42:
1612:
2356:) into a system. In a system, this heat energy can be transformed into work, and work can be transformed into heat through a cyclical process. Clausius writes that "The algebraic sum of all the transformations occurring in a cyclical process can only be less than zero, or, as an extreme case, equal to nothing." In other words, the equation
2347:
who intended to explain the relationship between the heat flow in a system and the entropy of the system and its surroundings. Clausius developed this in his efforts to explain entropy and define it quantitatively. In more direct terms, the theorem gives us a way to determine if a cyclical process is
2330:
at each infinitesimal stage of heat transfer. The
Clausius statement states that it is impossible to construct a device whose sole effect is the transfer of heat from a cool reservoir to a hot reservoir. Equivalently, heat spontaneously flows from a hot body to a cooler one, not the other way around.
2411:
being absolute temperature of the body when that energy is absorbed, is found to be true for any process that is cyclical and reversible. Clausius then took this a step further and determined that the following relation must be found true for any cyclical process that is possible, reversible or not.
4346:
process, there is no generation of entropy in each of the infinitesimal heat transfer processes since there is practically no temperature difference between the system and the thermal reservoirs (I.e., the system entropy change and the reservoirs entropy change is equal in magnitude and opposite in
2714:
If the amount of energy added by heating can be measured during the process, and the temperature can be measured during the process, then the
Clausius inequality can be used to determine whether the process is reversible or irreversible by carrying out the integration in the Clausius inequality. If
3985:
Because the total change in entropy for the system is zero in a thermodynamic cyclic process where all state functions of the system are reset or returned to initial values (values at the process starts) upon the completion of each cycle, if one adds all the infinitesimal steps of heat intake from
2351:
Clausius was one of the first to work on the idea of entropy and is even responsible for giving it that name. What is now known as the
Clausius theorem was first published in 1862 in Clausius' sixth memoir, "On the Application of the Theorem of the Equivalence of Transformations to Interior Work".
3720:
3203:
2803:
In the proof of the
Clausius theorem or inequality, a sign convention of heat is used; in the perspective of an object under consideration, when heat is absorbed by the object then the heat is positive, while when heat leaves from the object then the heat is negative.
4103:
2131:
2197:. This is because in a cyclic process the variation of a state function is zero per cycle, so the fact that this integral is equal to zero per cycle in a reversible process implies that there is some function (entropy) whose infinitesimal change is
4794:
4334:
1756:
4561:
3287:
1331:
2640:), which may vary depending on the path. In a cyclic process, therefore, the entropy of the system at the beginning of the cycle must equal to the entropy at the end of the cycle (because the entropy is a state function),
3594:
5071:
4910:
3891:
4989:
2797:
2288:
4160:
3358:
3029:
2461:
3077:
2632:, that the entropy is a state function: It depends only upon the state that the system is in, and not what path the system took to get there. This is in contrast to the amount of energy added as heat (𝛿
4399:
3509:
2623:
2020:
3940:
2709:
2398:
4215:
2731:
Because of the Second Law of
Thermodynamics, in each infinitesimal heat exchange process between the system and the reservoirs, the net change in entropy of the "universe", so to say, is
3817:
1948:
4618:
3589:
4442:
3069:
4482:
4255:
3979:
3420:
2901:
2227:
1792:
4019:
2028:
3545:
2572:
of the reservoir at a particular instant in time. Now that this is known, there must be a relation developed between the
Clausius inequality and entropy. The amount of entropy
2323:
as an infinitesimal change in entropy of a system (denoted by sys) under consideration applies not only to cyclic processes, but to any process that occurs in a closed system.
4011:
2566:
2321:
1908:
from the initial/final state to the same initial/final state (thermodynamic cycle). In principle, the closed integral can start and end at an arbitrary point along the path.
1898:
2931:
2516:
1848:
3750:
2839:
2164:
1166:
1111:
1056:
2667:
863:
816:
731:
684:
596:
549:
2539:
2487:
1871:
1819:
1641:
767:
635:
4680:
3772:
3381:
2862:
1001:
4825:
4665:
3450:
2958:
1342:
500:
4638:
4262:
2184:
1230:
839:
792:
707:
660:
572:
525:
2715:
integral result is equal to zero then it is a reversible process, while if greater than zero then an irreversible process (less than zero cannot be possible).
