
1221:
890:
457:
335:
2825:
4753:
4727:. On a practical human-scale level the Carnot cycle has proven a valuable model, as in advancing the development of the
4386:
2409:
1762:
273:
3144:
Evaluation of the above integral is particularly simple for a Carnot cycle. The amount of energy transferred as work is
5227:
3354:
and the total amount of heat transferred from the system to the cold reservoir (in the isothermal compression) will be
2707:), illustrated on a TS (temperature T–entropy S) diagram. The cycle takes place between a hot reservoir at temperature
1405:
1379:
900:
354:
5462:
4234:
3480:
306:
3879:
2019:
expansion. Heat (as an energy) is transferred reversibly from the hot temperature reservoir at constant temperature
1458:
929:
328:
4711:
2209:
Isothermal compression. Heat is transferred reversibly to the low temperature reservoir at a constant temperature
5500:
2788:
2687:
1627:
3259:
The total amount of heat transferred from the hot reservoir to the system (in the isothermal expansion) will be
1558:
90:
2262:
2084:
1665:
1453:
5536:
5220:
1533:
1306:
283:
1781:). When heat moves from the hot to the cold reservoir, the system applies work to the environment. The work
5447:
2928:, mathematically, for a reversible process, we may write the amount of work done over a cyclic process as:
2747:
2156:
918:
121:
111:
17:
4694:
4411:
2664:
126:
116:
2773:|, is the amount of energy exchanged between the system and the cold reservoir. The area in white,
5361:
5251:
2320:
2244:
to allow heat transfer from the gas to the cold reservoir. There is no change in temperature, it is an
2237:, and is thermally isolated from the hot reservoir. The gas temperature is infinitesimally higher than
1410:
1374:
152:
86:
1448:
5515:
1203:
951:
397:
210:
200:
5153:
4833:
Holubec Viktor and Ryabov Artem (2018). "Cycling Tames Power
Fluctuations near Optimum Efficiency".
5276:
5495:
5480:
5470:
5426:
2616:
1615:
1443:
1240:
1121:
1066:
1011:
943:
882:
418:
407:
73:
5196:
845:
798:
713:
666:
578:
531:
5148:
2061:> 0 is absorbed from the hot temperature reservoir, resulting in an increase in the entropy
1548:
1265:
349:
103:
78:
5087:
5080:
3664:{\displaystyle \eta ={\frac {W}{Q_{H}}}={\frac {Q_{H}-Q_{C}}{Q_{H}}}=1-{\frac {T_{C}}{T_{H}}}}
3091:, its integral over any closed loop is zero and it follows that the area inside the loop on a
1804:
1468:
749:
614:
5341:
5301:
1685:
1483:
1060:
373:
219:
68:
971:
5485:
5203:
5140:
4913:
4852:
4724:
4289:
4262:
4203:
3848:
3819:
3790:
3782:
3757:
3728:
3699:
3534:
2589:
2562:
1941:
1737:
1710:
1696:
1692:
system in creating a temperature difference through the application of work to the system.
1563:
1488:
1478:
278:
140:
8:
5490:
5391:
5243:
5207:
4925:
2700:
1995:
1661:
1508:
1270:
292:
258:
253:
166:
5144:
4917:
4856:
4357:
All reversible engines operating between the same heat reservoirs are equally efficient.
4259:
It can be seen from the above diagram that for any cycle operating between temperatures
2732:: A generalized thermodynamic cycle taking place between a hot reservoir at temperature
1503:
482:
5347:
5037:
4960:
4929:
4903:
4876:
4842:
4748:
4717:
3460:
3088:
2704:
2245:
2193:
2152:
2064:
1999:
1784:
1770:
1673:
1597:
1260:
1255:
1208:
824:
777:
692:
645:
557:
510:
440:
424:
311:
263:
248:
238:
47:
41:
2230:), the gas in the engine is in thermal contact with the cold reservoir at temperature
1981:
5510:
5332:
5168:
5117:
5091:
5064:
4978:
4974:
4949:"Effects of dark energy on the efficiency of charged AdS black holes as heat engines"
4933:
4868:
3060:{\displaystyle W=\oint PdV=\oint (dQ-dU)=\oint (TdS-dU)=\oint TdS-\oint dU=\oint TdS}
2174:
1704:
1592:
1553:
1543:
1115:
913:
741:
243:
233:
175:
5176:
5050:
4880:
4589:{\displaystyle \langle T_{C}\rangle ={\frac {1}{\Delta S}}\int _{Q_{\text{out}}}TdS}
4405:
5452:
5158:
5045:
4970:
4921:
4864:
4860:
4513:{\displaystyle \langle T_{H}\rangle ={\frac {1}{\Delta S}}\int _{Q_{\text{in}}}TdS}
4372:
2799:
diagram), in which the thermodynamic state is specified by a point on a graph with
1513:
1498:
1438:
1433:
1250:
1245:
895:
363:
228:
4414:
engine. So, real heat engines are even less efficient than indicated by
Equation
5475:
5109:
4948:
4801:
2412:
thermodynamic cycle (no net change in the system and its surroundings per cycle)
1463:
1311:
965:
606:
429:
190:
157:
5060:
5442:
5401:
5386:
5371:
5306:
5296:
5291:
5266:
5105:
5041:
4732:
4376:
2358:
2201:
2144:
2008:
1518:
1288:
388:
268:
205:
195:
63:
33:
2787:
The behavior of a Carnot engine or refrigerator is best understood by using a
5530:
5505:
5421:
5376:
5337:
5311:
5281:
4982:
4728:
4699:
4375:
becomes an inequality rather than an equality. Otherwise, since entropy is a
4334:
2259:) to the cold reservoir so the entropy of the system decreases by the amount
1778:
1689:
1587:
905:
474:
435:
147:
5411:
5406:
5381:
5327:
4872:
2692:
2393:
1931:{\displaystyle W=(T_{H}-T_{C})\Delta S=(T_{H}-T_{C}){\frac {Q_{H}}{T_{H}}}}
1538:
1523:
1473:
956:
4331:
diagram. For this figure, the curve indicates a vapor-liquid equilibrium (
3723:< 0 is the heat taken from the system (heat energy leaving the system),
2353:
because the isothermal compression decreases the multiplicity of the gas.
4784:
4783:
Planck, M. (1945). "equations 39, 40 and 65 in sections §90 & §137".
4439:
as the expression of the Carnot efficiency is still useful. Consider the
4389:
4368:
3752:> 0 is the heat put into the system (heat energy entering the system),
1677:
1493:
301:
5416:
5212:
5000:
4690:
2725:
2401:
2016:
1582:
1528:
5162:
5053:
4371:
of work into heat. In that case, the cycle is not reversible and the
3110:: A Carnot cycle taking place between a hot reservoir at temperature
1774:
180:
2476:{\displaystyle \Delta S_{H}+\Delta S_{C}=\Delta S_{\text{cycle}}=0,}
1965:
is heat transferred from the hot reservoir to the system per cycle.
4965:
4847:
4633:
For the Carnot cycle, or its equivalent, the average value ⟨
4364:
3694:
is the work done by the system (energy exiting the system as work),
2613:
are both smaller in magnitude and in fact are in the same ratio as
1296:
1213:
1005:
413:
185:
5001:"Power, efficiency, and fluctuations in steady-state heat engines"
4908:
4630:⟩, respectively, to estimate the efficiency a heat engine.
5051:
Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1963).
4440:
4316:
2800:
2192:= 0) between the system (the gas) and its surroundings. It is an
1766:
402:
4894:
N. A. Sinitsyn (2011). "Fluctuation
Relation for Heat Engines".
4410:
Carnot realized that, in reality, it is not possible to build a
3250:{\displaystyle W=\oint PdV=\oint TdS=(T_{H}-T_{C})(S_{B}-S_{A})}
3103:
4832:
4118:{\displaystyle Q_{C}=T_{C}(S_{A}-S_{B})=T_{C}\Delta S_{C}<0}
3448:{\displaystyle Q_{C}=T_{C}(S_{A}-S_{B})=T_{C}\Delta S_{C}<0}
2550:{\displaystyle {\frac {Q_{H}}{T_{H}}}=-{\frac {Q_{C}}{T_{C}}}.}
1700:
1668:
in 1824 and expanded upon by others in the 1830s and 1840s. By
4640:⟩ will equal the highest temperature available, namely
3133:
2255:< 0 (negative as leaving from the system, according to the
3929:
can be derived from the expressions above with the entropy:
2750:, the cycle cannot extend outside the temperature band from
2256:
1681:
378:
4017:{\displaystyle Q_{H}=T_{H}(S_{B}-S_{A})=T_{H}\Delta S_{H}}
3347:{\displaystyle Q_{H}=T_{H}(S_{B}-S_{A})=T_{H}\Delta S_{H}}
4661:. For other less efficient thermodynamic cycles, ⟨
5077:
5104:
4999:
Benenti, Giuliano; Casati, Giulio; Wang, Jiao (2020).