1688:
4489:
3211:
464:
2903:
to be positive or zero (i.e., non-negative) in this step (called the step 1 here) to fulfill the Second Law of
Thermodynamics, the temperature of the hot
1320:
2022:
interacting a thermodynamic system undergoing a thermodynamic cycle, the
Clausius inequality can be written as the following for expression clarity:
3591:) in an infinitesimal step (called the step 2), then again, for the Second Law of Thermodynamics to hold, one would have, in a very similar manner:
1353:
4997:
4833:
4915:
3822:
5295:
3986:
and heat expulsion to the reservoirs, signified by the previous two equations, with the temperature of each reservoir at each instant given by
2734:
2238:
923:
2669:, regardless of whether the process is reversible or irreversible. In irreversible cases, the net entropy is added to the system reservoirs
1634:
1221:
457:
335:
5095:
890:
4340:
273:
5230:
4114:
3715:{\displaystyle -dS_{{\text{Res}}_{2}}={\frac {\delta Q_{2}}{T_{\text{Cold}}}}\leq {\frac {\delta Q_{2}}{T_{2}}}=dS_{{\text{Sys}}_{2}}}
1405:
1379:
900:
354:
3292:
3198:{\displaystyle -dS_{{\text{Res}}_{1}}={\frac {\delta Q_{1}}{T_{\text{Hot}}}}\leq {\frac {\delta Q_{1}}{T_{1}}}=dS_{{\text{Sys}}_{1}}}
2963:
2418:
4667:
are work done by the heat engine and heat transferred from the hot thermal reservoir to the engine, respectively, can be derived by
1671:
306:
1458:
1950:
per thermodynamic cycle, meaning that the entropy of the reservoirs increases or does not change, and never decreases, per cycle.
4353:
2933:
needs to be equal to or greater than the temperature of the system at that instant; if the temperature of the system is given by
929:
328:
5300:
4578:
In the heat engine model with two thermal reservoirs (hot and cold reservoirs), the limit of the efficiency of any heat engine
1627:
2489:
is an infinitesimal amount of heat that is from the thermal reservoir interacting with the system and absorbed by the system (
3455:
1558:
90:
2582:
1453:
2727:
with which the system exchanges heat. At each instant of the process, the system is in contact with an external reservoir.
1956:
1533:
1306:
283:
4567:
4169:
3896:
2629:
2340:
2327:
918:
121:
111:
4668:
126:
116:
2723:
The temperature that enters in the denominator of the integrand in the
Clausius inequality is the temperature of the
4570:
at each infinitesimal stage of heat transfer, and is thus in a sense a weaker condition than the Second Law itself.
2672:
2362:
2348:
reversible or irreversible. The
Clausius theorem provides a quantitative formula for understanding the second law.
1905:
1448:
1410:
1374:
86:
4178:
2189:
In the special case of a reversible process, the equality holds, and the reversible case is used to introduce the
3777:
1914:
1203:
951:
397:
210:
200:
4581:
3554:
4405:
3034:
4448:
4221:
4098:{\displaystyle -\oint dS_{\text{Res}}=\oint {\frac {\delta Q}{T_{\text{surr}}}}\leq \oint dS_{\text{Sys}}=0.}
3945:
3386:
2867:
2126:{\displaystyle -\oint dS_{\text{Res}}=\oint \left(\sum _{n=1}^{N}{\frac {\delta Q_{n}}{T_{n}}}\right)\leq 0.}
5105:
5090:
1615:
1443:
1240:
943:
882:
418:
407:
73:
2200:
1764:
1548:
1468:
1265:
349:
103:
78:
5077:
This is the limit of heat engine efficiencies, and the equality of this expression is what is called the
5081:, that is the efficiency of all reversible heat engines and the maximum efficiency of all heat engines.
3514:
1483:
1060:
373:
219:
68:
3774:, signifying that heat is actually transferring (leaving) from the system to the cold reservoir, with
3989:
2711:
per thermodynamic cycle while in reversible cases, no entropy is created or added to the reservoirs.
2544:
2352:
Clausius sought to show a proportional relationship between entropy and the energy flow by heating (δ
2296:
1876:
1663:
1563:
1488:
1478:
278:
152:
140:
2909:
4343:
2492:
1824:
1679:
1508:
1503:
1270:
292:
258:
253:
166:
3725:
2814:
2139:
5100:
1821:
is an infinitesimal amount of heat that is taken from the reservoirs and absorbed by the system (
1597:
1260:
1208:
1121:
1066:
1011:
440:
424:
311:
263:
248:
238:
47:
41:
4789:{\displaystyle {{Q}_{1}}+{{Q}_{2}}=W\to \eta ={\frac {W}{{Q}_{1}}}=1+{\frac {{Q}_{2}}{{Q}_{1}}}}
2643:
1255:
845:
798:
713:
666:
578:
531:
5124:
3548:
2904:
2808:
2724:
2521:
2469:
1904:
of the reservoirs at a particular instant in time. The closed integral is carried out along a
1853:
1801:
1675:
1592:
1553:
1543:
1115:
913:
749:
741:
614:
243:
233:
175:
5234:
5130:
4168:
In summary, (the inequality in the third statement below, being obviously guaranteed by the
3754:
3363:
2844:
1513:
1498:
1438:
1433:
1250:
1245:
971:
363:
228:
4803:
4643:
3428:
2936:
4329:{\displaystyle \oint dS_{\text{Total}}=\oint dS_{\text{Res}}+\oint dS_{\text{Sys}}\geq 0.}
2344:
1463:
1311:
965:
606:
429:
190:
157:
5250:
482:
4623:
2190:
2169:
1751:{\displaystyle -\oint dS_{\text{Res}}=\oint {\frac {\delta Q}{T_{\text{surr}}}}\leq 0,}
1518:
1288:
824:
777:
692:
645:
557:
510:
388:
268:
205:
195:
63:
33:
17:
4556:{\displaystyle \oint dS_{\text{Total}}=\oint dS_{\text{Res}}+\oint dS_{\text{Sys}}=0.}
3282:{\textstyle \left|dS_{{\text{Res}}_{1}}\right|={\frac {\delta Q_{1}}{T_{\text{Hot}}}}}
5289:
1587:
905:
474:
435:
147:
5078:
1538:
1523:
1473:
956:
4671:(i.e., the law of conservation of energy) and the Clausius theorem or inequality.
5269:
2569:
1901:
1667:
1493:
301:
1582:
1528:
3893:
is equal to or greater than the magnitude of the entropy loss of the system
180:
5066:{\displaystyle \eta ={\frac {W}{{Q}_{1}}}\leq 1-{\frac {{T}_{2}}{{T}_{1}}}}
4905:{\displaystyle {\frac {{Q}_{1}}{{T}_{1}}}+{\frac {{Q}_{2}}{{T}_{2}}}\leq 0}
3886:{\textstyle dS_{{\text{Res}}_{2}}=-{\frac {\delta Q_{2}}{T_{\text{cold}}}}}
5251:"The Clausius Inequality And The Mathematical Statement Of The Second Law"
4984:{\displaystyle {\frac {{Q}_{2}}{{Q}_{1}}}\leq -{\frac {{T}_{2}}{{T}_{1}}}}
1296:
1213:
1005:
413:
185:
3208:
This means the magnitude of the entropy "loss" from the hot reservoir,
2792:{\displaystyle dS_{\text{Total}}=dS_{\text{Sys}}+dS_{\text{Res}}\geq 0}
2283:{\displaystyle dS_{\text{sys}}\geq {\frac {\delta Q}{T_{\text{surr}}}}}
2194:
1795:
402:
4991:. By substituting this inequality to the above equation results in,
2799:, where Sys and Res stand for System and Reservoir, respectively.
1873:< 0 if heat is leaving from the system to the reservoirs) and
4155:{\displaystyle \oint {\frac {\delta Q}{T_{\text{surr}}}}\leq 0,}
3353:{\textstyle dS_{{\text{Sys}}_{1}}={\frac {\delta Q_{1}}{T_{1}}}}
3024:{\textstyle dS_{{\text{Sys}}_{1}}={\frac {\delta Q_{1}}{T_{1}}}}
2541:< 0 if heat is leaving from the system to the reservoir) and
378:
2456:{\displaystyle \oint {\frac {\delta Q}{T_{\text{surr}}}}\leq 0}
3722:
Here, the amount of heat 'absorbed' by the system is given by
3289:
is equal to or less than the magnitude of the entropy "gain"
2339:
The Clausius theorem is a mathematical representation of the
3819:. The magnitude of the entropy gained by the cold reservoir
1953:
For multiple thermal reservoirs with different temperatures
4827:
is heat transferred from the engine to the cold reservoir.