4193:{\displaystyle \Delta S_{C}=S_{A}-S_{B}=-\Delta S_{H}}
3457:
Due to energy conservation, the net heat transferred,
2911:
an upper portion of the cycle and a lower portion. In
2028:
to the gas at a temperature infinitesimally less than
5016:
However, fluctuations make impractical such engines.
4823:. 7th ed. New York: McGraw-Hill, 2011. p. 299. Print.
4525:
4449:
4292:
4265:
4206:
4131:
4030:
3935:
3882:
3851:
3822:
3793:
3760:
3731:
3702:
3562:
3537:
3483:
3463:
3360:
3265:
3152:
2939:
2828:
2619:
2592:
2565:
2489:
2418:
2323:
2265:
2087:
2067:
1944:
1830:
1807:
1787:
1740:
1713:
1124:
1069:
1014:
974:
848:
827:
801:
780:
752:
716:
695:
669:
648:
617:
581:
560:
534:
513:
485:
4720:
is, ultimately, a theoretical construct based on an
4712:
Carnot heat engine § As a macroscopic construct
4313:, none can exceed the efficiency of a Carnot cycle.
2887:{\displaystyle Q=\int _{A}^{B}dQ=\int _{A}^{B}T\,dS}
4200:, a minus sign appears in the final expression for
5079:
5052:
4588:
4512:
4399:
4305:
4278:
4212:
4192:
4117:
4016:
3921:
3864:
3835:
3806:
3773:
3744:
3715:
3663:
3543:
3521:
3469:
3447:
3346:
3249:
3059:
2886:
2681:
2647:
2605:
2578:
2549:
2475:
2345:
2309:
2131:
2073:
1957:
1930:
1816:
1793:
1753:
1726:
1160:
1105:
1050:
995:
857:
833:
810:
786:
761:
725:
701:
678:
654:
629:
590:
566:
543:
519:
494:
4998:
4702:was based on the principles of the Carnot cycle.
3814:is the absolute temperature of the hot reservoir.
5528:
5086:(3rd ed.). John Wiley & Sons. pp.
2159:) expansion of the gas (isentropic work output).
4689:. This can help illustrate, for example, why a
5059:. Addison-Wesley Publishing Company. pp.
4893:
3922:{\displaystyle \eta =1-{\frac {T_{C}}{T_{H}}}}
2676:
5228:
4705:
1635:
4778:
4776:
4774:
4539:
4526:
4463:
4450:
2658:
2357:
2200:
2143:
2007:
2184:. The entropy remains unchanged as no heat
5235:
5221:
4947:Liu, Hang; Meng, Xin-He (18 August 2017).
2807:) as the horizontal axis and temperature (
1982:Carnot cycle from The Mechanical Universe
1642:
1628:
40:
5152:
5078:Halliday, David; Resnick, Robert (1978).
4964:
4907:
4846:
4771:
2877:
5242:
4819:Çengel, Yunus A., and Michael A. Boles.
4793:
4315:
4228:
3132:
3102:
2724:
2691:
2392:
2310:{\displaystyle \Delta S_{C}=Q_{C}/T_{C}}
2132:{\displaystyle \Delta S_{H}=Q_{H}/T_{H}}
5397:Homogeneous charge compression ignition
5116:(2nd ed.). W. H. Freeman Company.
5033:Reflections on the Motive Power of Fire
4946:
4821:Thermodynamics: An Engineering Approach
4789:. Dover Publications. pp. 75, 135.
14:
5529:
5130:
4782:
2257:universal convention in thermodynamics
2040:isothermal heat addition or absorption
5216:
5167:American Institute of Physics, 2011.
4799:
2002:, consisting of the following steps:
1688:, or conversely, the efficiency of a
4345:is a formal statement of this fact:
3876:The expression with the temperature
3553:
3117:and a cold reservoir at temperature
2930:
2819:
2739:and a cold reservoir at temperature
2714:and a cold reservoir at temperature
2663:When a Carnot cycle is plotted on a
1672:, it provides an upper limit on the
4940:
4754:Reversible process (thermodynamics)
4248:
3140:: A visualization of a Carnot cycle
3128:
24:
5042:The Steam-Engine and Other Engines
4551:
4475:
4320:
4177:
4132:
4096:
4001:
3426:
3331:
3137:
3107:
2923:
2813:
2729:
2696:
2669:
2451:
2435:
2419:
2400:: A Carnot cycle illustrated on a
2397:
2376:
2370:
2324:
2266:
2226:
2220:
2169:
2163:
2088:
2050:
2044:
1866:
1808:
849:
802:
717:
670:
582:
535:
355:Intensive and extensive properties
25:
5548:
5190:
4809:. Dover Publications. p. 48.
4800:Fermi, E. (1956). "equation 64".
4255:Carnot's theorem (thermodynamics)
4235:heat pump and refrigeration cycle
3477:, is equal to the work performed
2699:: A Carnot cycle as an idealized
2346:{\displaystyle \Delta S_{C}<0}
1976:
1703:in the form of heat between two
1611:
1610:
930:Table of thermodynamic equations
5055:The Feynman Lectures on Physics
4953:The European Physical Journal C
4400:Efficiency of real heat engines
3522:{\displaystyle W=Q=Q_{H}-Q_{C}}
2682:The temperature–entropy diagram
2042:.) During this step (1 to 2 on
1994:A Carnot cycle as an idealized
1406:Maxwell's thermodynamic surface
5135:. AIP Conference Proceedings.
4992:
4975:10.1140/epjc/s10052-017-5134-9
4926:10.1088/1751-8113/44/40/405001
4887:
4865:10.1103/PhysRevLett.121.120601
4826:
4813:
4080:
4054:
3985:
3959:
3410:
3384:
3315:
3289:
3244:
3218:
3215:
3189:
3012:
2991:
2982:
2964:
1901:
1875:
1863:
1837:
1140:
1128:
1085:
1073:
1030:
1018:
990:
978:
13:
1:
4759:
4682:⟩ will be higher than
4433:is an idealization, Equation
4406:Heat Engine § Efficiency
3872:is the minimum system entropy
3843:is the maximum system entropy
1664:proposed by French physicist
1307:Mechanical equivalent of heat
27:Idealized thermodynamic cycle
5199:article on the Carnot cycle.
4668:⟩ will be lower than
4654:⟩ the lowest, namely
4412:thermodynamically reversible
4385:gives the efficiency of any
2748:second law of thermodynamics
2408:In this case, since it is a
2216:(isothermal heat rejection).
919:Onsager reciprocal relations
7:
5302:Stirling (pseudo/adiabatic)
4742:
4612:
4435:
4422:
4416:
4381:
4351:
3677:
3073:
2900:
2789:temperature–entropy diagram
2688:Temperature–entropy diagram
2677:Properties and significance
2648:{\displaystyle Q_{H}/T_{H}}
1411:Entropy as energy dispersal
1222:"Perpetual motion" machines
1161:{\displaystyle G(T,p)=H-TS}
1106:{\displaystyle A(T,V)=U-TS}
1051:{\displaystyle H(S,p)=U+pV}
10:
5553:
4786:Treatise on Thermodynamics
4709:
4706:As a macroscopic construct
4403:
4252:
3785:of the cold reservoir, and
2685:
858:{\displaystyle \partial T}
811:{\displaystyle \partial V}
726:{\displaystyle \partial p}
679:{\displaystyle \partial V}
591:{\displaystyle \partial T}
544:{\displaystyle \partial S}
5461:
5435:
5360:
5320:
5261:
5250:
5204:Carnot Cycle on Ideal Gas
5178:. Full article (24 pages
2764:. The area in red, |
2659:The pressure–volume graph
2161:For this step (2 to 3 on
2081:of the gas by the amount
1989:
1975:
1970:
1680:during the conversion of
1332:An Inquiry Concerning the
5044:edition 3, page 62, via
2811:) as the vertical axis (
2218:In this step (3 to 4 on
1817:{\displaystyle \Delta S}
1345:Heterogeneous Substances
762:{\displaystyle \alpha =}
630:{\displaystyle \beta =-}
4896:J. Phys. A: Math. Theor
2665:pressure–volume diagram
2366:Isentropic compression.