4394:{\displaystyle \oint {\frac {\delta Q_{\text{rev}}}{T}}=0,}
1850:
if heat from the reservoirs is absorbed by the system, and
5210:. 4th ed. New York: McGraw-Hill Book Company, 1957. Print.
4674:
In respecting the abovementioned sign convention of heat,
2518:
if heat from the reservoir is absorbed by the system, and
1798:
change in the external thermal reservoirs (surroundings),
4566:
The Clausius inequality is a consequence of applying the
2326:
The Clausius inequality is a consequence of applying the
4347:
sign at any instant.), so the following equality holds,
3031:
as the entropy change in the system at the instant, and
3504:{\displaystyle \left|\delta Q_{2}\right|=-\delta Q_{2}}
5203:. Washington, D.C.: Scripta Book Company, 1975. Print.
3825:
3295:
3214:
2966:
2618:{\displaystyle \Delta S{=}\oint {\frac {\delta Q}{T}}}
5000:
4918:
4836:
4806:
4683:
4646:
4626:
4584:
4492:
4451:
4408:
4356:
4265:
4224:
4181:
4117:
4022:
3992:
3948:
3899:
3780:
3757:
3728:
3597:
3557:
3517:
3458:
3431:
3389:
3366:
3080:
3037:
2939:
2912:
2870:
2847:
2817:
2737:
2675:
2646:
2585:
2547:
2524:
2495:
2472:
2421:
2407:
being energy flow into the system due to heating and
2365:
2299:
2241:
2203:
2172:
2142:
2031:
2015:{\displaystyle \left(T_{1},T_{2},\dots ,T_{N}\right)}
1959:
1917:
1911:
The Clausius theorem or inequality obviously implies
1879:
1856:
1827:
1804:
1767:
1691:
1124:
1069:
1014:
974:
848:
827:
801:
780:
752:
716:
695:
669:
648:
617:
581:
560:
534:
513:
485:
5189:. New York: Philosophical Library Inc., 1969. Print.
2576:added to the system during the cycle is defined as
5065:
4983:
4904:
4819:
4788:
4659:
4632:
4612:
4555:
4476:
4436:
4393:
4328:
4249:
4209:
4154:
4097:
4005:
3973:
3935:{\displaystyle \left|dS_{{\text{Sys}}_{2}}\right|}
3934:
3885:
3811:
3766:
3744:
3714:
3583:
3539:
3503:
3444:
3414:
3375:
3352:
3281:
3197:
3063:
3023:
2952:
2925:
2895:
2856:
2833:
2791:
2703:
2661:
2617:
2560:
2533:
2510:
2481:
2455:
2392:
2315:
2282:
2221:
2178:
2158:
2125:
2014:
1942:
1892:
1865:
1842:
1813:
1786:
1750:
1160:
1105:
1050:
995:
857:
833:
810:
786:
761:
725:
701:
678:
654:
629:
590:
566:
543:
519:
494:
5196:. Englewood Cliffs: Prentice-Hall, 1966. Print.
2807:When the system takes heat from a hotter (hot)
2704:{\displaystyle (\Delta S_{\text{surr}}>0)}
2393:{\displaystyle \oint {\frac {\delta Q}{T}}=0}
1635:
8:
4210:{\displaystyle \oint dS_{\text{Res}}\geq 0,}
4165:which was to be proven (and is now proven).
2412:This relation is the "Clausius inequality",
2166:is an infinitesimal heat from the reservoir
3812:{\displaystyle dS_{{\text{Sys}}_{2}}\leq 0}
3383:) by the system, so the net entropy change
1943:{\displaystyle \oint dS_{\text{Res}}\geq 0}
27:Version of the second law of thermodynamics
4613:{\displaystyle \eta ={\frac {W}{{Q}_{1}}}}
4172:, which is the basis of our calculation),
3425:Similarly, when the system at temperature
1642:
1628:
1191:
343:
162:
40:
29:
5217:. London: Taylor and Francis, 1867. eBook
5055:
5050:
5043:
5038:
5035:
5018:
5013:
5007:
4999:
4973:
4968:
4961:
4956:
4953:
4939:
4934:
4927:
4922:
4919:
4917:
4888:
4883:
4876:
4871:
4868:
4857:
4852:
4845:
4840:
4837:
4835:
4811:
4805:
4778:
4773:
4766:
4761:
4758:
4741:
4736:
4730:
4708:
4703:
4701:
4691:
4686:
4684:
4682:
4651:
4645:
4625:
4602:
4597:
4591:
4583:
4541:
4522:
4503:
4491:
4462:
4450:
4419:
4407:
4370:
4360:
4355:
4314:
4295:
4276:
4264:
4235:
4223:
4192:
4180:
4135:
4121:
4116:
4083:
4062:
4048:
4036:
4021:
3997:
3991:
3963:
3958:
3956:
3947:
3919:
3914:
3912:
3898:
3875:
3864:
3854:
3840:
3835:
3833:
3824:
3795:
3790:
3788:
3779:
3756:
3736:
3727:
3704:
3699:
3697:
3679:
3668:
3658:
3647:
3636:
3626:
3615:
3610:
3608:
3596:
3584:{\displaystyle T_{\text{Cold}}\leq T_{2}}
3575:
3562:
3556:
3525:
3516:
3495:
3471:
3457:
3436:
3430:
3404:
3399:
3397:
3388:
3365:
3342:
3331:
3321:
3310:
3305:
3303:
3294:
3271:
3260:
3250:
3234:
3229:
3227:
3213:
3187:
3182:
3180:
3162:
3151:
3141:
3130:
3119:
3109:
3098:
3093:
3091:
3079:
3055:
3042:
3036:
3013:
3002:
2992:
2981:
2976:
2974:
2965:
2944:
2938:
2917:
2911:
2885:
2880:
2878:
2869:
2846:
2825:
2816:
2777:
2761:
2745:
2736:
2686:
2674:
2645:
2628:It has been determined, as stated in the
2600:
2592:
2584:
2552:
2546:
2523:
2494:
2471:
2439:
2425:
2420:
2369:
2364:
2307:
2298:
2272:
2258:
2249:
2240:
2232:The generalized "inequality of Clausius"
2204:
2202:
2171:
2150:
2141:
2104:
2093:
2083:
2077:
2066:
2045:
2030:
2001:
1982:
1969:
1958:
1928:
1916:
1884:
1878:
1855:
1826:
1803:
1778:
1766:
1731:
1717:
1705:
1690:
1123:
1068:
1013:
973:
847:
826:
800:
779:
751:
715:
694:
668:
647:
616:
580:
559:
533:
512:
484:
5145:. 3rd ed., p. 120, Academic Press, 2008.
4437:{\displaystyle \oint dS_{\text{Res}}=0,}
3064:{\displaystyle T_{\text{Hot}}\geq T_{1}}
5171:. 6th ed., Pearson/Prentice Hall, 2005.
5117:
4477:{\displaystyle \oint dS_{\text{Sys}}=0}
4250:{\displaystyle \oint dS_{\text{Sys}}=0}
3981:is also zero or positive in this case.
3974:{\displaystyle dS_{{\text{Total}}_{2}}}
3415:{\displaystyle dS_{{\text{Total}}_{1}}}
2896:{\displaystyle dS_{{\text{Total}}_{1}}}
1388:
1365:
1319:
1279:
1229:
1194:
387:
362:
291:
218:
165:
32:
5271:The Mechanical Theory of Heat (eBook)
5169:Physics: Principles with Applications
2222:{\displaystyle {\frac {\delta Q}{T}}}
1787:{\displaystyle \oint dS_{\text{Res}}}
7:
3540:{\displaystyle \delta Q_{2}\leq 0}
2679:
2647:
2586:
1682:, the following inequality holds.
849:
802:
717:
670:
582:
535:
355:Intensive and extensive properties
25:
5185:Morton, A. S., and P.J. Beckett.