4590:
4514:
4339:
4307:
4280:
4214:
4194:
4119:
4018:
3923:
3866:
3837:
3808:
3775:
3746:
3717:
3665:
3545:
3523:
3471:
3449:
3348:
3251:
3141:
3125:
3061:
2888:
2784:
2722:
2649:
2607:
2580:
2551:
2477:
2405:
2362:
2347:
2311:
2205:
2148:
2133:
2075:
2012:
1959:
1932:
1818:
1795:
1755:
1728:
1162:
1107:
1052:
997:
996:{\displaystyle U(S,V)}
859:
835:
812:
788:
763:
727:
703:
680:
656:
631:
592:
568:
545:
521:
496:
475:Specific heat capacity
79:Quantum thermodynamics
4623:⟩ and ⟨
4591:
4515:
4319:
4308:
4306:{\displaystyle T_{C}}
4281:
4279:{\displaystyle T_{H}}
4229:Reversed Carnot cycle
4215:
4213:{\displaystyle \eta }
4195:
4120:
4019:
3924:
3867:
3865:{\displaystyle S_{A}}
3838:
3836:{\displaystyle S_{B}}
3809:
3807:{\displaystyle T_{H}}
3776:
3774:{\displaystyle T_{C}}
3747:
3745:{\displaystyle Q_{H}}
3718:
3716:{\displaystyle Q_{C}}
3666:
3546:
3544:{\displaystyle \eta }
3524:
3472:
3450:
3349:
3252:
3136:
3106:
3062:
2889:
2728:
2695:
2650:
2608:
2606:{\displaystyle T_{C}}
2581:
2579:{\displaystyle Q_{C}}
2552:
2478:
2396:
2361:
2348:
2312:
2204:
2147:
2134:
2076:
2011:
1960:
1958:{\displaystyle Q_{H}}
1933:
1819:
1796:
1756:
1754:{\displaystyle T_{C}}
1729:
1727:{\displaystyle T_{H}}
1695:In a Carnot cycle, a
1343:On the Equilibrium of
1163:
1108:
1061:Helmholtz free energy
1053:
998:
860:
836:
813:
789:
764:
728:
704:
681:
657:
632:
593:
569:
546:
522:
497:
5537:Thermodynamic cycles
5486:Regenerative cooling
5364:combustion / thermal
5263:Without phase change
5254:combustion / thermal
5244:Thermodynamic cycles
4725:thermodynamic system
4523:
4447:
4290:
4263:
4204:
4129:
4028:
3933:
3880:
3849:
3820:
3791:
3783:absolute temperature
3758:
3729:
3700:
3560:
3535:
3481:
3461:
3358:
3263:
3150:
2937:
2826:
2617:
2590:
2563:
2487:
2416:
2321:
2263:
2157:reversible adiabatic
2085:
2065:
1942:
1828:
1805:
1785:
1738:
1711:
1699:or engine transfers
1678:thermodynamic engine
1356:Motive Power of Fire
1122:
1067:
1012:
972:
924:Bridgman's equations
901:Fundamental relation
846:
825:
799:
778:
750:
714:
693:
667:
646:
615:
579:
558:
532:
511:
483:
5208:Wolfram Mathematica
5145:2011AIPC.1411..327K
4918:2011JPhA...44N5001S
4857:2018PhRvL.121l0601H
2873:
2849:
2701:thermodynamic cycle
1996:thermodynamic cycle
1662:thermodynamic cycle
1334:Source ... Friction
1266:Loschmidt's paradox
458:Material properties
336:Conjugate variables
4749:Carnot heat engine
4718:Carnot heat engine
4586:
4510:
4340:
4303:
4276:
4210:
4190:
4115:
4014:
3919:
3862:
3833:
3804:
3771:
3742:
3713:
3661:
3551:is defined to be:
3541:
3519:
3467:
3445:
3344:
3247:
3142:
3126:
3089:exact differential
3057:
2884:
2859:
2835:
2785:
2723:
2705:Carnot heat engine
2645:
2603:
2576:
2547:
2473:
2406:
2363:
2343:
2307:
2246:isothermal process
2206:
2194:isentropic process
2149:
2129:
2071:
2013:
2000:Carnot heat engine
1955:
1928:
1824:per cycle such as
1814:
1791:
1751:
1724:
1705:thermal reservoirs
1598:Order and disorder
1354:Reflections on the
1261:Heat death paradox
1158:
1103:
1048:
993:
855:
831:
808:
784:
759:
723:
699:
676:
652:
627:
588:
564:
541:
517:
495:{\displaystyle c=}
492:
465:Property databases
441:Reduced properties
425:Chemical potential
389:Functions of state
312:Thermal efficiency
48:Carnot heat engine
5524:
5523:
5501:Vapor-compression
5427:Staged combustion
5356:
5355:
5321:With phase change
5173:978-0-7354-0985-9
5163:10.1063/1.3665247
5123:978-0-7167-1088-2
5097:978-0-471-02456-9
5070:978-0-201-02116-5
5008:Physical Review E
4572:
4558:
4496:
4482:
3917:
3685:
3684:
3659:
3626:
3584:
3470:{\displaystyle Q}
3081:
3080:
2908:
2907:
2542:
2512:
2461:
2175:adiabatic process
2074:{\displaystyle S}
1987:
1986:
1926:
1794:{\displaystyle W}
1676:of any classical
1652:
1651:
1593:Self-organization
1418:
1417:
1116:Gibbs free energy
914:Maxwell relations
872:
871:
868:
867:
834:{\displaystyle V}
787:{\displaystyle 1}
742:Thermal expansion
736:
735:
702:{\displaystyle V}
655:{\displaystyle 1}
601:
600:
567:{\displaystyle N}
520:{\displaystyle T}
448:
447:
364:Process functions
350:Property diagrams
329:System properties
319:
318:
284:Endoreversibility
176:Equation of state
16:(Redirected from
5544:
5496:Vapor absorption
5259:
5258:
5237:
5230:
5223:
5214:
5213:
5166:
5156:
5127:
5110:Kroemer, Herbert
5101:
5085:
5074:
5058:
5046:Internet Archive
5019:
5018:
5005:
4996:
4990:
4989:
4968:
4944:
4938:
4937:
4911:
4891:
4885:
4884:
4850:
4830:
4824:
4817:
4811:
4810:
4808:
4797:
4791:
4790:
4780:
4737:Carnot's theorem
4595:
4593:
4592:
4587:
4576:
4575:
4574:
4573:
4570:
4559:
4557:
4546:
4538:
4537:
4519:
4517:
4516:
4511:
4500:
4499:
4498:
4497:
4494:
4483:
4481:
4470:
4462:
4461:
4373:Clausius theorem
4343:Carnot's theorem
4322:
4312:
4310:
4309:
4304:
4302:
4301:
4285:
4283:
4282:
4277:
4275:
4274:
4249:Carnot's theorem
4219:
4217:
4216:
4211:
4199:
4197:
4196:
4191:
4189:
4188:
4170:
4169:
4157:
4156:
4144:
4143:
4124:
4122:
4121:
4116:
4108:
4107:
4095:
4094:
4079:
4078:
4066:
4065:
4053:
4052:
4040:
4039:
4023:
4021:
4020:
4015:
4013:
4012:
4000:
3999:
3984:
3983:
3971:
3970:
3958:
3957:
3945:
3944:
3928:
3926:
3925:
3920:
3918:
3916:
3915:
3906:
3905:
3896:
3871:
3869:
3868:
3863:
3861:
3860:
3842:
3840:
3839:
3834:
3832:
3831:
3813:
3811:
3810:
3805:
3803:
3802:
3780:
3778:
3777:
3772:
3770:
3769:
3751:
3749:
3748:
3743:
3741:
3740:
3722:
3720:
3719:
3714:
3712:
3711:
3693:
3679:
3670:
3668:
3667:
3662:
3660:
3658:
3657:
3648:
3647:
3638:
3627:
3625:
3624:
3615:
3614:
3613:
3601:
3600:
3590:
3585:
3583:
3582:
3570:
3554:
3550:
3548:
3547:
3542:
3528:
3526:
3525:
3520:
3518:
3517:
3505:
3504:
3476:
3474:
3473:
3468:
3454:
3452:
3451:
3446:
3438:
3437:
3425:
3424:
3409:
3408:
3396:
3395:
3383:
3382:
3370:
3369:
3353:
3351:
3350:
3345:
3343:
3342:
3330:
3329:
3314:
3313:
3301:
3300:
3288:
3287:
3275:
3274:
3256:
3254:
3253:
3248:
3243:
3242:
3230:
3229:
3214:
3213:
3201:
3200:
3139:
3129:The Carnot cycle
3109:
3075:
3066:
3064:
3063:
3058:
2931:
2927:
2902:
2893:
2891:
2890:
2885:
2872:
2867:
2848:
2843:
2820:
2772:
2731:
2698:
2654:
2652:
2651:
2646:
2644:
2643:
2634:
2629:
2628:
2612:
2610:
2609:
2604:
2602:
2601:
2585:
2583:
2582:
2577:
2575:
2574:
2559:This is true as
2556:
2554:
2553:
2548:
2543:
2541:
2540:
2531:
2530:
2521:
2513:
2511:
2510:
2501:
2500:
2491:
2482:
2480:
2479:
2474:
2463:
2462:
2459:
2447:
2446:
2431:
2430:
2399:
2352:
2350:
2349:
2344:
2336:
2335:
2316:
2314:
2313:
2308:
2306:
2305:
2296:
2291:
2290:
2278:
2277:
2138:
2136:
2135:
2130:
2128:
2127:
2118:
2113:
2112:
2100:
2099:
2080:
2078:
2077:
2072:
1980:
1979:
1968:
1967:
1964:
1962:
1961:
1956:
1954:
1953:
1937:
1935:
1934:
1929:
1927:
1925:
1924:
1915:
1914:
1905:
1900:
1899:
1887:
1886:
1862:
1861:
1849:
1848:
1823:
1821:
1820:
1815:
1800:
1798:
1797:
1792:
1760:
1758:
1757:
1752:
1750:
1749:
1733:
1731:
1730:
1725:
1723:
1722:
1707:at temperatures
1670:Carnot's theorem
1644:
1637:
1630:
1614:
1613:
1321:Key publications
1302:
1301:("living force")
1251:Brownian ratchet
1246:Entropy and life
1241:Entropy and time
1192:
1191:
1167:
1165:
1164:
1159:
1112:
1110:
1109:
1104:
1057:
1055:
1054:
1049:
1002:
1000:
999:
994:
896:Clausius theorem
891:Carnot's theorem
864:
862:
861:
856:
840:
838:
837:
832:
817:
815:
814:
809:
793:
791:
790:
785:
772:
771:
768:
766:
765:
760:
732:
730:
729:
724:
708:
706:
705:
700:
685:
683:
682:
677:
661:
659:
658:
653:
640:
639:
636:
634:
633:
628:
597:
595:
594:
589:
573:
571:
570:
565:
550:
548:
547:
542:
526:
524:
523:
518:
505:
504:
501:
499:
498:
493:
471:
470:
344:
343:
163:
162:
44:
30:
29:
21:
5552:
5551:
5547:
5546:
5545:
5543:
5542:
5541:
5527:
5526:
5525:
5520:
5457:
5431:
5363:
5352:
5342:Organic Rankine
5316:
5270:
5267:hot air engines
5264:
5253:
5246:
5241:
5193:
5175:. Abstract at:
5154:10.1.1.405.1945
5124:
5114:Thermal Physics
5106:Kittel, Charles
5098:
5071:
5023:
5022:
5003:
4997:
4993:
4945:
4941:
4892:
4888:
4835:Phys. Rev. Lett
4831:
4827:
4818:
4814:
4806:
4798:
4794:
4781:
4772:
4762:
4745:
4735:. As such, per
4714:
4708:
4687:
4680:
4673:
4666:
4659:
4652:
4645:
4638:
4628:
4621:
4608:
4601:
4569:
4565:
4564:
4560:
4550:
4545:
4533:
4529:
4524:
4521:
4520:
4493:
4489:
4488:
4484:
4474:
4469:
4457:
4453:
4448:
4445:
4444:
4408:
4402:
4349:Thus, Equation
4297:
4293:
4291:
4288:
4287:
4270:
4266:
4264:
4261:
4260:
4257:
4251:
4231:
4223:
4205:
4202:
4201:
4184:
4180:
4165:
4161:
4152:
4148:
4139:
4135:
4130:
4127:
4126:
4103:
4099:
4090:
4086:
4074:
4070:
4061:
4057:
4048:
4044:
4035:
4031:
4029:
4026:
4025:
4008:
4004:
3995:
3991:
3979:
3975:
3966:
3962:
3953:
3949:
3940:
3936:
3934:
3931:
3930:
3911:
3907:
3901:
3897:
3895:
3881:
3878:
3877:
3856:
3852:
3850:
3847:
3846:
3827:
3823:
3821:
3818:
3817:
3798:
3794:
3792:
3789:
3788:
3765:
3761:
3759:
3756:
3755:
3736:
3732:
3730:
3727:
3726:
3707:
3703:
3701:
3698:
3697:
3691:
3653:
3649:
3643:
3639:
3637:
3620:
3616:
3609:
3605:
3596:
3592:
3591:
3589:
3578:
3574:
3569:
3561:
3558:
3557:
3536:
3533:
3532:
3531:The efficiency
3513:
3509:
3500:
3496:
3482:
3479:
3478:
3462:
3459:
3458:
3433:
3429:
3420:
3416:
3404:
3400:
3391:
3387:
3378:
3374:
3365:
3361:
3359:
3356:
3355:
3338:
3334:
3325:
3321:
3309:
3305:
3296:
3292:
3283:
3279:
3270:
3266:
3264:
3261:
3260:
3238:
3234:
3225:
3221:
3209:
3205:
3196:
3192:
3151:
3148:
3147:
3131:
3123:
3116:
2938:
2935:
2934:
2921:
2868:
2863:
2844:
2839:
2827:
2824:
2823:
2782:
2778:
2771:
2765:
2763:
2756:
2745:
2738:
2720:
2713:
2703:performed by a
2690:
2684:
2679:
2661:
2639:
2635:
2630:
2624:
2620:
2618:
2615:
2614:
2597:
2593:
2591:
2588:
2587:
2570:
2566:
2564:
2561:
2560:
2536:
2532:
2526:
2522:
2520:
2506:
2502:
2496:
2492:
2490:
2488:
2485:
2484:
2458:
2454:
2442:
2438:
2426:
2422:
2417:
2414:
2413:
2391:
2385:
2331:
2327:
2322:
2319:
2318:
2301:
2297:
2292:
2286:
2282:
2273:
2269:
2264:
2261:
2260:
2254:
2243:
2236:
2215:
2183:
2123:
2119:
2114:
2108:
2104:
2095:
2091:
2086:
2083:
2082:
2066:
2063:
2062:
2060:
2036:
2027:
1998:performed by a
1992:
1977:
1971:External videos
1949:
1945:
1943:
1940:
1939:
1920:
1916:
1910:
1906:
1904:
1895:
1891:
1882:
1878:
1857:
1853:
1844:
1840:
1829:
1826:
1825:
1806:
1803:
1802:
1786:
1783:
1782:
1745:
1741:
1739:
1736:
1735:
1718:
1714:
1712:
1709:
1708:
1648:
1603:
1602:
1578:
1570:
1569:
1568:
1428:
1420:
1419:
1398:
1384:
1359:
1355:
1348:
1344:
1337:
1333:
1300:
1293:
1275:
1256:Maxwell's demon
1218:
1189:
1188:
1172:
1171:
1170:
1123:
1120:
1119:
1118:
1068:
1065:
1064:
1063:
1013:
1010:
1009:
1008:
973:
970:
969:
968:
966:Internal energy
961:
946:
936:
935:
910:
885:
875:
874:
873:
847:
844:
843:
826:
823:
822:
800:
797:
796:
779:
776:
775:
751:
748:
747:
715:
712:
711:
694:
691:
690:
668:
665:
664:
647:
644:
643:
616:
613:
612:
607:Compressibility
580:
577:
576:
559:
556:
555:
533:
530:
529:
512:
509:
508:
484:
481:
480:
460:
450:
449:
430:Particle number
383:
342:
331:
321:
320:
279:Irreversibility
191:State of matter
158:Isolated system
143:
133:
132:
131:
106:
96:
95:
91:Non-equilibrium
83:
58:
50:
28:
23:
22:
15:
12:
11:
5:
5550:
5540:
5539:
5522:
5521:
5519:
5518:
5513:
5508:
5503:
5498:
5493:
5488:
5483:
5478:
5473:
5467:
5465:
5459:
5458:
5456:
5455:
5450:
5445:
5439:
5437:
5433:
5432:
5430:
5429:
5424:
5419:
5414:
5409:
5404:
5399:
5394:
5389:
5384:
5379:
5374:
5368:
5366:
5358:
5357:
5354:
5353:
5351:
5350:
5345:
5335:
5330:
5324:
5322:
5318:
5317:
5315:
5314:
5309:
5304:
5299:
5294:
5289:
5284:
5279:
5273:
5271:
5262:
5256:
5248:
5247:
5240:
5239:
5232:
5225:
5217:
5211:
5210:
5202:S. M. Blinder
5200:
5192:
5191:External links
5189:
5188:
5187:
5186:
5185:
5139:(1): 327–350.