5096:Carnot's theorem (thermodynamics)
2864:), for the net change in entropy
1611:
1610:
930:Table of thermodynamic equations
4669:the first law of thermodynamics
4006:{\displaystyle T_{\text{surr}}}
2561:{\displaystyle T_{\text{surr}}}
2316:{\displaystyle dS_{\text{sys}}}
1893:{\displaystyle T_{\text{surr}}}
1406:Maxwell's thermodynamic surface
5229:Judith McGovern (2004-03-17).
4721:
2926:{\displaystyle T_{\text{Hot}}}
2698:
2676:
1140:
1128:
1085:
1073:
1030:
1018:
990:
978:
1:
5296:Eponymous theorems of physics
5231:"Proof of Clausius's theorem"
5215:The Mechanical Theory of Heat
2511:{\displaystyle \delta Q>0}
1843:{\displaystyle \delta Q>0}
1307:Mechanical equivalent of heat
5201:Principles of Thermodynamics
5194:Thermodynamics for Engineers
4568:second law of thermodynamics
4170:second law of thermodynamics
3942:, so the net entropy change
3745:{\displaystyle \delta Q_{2}}
2834:{\displaystyle \delta Q_{1}}
2630:second law of thermodynamics
2341:second law of thermodynamics
2328:second law of thermodynamics
2159:{\displaystyle \delta Q_{n}}
919:Onsager reciprocal relations
5158:. 2nd ed., CRC Press, 1993.
2811:by an infinitesimal amount
1676:external thermal reservoirs
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}
5317:
2725:external thermal reservoir
2662:{\displaystyle \Delta S=0}
1906:thermodynamic process path
858:{\displaystyle \partial T}
811:{\displaystyle \partial V}
726:{\displaystyle \partial p}
679:{\displaystyle \partial V}
591:{\displaystyle \partial T}
544:{\displaystyle \partial S}
5268:Clausius, Rudolf (1867).
3452:expels heat in magnitude
1332:An Inquiry Concerning the
4830:The Clausius inequality
2534:{\displaystyle \delta Q}
2482:{\displaystyle \delta Q}
1866:{\displaystyle \delta Q}
1814:{\displaystyle \delta Q}
1345:Heterogeneous Substances
762:{\displaystyle \alpha =}
630:{\displaystyle \beta =-}
5208:Heat and Thermodynamics
5106:Introduction to entropy
5091:Kelvin-Planck statement
3547:) into a colder (cold)
1674:) exchanging heat with
1656:Clausius theorem (1855)
18:Clausius's theorem
5301:Laws of thermodynamics
5067:
4985:
4906:
4821:
4790:
4661:
4634:
4614:
4574:Heat engine efficiency
4557:
4484:(as a cyclic process),
4478:
4438:
4395:
4330:
4257:(as a cyclic process),
4251:
4211:
4156:
4099:
4007:
3975:
3936:
3887:
3813:
3768:
3767:{\displaystyle \leq 0}
3746:
3716:
3585:
3541:
3505:
3446:
3416:
3377:
3376:{\displaystyle \geq 0}
3354:
3283:
3199:
3065:
3025:
2960:at that instant, then
2954:
2927:
2897:
2858:
2857:{\displaystyle \geq 0}
2835:
2793:
2705:
2663:
2619:
2562:
2535:
2512:
2483:
2457:
2394:
2343:. It was developed by
2317:
2284:
2223:
2180:
2160:
2127:
2082:
2016:
1944:
1894:
1867:
1844:
1815:
1788:
1752:
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
5167:Giancoli, Douglas C.
5068:
4986:
4907:
4822:
4820:{\displaystyle Q_{2}}
4791:
4662:
4660:{\displaystyle Q_{1}}
4635:
4615:
4558:
4479:
4439:
4396:
4331:
4252:
4212:
4157:
4100:
4008:
3976:
3937:
3888:
3814:
3769:
3747:
3717:
3586:
3542:
3506:
3447:
3445:{\displaystyle T_{2}}
3422:is zero or positive.