5133:AIP Conf. Proc
5128:
5122:
5102:
5096:
5075:
5069:
5048:
5035:
5031:Carnot, Sadi,
5027:
5021:
5020:
4991:
4939:
4902:(40): 405001.
4886:
4841:(12): 120601.
4825:
4812:
4803:Thermodynamics
4792:
4769:
4768:
4767:
4766:
4761:
4758:
4757:
4756:
4751:
4744:
4741:
4710:Main article:
4707:
4704:
4685:
4678:
4675:, and ⟨
4671:
4664:
4657:
4650:
4647:, and ⟨
4643:
4636:
4626:
4619:
4606:
4599:
4585:
4582:
4579:
4568:
4563:
4556:
4553:
4549:
4544:
4541:
4536:
4532:
4528:
4509:
4506:
4503:
4492:
4487:
4480:
4477:
4473:
4468:
4465:
4460:
4456:
4452:
4443:temperatures,
4431:Carnot's cycle
4401:
4398:
4377:state function
4300:
4296:
4273:
4269:
4253:Main article:
4250:
4247:
4230:
4227:
4209:
4187:
4183:
4179:
4176:
4173:
4168:
4164:
4160:
4155:
4151:
4147:
4142:
4138:
4134:
4114:
4111:
4106:
4102:
4098:
4093:
4089:
4085:
4082:
4077:
4073:
4069:
4064:
4060:
4056:
4051:
4047:
4043:
4038:
4034:
4011:
4007:
4003:
3998:
3994:
3990:
3987:
3982:
3978:
3974:
3969:
3965:
3961:
3956:
3952:
3948:
3943:
3939:
3914:
3910:
3904:
3900:
3894:
3891:
3888:
3885:
3874:
3873:
3859:
3855:
3844:
3830:
3826:
3815:
3801:
3797:
3786:
3768:
3764:
3753:
3739:
3735:
3724:
3710:
3706:
3695:
3683:
3682:
3673:
3671:
3656:
3652:
3646:
3642:
3636:
3633:
3630:
3623:
3619:
3612:
3608:
3604:
3599:
3595:
3588:
3581:
3577:
3573:
3568:
3565:
3540:
3516:
3512:
3508:
3503:
3499:
3495:
3492:
3489:
3486:
3466:
3444:
3441:
3436:
3432:
3428:
3423:
3419:
3415:
3412:
3407:
3403:
3399:
3394:
3390:
3386:
3381:
3377:
3373:
3368:
3364:
3341:
3337:
3333:
3328:
3324:
3320:
3317:
3312:
3308:
3304:
3299:
3295:
3291:
3286:
3282:
3278:
3273:
3269:
3246:
3241:
3237:
3233:
3228:
3224:
3220:
3217:
3212:
3208:
3204:
3199:
3195:
3191:
3188:
3185:
3182:
3179:
3176:
3173:
3170:
3167:
3164:
3161:
3158:
3155:
3130:
3127:
3121:
3114:
3079:
3078:
3069:
3067:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
3023:
3020:
3017:
3014:
3011:
3008:
3005:
3002:
2999:
2996:
2993:
2990:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2906:
2905:
2896:
2894:
2883:
2880:
2876:
2871:
2866:
2862:
2858:
2855:
2852:
2847:
2842:
2838:
2834:
2831:
2780:
2769:
2761:
2754:
2743:
2736:
2718:
2711:
2686:Main article:
2683:
2680:
2678:
2675:
2660:
2657:
2642:
2638:
2633:
2627:
2623:
2600:
2596:
2573:
2569:
2546:
2539:
2535:
2529:
2525:
2519:
2516:
2509:
2505:
2499:
2495:
2472:
2469:
2466:
2457:
2453:
2450:
2445:
2441:
2437:
2434:
2429:
2425:
2421:
2390:
2389:
2383:
2355:
2342:
2339:
2334:
2330:
2326:
2304:
2300:
2295:
2289:
2285:
2281:
2276:
2272:
2268:
2252:
2241:
2234:
2213:
2198:
2181:
2141:
2126:
2122:
2117:
2111:
2107:
2103:
2098:
2094:
2090:
2070:
2058:
2032:
2023:
2004:
1991:
1988:
1985:
1984:
1973:
1972:
1952:
1948:
1923:
1919:
1913:
1909:
1903:
1898:
1894:
1890:
1885:
1881:
1877:
1874:
1871:
1868:
1865:
1860:
1856:
1852:
1847:
1843:
1839:
1836:
1833:
1813:
1810:
1790:
1748:
1744:
1721:
1717:
1650:
1649:
1647:
1646:
1639:
1632:
1624:
1621:
1620:
1619:
1618:
1605:
1604:
1601:
1600:
1595:
1590:
1585:
1579:
1576:
1575:
1572:
1571:
1567:
1566:
1561:
1556:
1551:
1546:
1541:
1536:
1531:
1526:
1521:
1516:
1511:
1506:
1501:
1496:
1491:
1486:
1481:
1476:
1471:
1466:
1461:
1456:
1451:
1446:
1441:
1436:
1430:
1429:
1426:
1425:
1422:
1421:
1416:
1415:
1414:
1413:
1408:
1400:
1399:
1397:
1396:
1393:
1389:
1386:
1385:
1383:
1382:
1377:
1375:Thermodynamics
1371:
1368:
1367:
1363:
1362:
1361:
1360:
1351:
1349:
1340:
1338:
1329:
1324:
1323:
1317:
1316:
1315:
1314:
1309:
1304:
1292:
1291:
1289:Caloric theory
1285:
1282:
1281:
1277:
1276:
1274:
1273:
1268:
1263:
1258:
1253:
1248:
1243:
1237:
1234:
1233:
1227:
1226:
1225:
1224:
1217:
1216:
1211:
1206:
1200:
1197:
1196:
1190:
1187:
1186:
1183:
1179:
1178:
1177:
1174:
1173:
1169:
1168:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1113:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1058:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1003:
992:
989:
986:
983:
980:
977:
962:
960:
959:
954:
948:
947:
942:
941:
938:
937:
934:
933:
926:
921:
916:
909:
908:
903:
898:
893:
887:
886:
881:
880:
877:
876:
870:
869:
866:
865:
854:
851:
841:
830:
819:
818:
807:
804:
794:
783:
769:
758:
755:
745:
738:
737:
734:
733:
722:
719:
709:
698:
687:
686:
675:
672:
662:
651:
637:
626:
623:
620:
610:
603:
602:
599:
598:
587:
584:
574:
563:
552:
551:
540:
537:
527:
516:
502:
491:
488:
478:
469:
468:
467:
461:
456:
455:
452:
451:
446:
445:
444:
443:
438:
433:
422:
411:
392:
391:
385:
384:
382:
381:
376:
370:
367:
366:
360:
359:
358:
357:
352:
333:
332:
327:
326:
323:
322:
317:
316:
315:
314:
309:
304:
296:
295:
289:
288:
287:
286:
281:
276:
271:
269:Free expansion
266:
261:
256:
251:
246:
241:
236:
231:
223:
222:
216:
215:
214:
213:
208:
206:Control volume
203:
198:
196:Phase (matter)
193:
188:
183:
178:
170:
169:
161:
160:
155:
150:
144:
139:
138:
135:
134:
130:
129:
124:
119:
114:
108:
107:
102:
101:
98:
97:
94:
93:
82:
81:
76:
71:
66:
60:
59:
56:
55:
52:
51:
46:The classical
45:
37:
36:
34:Thermodynamics
26:
9:
6:
4:
3:
2:
5549:
5538:
5535:
5534:
5532:
5517:
5514:
5512:
5509:
5507:
5504:
5502:
5499:
5497:
5494:
5492:
5491:Transcritical
5489:
5487:
5484:
5482:
5479:
5477:
5474:
5472:
5471:Hampson–Linde
5469:
5468:
5466:
5464:
5463:Refrigeration
5460:
5454:
5451:
5449:
5446:
5444:
5441:
5440:
5438:
5434:
5428:
5425:
5423:
5420:
5418:
5415:
5413:
5410:
5408:
5405:
5403:
5400:
5398:
5395:
5393:
5392:Gas-generator
5390:
5388:
5385:
5383:
5380:
5378:
5377:Brayton/Joule
5375:
5373:
5370:
5369:
5367:
5365:
5359:
5349:
5346:
5343:
5339:
5336:
5334:
5331:
5329:
5326:
5325:
5323:
5319:
5313:
5310:
5308:
5305:
5303:
5300:
5298:
5295:
5293:
5290:
5288:
5285:
5283:
5282:Brayton/Joule
5280:
5278:
5275:
5274:
5272:
5268:
5260:
5257:
5255:
5249:
5245:
5238:
5233:
5231:
5226:
5224:
5219:
5218:
5215:
5209:
5205:
5201:
5198:
5195:
5194:
5183:
5180:
5177:
5174:
5170:
5164:
5160:
5155:
5150:
5146:
5142:
5138:
5134:
5129:
5125:
5119:
5115:
5111:
5107:
5103:
5099:
5093:
5089:
5084:
5083:
5076:
5072:
5066:
5062:
5057:
5056:
5049:
5047:
5043:
5039:
5036:
5034:
5030:
5029:
5028:
5025:
5024:
5017:
5013:
5009:
5002:
4995:
4988:
4984:
4980:
4976:
4972:
4967:
4962:
4958:
4954:
4950:
4943:
4935:
4931:
4927:
4923:
4919:
4915:
4910:
4905:
4901:
4897:
4890:
4882:
4878:
4874:
4870:
4866:
4862:
4858:
4854:
4849:
4844:
4840:
4836:
4829:
4822:
4816:
4805:
4804:
4796:
4788:
4787:
4779:
4777:
4775:
4770:
4764:
4763:
4755:
4752:
4750:
4747:
4746:
4740:
4738:
4734:
4730:
4729:diesel engine
4726:
4723:
4719:
4713:
4703:
4701:
4700:diesel engine
4696:
4692:
4688:
4681:
4674:
4667:
4660:
4653:
4646:
4639:
4631:
4629:
4622:
4615:
4614:
4609:
4602:
4583:
4580:
4577:
4566:
4561:
4554:
4547:
4542:
4534:
4530:
4507:
4504:
4501:
4490:
4485:
4478:
4471:
4466:
4458:
4454:
4442:
4438:
4437:
4432:
4427:
4425:
4424:
4419:
4418:
4413:
4407:
4397:
4393:
4391:
4388:
4384:
4383:
4378:
4374:
4370:
4366:
4361:
4358:
4354:
4353:
4348:
4344:
4337:
4336:
4335:Rankine cycle
4330:
4326:
4318:
4314:
4298:
4294:
4271:
4267:
4256:
4246:
4244:
4240:
4236:
4226:
4221:
4207:
4185:
4181:
4174:
4171:
4166:
4162:
4158:
4153:
4149:
4145:
4140:
4136:
4112:
4109:
4104:
4100:
4091:
4087:
4083:
4075:
4071:
4067:
4062:
4058:
4049:
4045:
4041:
4036:
4032:
4009:
4005:
3996:
3992:
3988:
3980:
3976:
3972:
3967:
3963:
3954:
3950:
3946:
3941:
3937:
3912:
3908:
3902:
3898:
3892:
3889:
3886:
3883:
3857:
3853:
3845:
3828:
3824:
3816:
3799:
3795:
3787:
3784:
3766:
3762:
3754:
3737:
3733:
3725:
3708:
3704:
3696:
3690:
3689:
3688:
3681:
3674:
3672:
3654:
3650:
3644:
3640:
3634:
3631:
3628:
3621:
3617:
3610:
3606:
3602:
3597:
3593:
3586:
3579:
3575:
3571:
3566:
3563:
3556:
3555:
3552:
3538:
3529:
3514:
3510:
3506:
3501:
3497:
3493:
3490:
3487:
3484:
3464:
3455:
3442:
3439:
3434:
3430:
3421:
3417:
3413:
3405:
3401:
3397:
3392:
3388:
3379:
3375:
3371:
3366:
3362:
3339:
3335:
3326:
3322:
3318:
3310:
3306:
3302:
3297:
3293:
3284:
3280:
3276:
3271:
3267:
3257:
3239:
3235:
3231:
3226:
3222:
3210:
3206:
3202:
3197:
3193:
3186:
3183:
3180:
3177:
3174:
3171:
3168:
3165:
3162:
3159:
3156:
3153:
3145:
3135:
3120:
3113:
3105:
3101:
3098:
3094:
3090:
3086:
3077:
3070:
3068:
3054:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3021:
3018:
3015:
3009:
3006:
3003:
3000:
2997:
2994:
2988:
2985:
2979:
2976:
2973:
2970:
2967:
2961:
2958:
2955:
2952:
2949:
2946:
2943:
2940:
2933:
2932:
2929:
2926:
2925:
2918:
2914:
2904:
2897:
2895:
2881:
2878:
2874:
2869:
2864:
2860:
2856:
2853:
2850:
2845:
2840:
2836:
2832:
2829:
2822:
2821:
2818:
2816:
2815:
2810:
2806:
2802:
2798:
2794:
2790:
2776:
2768:
2760:
2753:
2749:
2742:
2735:
2727:
2717:
2710:
2706:
2702:
2694:
2689:
2674:
2672:
2671:
2666:
2656:
2640:
2636:
2631:
2625:
2621:
2598:
2594:
2571:
2567:
2557:
2544:
2537:
2533:
2527:
2523:
2517:
2514:
2507:
2503:
2497:
2493:
2470:
2467:
2464:
2455:
2448:
2443:
2439:
2432:
2427:
2423:
2411:
2403:
2395:
2388:
2386:
2379:
2378:
2373:
2372:
2367:
2360:
2356:
2354:
2340:
2337:
2332:
2328:
2302:
2298:
2293:
2287:
2283:
2279:
2274:
2270:
2258:
2251:
2247:
2240:
2233:
2229:
2228:
2223:
2222:
2217:
2212:
2203:
2199:
2197:
2195:
2191:
2187:
2180:
2176:
2172:
2171:
2166:
2165:
2160:
2158:
2154:
2146:
2142:
2140:
2124:
2120:
2115:
2109:
2105:
2101:
2096:
2092:
2068:
2057:
2053:
2052:
2047:
2046:
2041:
2037:
2035:
2031:
2026:
2022:
2018:
2010:
2006:
2005:
2003:
2001:
1997:
1983:
1974:
1969:
1966:
1950:
1946:
1921:
1917:
1911:
1907:
1896:
1892:
1888:
1883:
1879:
1872:
1869:
1858:
1854:
1850:
1845:
1841:
1834:
1831:
1811:
1788:
1780:
1779:refrigeration
1776:
1772:
1768:
1764:
1746:
1742:
1719:
1715:
1706:
1702:
1698:
1693:
1691:
1690:refrigeration
1687:
1683:
1679:
1675:
1671:
1667:
1663:
1659:
1654:
1645:
1640:
1638:
1633:
1631:
1626:
1625:
1623:
1622:
1617:
1609:
1608:
1607:
1606:
1599:
1596:
1594:
1591:
1589:
1588:Self-assembly
1586:
1584:
1581:
1580:
1574:
1573:
1565:
1562:
1560:
1559:van der Waals
1557:
1555:
1552:
1550:
1547:
1545:
1542:
1540:
1537:
1535:
1532:
1530:
1527:
1525:
1522:
1520:
1517:
1515:
1512:
1510:
1507:
1505:
1502:
1500:
1497:
1495:
1492:
1490:
1487:
1485:
1484:von Helmholtz
1482:
1480:
1477:
1475:
1472:
1470:
1467:
1465:
1462:
1460:
1457:
1455:
1452:
1450:
1447:
1445:
1442:
1440:
1437:
1435:
1432:
1431:
1424:
1423:
1412:
1409:
1407:
1404:
1403:
1402:
1401:
1394:
1391:
1390:
1388:
1387:
1381:
1378:
1376:
1373:
1372:
1370:
1369:
1365:
1364:
1358:
1357:
1350:
1347:
1346:
1339:
1336:
1335:
1328:
1327:
1326:
1325:
1322:
1319:
1318:
1313:
1310:
1308:
1305:
1303:
1299:
1295:
1294:
1290:
1287:
1286:
1284:
1283:
1279:
1278:
1272:
1269:
1267:
1264:
1262:
1259:
1257:
1254:
1252:
1249:
1247:
1244:
1242:
1239:
1238:
1236:
1235:
1232:
1229:
1228:
1223:
1220:
1219:
1215:
1212:
1210:
1207:
1205:
1202:
1201:
1199:
1198:
1194:
1193:
1184:
1181:
1180:
1176:
1175:
1155:
1152:
1149:
1146:
1143:
1137:
1134:
1131:
1125:
1117:
1114:
1100:
1097:
1094:
1091:
1088:
1082:
1079:
1076:
1070:
1062:
1059:
1045:
1042:
1039:
1036:
1033:
1027:
1024:
1021:
1015:
1007:
1004:
987:
984:
981:
975:
967:
964:
963:
958:
955:
953:
950:
949:
945:
940:
939:
932:
931:
927:
925:
922:
920:
917:
915:
912:
911:
907:
906:Ideal gas law
904:
902:
899:
897:
894:
892:
889:
888:
884:
879:
878:
852:
842:
828:
821:
820:
805:
795:
781:
774:
773:
770:
756:
753:
746:
743:
740:
739:
720:
710:
696:
689:
688:
673:
663:
649:
642:
641:
638:
624:
621:
618:
611:
608:
605:
604:
585:
575:
561:
554:
553:
538:
528:
514:
507:
506:
503:
489:
486:
479:
476:
473:
472:
466:
463:
462:
459:
454:
453:
442:
439:
437:
436:Vapor quality
434:
432:
431:
426:
423:
421:
420:
415:
412:
409:
405:
404:
399:
396:
395:
394:
393:
390:
387:
386:
380:
377:
375:
372:
371:
369:
368:
365:
362:
361:
356:
353:
351:
348:
347:
346:
345:
341:
337:
330:
325:
324:
313:
310:
308:
305:
303:
300:
299:
298:
297:
294:
291:
290:
285:
282:
280:
277:
275:
274:Reversibility
272:
270:
267:
265:
262:
260:
257:
255:
252:
250:
247:
245:
242:
240:
237:
235:
232:
230:
227:
226:
225:
224:
221:
218:
217:
212:
209:
207:
204:
202:
199:
197:
194:
192:
189:
187:
184:
182:
179:
177:
174:
173:
172:
171:
168:
165:
164:
159:
156:
154:
151:
149:
148:Closed system
146:
145:
142:
137:
136:
128:
125:
123:
120:
118:
115:
113:
110:
109:
105:
100:
99:
92:
88:
85:
84:
80:
77:
75:
72:
70:
67:
65:
62:
61:
54:
53:
49:
43:
39:
38:
35:
32:
31:
19:
5348:Regenerative
5286:
5277:Bell Coleman
5197:Hyperphysics
5136:
5132:
5113:
5081:
5054:
5038:Ewing, J. A.
5015:
5011:
5007:
4994:
4986:
4956:
4952:
4942:
4899:
4895:
4889:
4838:
4834:
4828:
4820:
4815:
4802:
4795:
4785:
4721:
4715:
4683:
4676:
4669:
4662:
4655:
4648:
4641:
4634:
4632:
4624:
4617:
4611:
4610:in Equation
4604:
4597:
4434:
4430:
4428:
4421:
4415:
4409:
4394:
4380:
4362:
4356:
4350:
4346:
4342:
4341:
4332:
4328:
4324:
4258:
4242:
4238:
4232:
4222:
3875:
3686:
3675:
3530:
3456:
3258:
3146:
3143:
3118:
3111:
3096:
3092:
3084:
3082:
3071:
2922:
2916:
2912:
2909:
2898:
2812:
2808:
2804:
2796:
2792:
2786:
2774:
2766:
2758:
2751:
2740:
2733:
2715:
2708:
2668:
2662:
2558:
2407:
2381:
2375:
2374:, D to A on
2369:
2365:
2364:
2249:
2238:
2231:
2225:
2224:, C to D on
2219:
2210:
2208:
2207:
2189:
2185:
2178:
2168:
2167:, B to C in
2162:
2151:
2150:
2055:
2049:
2048:, A to B in
2043:
2039:
2033:
2029:
2024:
2020:
2015:
2014:
1993:
1694:
1660:is an ideal
1658:Carnot cycle
1657:
1655:
1653:
1449:Carathéodory
1380:Heat engines
1352:
1341:
1330:
1312:Motive power
1297:
957:Free entropy
928:
428:
427: /
417:
416: /
408:introduction
401:
400: /
339:
302:Heat engines
89: /
18:Engine cycle
5516:Ionocaloric
5511:Vuilleumier
5333:Hygroscopic
5206:powered by
5181:), also at
4695:regenerator
4616:by ⟨
4390:heat engine
4369:dissipation
4367:leading to
2368:(4 to 1 on
2188:transfers (
1666:Sadi Carnot
1271:Synergetics
952:Free energy
398:Temperature
259:Quasistatic
254:Isenthalpic
211:Instruments
201:Equilibrium
153:Open system
87:Equilibrium
69:Statistical
5481:Pulse tube
5453:Mixed/dual
5061:Chapter 44
4966:1704.04363
4959:(8): 556.
4848:1805.00848
4760:References
4404:See also:
4387:reversible
2410:reversible
2402:PV diagram
2153:Isentropic
2017:Isothermal
1763:reversible
1674:efficiency
1583:Nucleation
1427:Scientists
1231:Philosophy
944:Potentials
307:Heat pumps
264:Polytropic
249:Isentropic
239:Isothermal
5476:Kleemenko
5362:Internal
5149:CiteSeerX
4983:1434-6052
4934:119261929
4909:1111.7014
4722:idealized
4562:∫
4552:Δ
4540:⟩
4527:⟨
4486:∫
4476:Δ
4464:⟩
4451:⟨
4429:Although
4208:η
4178:Δ
4175:−
4159:−
4133:Δ
4097:Δ
4068:−
4002:Δ
3973:−
3893:−
3884:η
3635:−
3603:−
3564:η
3539:η
3507:−
3427:Δ
3398:−
3332:Δ
3303:−
3232:−
3203:−
3175:∮
3160:∮
3046:∮
3034:∮
3031:−
3019:∮
3004:−
2989:∮
2974:−
2962:∮
2947:∮
2861:∫
2837:∫
2746:. By the
2518:−
2452:Δ
2436:Δ
2420:Δ
2325:Δ
2267:Δ
2089:Δ
1889:−
1867:Δ
1851:−
1809:Δ
1775:heat pump
1771:conserved
1564:Waterston
1514:von Mayer
1469:de Donder
1459:Clapeyron
1439:Boltzmann
1434:Bernoulli
1395:Education
1366:Timelines
1150:−
1095:−
883:Equations
850:∂
803:∂
754:α
718:∂
671:∂
625:−
619:β
583:∂
536:∂
244:Adiabatic
234:Isochoric
220:Processes
181:Ideal gas
64:Classical
5531:Category
5443:Combined
5402:Humphrey
5387:Expander
5372:Atkinson
5307:Stoddard
5297:Stirling
5292:Ericsson
5252:External
5112:(1980).
4881:52943273
4873:30296120
4743:See also
4691:reheater
4365:friction
4321:Figure 6
4125:. Since
3138:Figure 5
3108:Figure 4
2924:Figure 1
2814:Figure 2
2730:Figure 3
2697:Figure 2
2670:Figure 1
2398:Figure 1
2377:Figure 2
2371:Figure 1
2227:Figure 2
2221:Figure 1
2170:Figure 2
2164:Figure 1
2051:Figure 2
2045:Figure 1
1938:, where
1616:Category
1554:Thompson
1464:Clausius
1444:Bridgman
1298:Vis viva
1280:Theories
1214:Gas laws
1006:Enthalpy
414:Pressure
229:Isobaric
186:Real gas
74:Chemical
57:Branches
5506:Siemens
5422:Scuderi
5338:Rankine
5141:Bibcode
5088:541–548
5082:Physics
5040:(1910)
5026:Sources
4914:Bibcode
4853:Bibcode
4441:average
3781:is the
2801:entropy
1767:entropy
1539:Smeaton
1534:Rankine
1524:Onsager
1509:Maxwell
1504:Massieu
1209:Entropy
1204:General
1195:History
1185:Culture
1182:History
406: (
403:Entropy
340:italics
141:Systems
5412:Miller
5407:Lenoir
5382:Diesel
5328:Kalina
5312:Manson
5287:Carnot
5171:
5151:
5120:
5094:
5067:
4981:
4932:
4879:
4871:
3687:where
3087:is an
3083:Since
1990:Stages
1765:, and
1701:energy
1697:system
1529:Planck
1519:Nernst
1494:Kelvin
1454:Carnot
744:
609:
477:
419:Volume
334:Note:
293:Cycles
122:Second
112:Zeroth
5436:Mixed
5014:(4).
5004:(PDF)
4961:arXiv
4930:S2CID
4904:arXiv
4877:S2CID
4843:arXiv
4807:(PDF)
4765:Notes
4693:or a
2460:cycle
1684:into
1577:Other
1544:Stahl
1499:Lewis
1489:Joule
1479:Gibbs
1474:Duhem
167:State
127:Third
117:First
5448:HEHC
5417:Otto
5169:ISBN
5137:1411
5118:ISBN
5092:ISBN
5065:ISBN
4979:ISSN
4869:PMID
4733:work
4716:The
4603:and
4333:See
4286:and
4110:<
4024:and
3440:<
2586:and
2483:or,
2338:<
1734:and
1686:work
1682:heat
1549:Tait
379:Heat
374:Work
104:Laws
5159:doi
5012:102
4971:doi
4922:doi
4861:doi
4839:121
4571:out
2757:to
1777:or
1769:is
1392:Art
338:in
5533::
5157:.
5147:.
5108:;
5090:.
5063:.
5010:.
5006:.
4985:.
4977:.
4969:.
4957:77
4955:.
4951:.
4928:.
4920:.
4912:.
4900:44
4898:.
4875:.
4867:.
4859:.
4851:.
4837:.
4773:^
4495:in
4392:.
4220:.
3085:dU
2779:Q
2655:.
2317:.
2196:.
2139:.
1656:A
5344:)
5340:(
5269:)
5265:(
5236:e
5229:t
5222:v
5184:.
5165:.
5161::
5143::
5126:.
5100:.
5073:.
4973::
4963::
4936:.
4924::
4916::
4906::
4883:.
4863::
4855::
4845::
4686:C
4684:T
4679:C
4677:T
4672:H
4670:T
4665:H
4663:T
4658:C
4656:T
4651:C
4649:T
4644:H
4642:T
4637:H
4635:T
4627:C
4625:T
4620:H
4618:T
4613:3
4607:C
4605:T
4600:H
4598:T
4584:S
4581:d
4578:T
4567:Q
4555:S
4548:1
4543:=
4535:C
4531:T
4508:S
4505:d
4502:T
4491:Q
4479:S
4472:1
4467:=
4459:H
4455:T
4436:3
4423:3
4417:3
4382:3
4352:3
4329:S
4327:–
4325:T
4299:C
4295:T
4272:H
4268:T
4243:V
4241:–
4239:P
4186:H
4182:S
4172:=
4167:B
4163:S
4154:A
4150:S
4146:=
4141:C
4137:S
4113:0
4105:C
4101:S
4092:C
4088:T
4084:=
4081:)
4076:B
4072:S
4063:A
4059:S
4055:(
4050:C
4046:T
4042:=
4037:C
4033:Q
4010:H
4006:S
3997:H
3993:T
3989:=
3986:)
3981:A
3977:S
3968:B
3964:S
3960:(
3955:H
3951:T
3947:=
3942:H
3938:Q
3913:H
3909:T
3903:C
3899:T
3890:1
3887:=
3858:A
3854:S
3829:B
3825:S
3800:H
3796:T
3767:C
3763:T
3738:H
3734:Q
3709:C
3705:Q
3692:W
3680:)
3678:3
3676:(
3655:H
3651:T
3645:C
3641:T
3632:1
3629:=
3622:H
3618:Q
3611:C
3607:Q
3598:H
3594:Q
3587:=
3580:H
3576:Q
3572:W
3567:=
3515:C
3511:Q
3502:H
3498:Q
3494:=
3491:Q
3488:=
3485:W
3465:Q
3443:0
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3431:S
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3418:T
3414:=
3411:)
3406:B
3402:S
3393:A
3389:S
3385:(
3380:C
3376:T
3372:=
3367:C
3363:Q
3340:H
3336:S
3327:H
3323:T
3319:=
3316:)
3311:A
3307:S
3298:B
3294:S
3290:(
3285:H
3281:T
3277:=
3272:H
3268:Q
3245:)
3240:A
3236:S
3227:B
3223:S
3219:(
3216:)
3211:C
3207:T
3198:H
3194:T
3190:(
3187:=
3184:S
3181:d
3178:T
3172:=
3169:V
3166:d
3163:P
3157:=
3154:W
3124:.
3122:C
3119:T
3115:H
3112:T
3097:S
3095:–
3093:T
3076:)
3074:2
3072:(
3055:S
3052:d
3049:T
3043:=
3040:U
3037:d
3028:S
3025:d
3022:T
3016:=
3013:)
3010:U
3007:d
3001:S
2998:d
2995:T
2992:(
2986:=
2983:)
2980:U
2977:d
2971:Q
2968:d
2965:(
2959:=
2956:V
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2950:P
2944:=
2941:W
2917:S
2915:-
2913:T
2903:)
2901:1
2899:(
2882:S
2879:d
2875:T
2870:B
2865:A
2857:=
2854:Q
2851:d
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2841:A
2833:=
2830:Q
2809:T
2805:S
2803:(
2797:S
2795:–
2793:T
2791:(
2781:C
2775:W
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2667:(
2641:H
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2632:/
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2515:=
2508:H
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2471:,
2468:0
2465:=
2456:S
2449:=
2444:C
2440:S
2433:+
2428:H
2424:S
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2382:T
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2333:C
2329:S
2303:C
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2294:/
2288:C
2284:Q
2280:=
2275:C
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2253:C
2250:Q
2242:C
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2190:Q
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2155:(
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2116:/
2110:H
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2102:=
2097:H
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2069:S
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2030:T
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1951:H
1947:Q
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1918:T
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1902:)
1897:C
1893:T
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1880:T
1876:(
1873:=
1870:S
1864:)
1859:C
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1842:T
1838:(
1835:=
1832:W
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1747:C
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1629:v
1156:S
1153:T
1147:H
1144:=
1141:)
1138:p
1135:,
1132:T
1129:(
1126:G
1101:S
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1092:U
1089:=
1086:)
1083:V
1080:,
1077:T
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1071:A
1046:V
1043:p
1040:+
1037:U
1034:=
1031:)
1028:p
1025:,
1022:S
1019:(
1016:H
991:)
988:V
985:,
982:S
979:(
976:U
853:T
829:V
806:V
782:1
757:=
721:p
697:V
674:V
650:1
622:=
586:T
562:N
539:S
515:T
490:=
487:c
410:)
20:)
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