3417:
3378:
3355:
3284:
3200:
3066:
3026:
2955:
2953:{\displaystyle T_{1}}
2928:
2898:
2859:
2836:
2794:
2706:
2664:
2620:
2563:
2536:
2513:
2484:
2458:
2395:
2318:
2285:
2224:
2181:
2161:
2128:
2062:
2017:
1945:
1895:
1868:
1845:
1816:
1789:
1753:
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:
5187:Basic Thermodynamics
4998:
4916:
4912:can be expressed as
4834:
4804:
4681:
4644:
4624:
4582:
4490:
4449:
4406:
4354:
4263:
4222:
4179:
4115:
4020:
3990:
3946:
3897:
3823:
3778:
3755:
3726:
3595:
3555:
3515:
3456:
3429:
3387:
3364:
3293:
3212:
3078:
3035:
2964:
2937:
2910:
2868:
2845:
2815:
2735:
2673:
2644:
2583:
2545:
2522:
2493:
2470:
2419:
2363:
2297:
2239:
2201:
2170:
2140:
2029:
1957:
1915:
1877:
1854:
1825:
1802:
1765:
1689:
1664:thermodynamic system
1662:, states that for a
1658:, also known as the
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:
3071:forces us to have:
1680:thermodynamic cycle
1660:Clausius inequality
1334:Source ... Friction
1266:Loschmidt's paradox
458:Material properties
336:Conjugate variables
5213:Clausius, Rudolf.
5206:Zemansky, Mark W.
5199:Hsieh, Jui Sheng.
5154:Finn, Colin B. P.
5143:Physical Chemistry
5101:Carnot heat engine
5063:
4981:
4902:
4817:
4786:
4657:
4630:
4610:
4553:
4474:
4434:
4391:
4326:
4247:
4207:
4152:
4095:
4003:
3971:
3932:
3883:
3809:
3764:
3742:
3712:
3581:
3537:
3501:
3442:
3412:
3373:
3350:
3279:
3195:
3061:
3021:
2950:
2923:
2893:
2854:
2831:
2789:
2701:
2659:
2615:
2558:
2531:
2508:
2479:
2453:
2390:
2313:
2280:
2219:
2176:
2156:
2123:
2012:
1940:
1890:
1863:
1840:
1811:
1784:
1748:
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
5079:Carnot efficiency
5061:
5024:
4979:
4945:
4894:
4863:
4784:
4747:
4633:{\displaystyle W}
4608:
4544:
4525:
4506:
4465:
4422:
4380:
4373:
4317:
4298:
4279:
4238:
4195:
4141:
4138:
4086:
4068:
4065:
4039:
4000:
3961:
3917:
3881:
3878:
3838:
3793:
3702:
3685:
3653:
3650:
3613:
3565:
3402:
3348:
3308:
3277:
3274:
3232:
3185:
3168:
3136:
3133:
3096:
3045:
3019:
2979:
2920:
2883:
2780:
2764:
2748:
2689:
2636:) and as work (𝛿
2613:
2555:
2445:
2442:
2382:
2310:
2278:
2275:
2252:
2217:
2179:{\displaystyle n}
2110:
2048:
1931:
1887:
1781:
1737:
1734:
1708:
1678:and undergoing a
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
5308:
5282:
5280:
5278:
5264:
5262:
5260:
5255:
5246:
5244:
5242:
5237:on July 19, 2011
5233:. Archived from
5192:Saad, Michel A.
5172:
5165:
5159:
5152:
5146:
5141:Mortimer, R. G.
5139:
5133:
5131:Wolfram Research
5126:Clausius theorem
5122:
5072:
5070:
5069:
5064:
5062:
5060:
5059:
5054:
5048:
5047:
5042:
5036:
5025:
5023:
5022:
5017:
5008:
4990:
4988:
4987:
4982:
4980:
4978:
4977:
4972:
4966:
4965:
4960:
4954:
4946:
4944:
4943:
4938:
4932:
4931:
4926:
4920:
4911:
4909:
4908:
4903:
4895:
4893:
4892:
4887:
4881:
4880:
4875:
4869:
4864:
4862:
4861:
4856:
4850:
4849:
4844:
4838:
4826:
4824:
4823:
4818:
4816:
4815:
4795:
4793:
4792:
4787:
4785:
4783:
4782:
4777:
4771:
4770:
4765:
4759:
4748:
4746:
4745:
4740:
4731:
4714:
4713:
4712:
4707:
4697:
4696:
4695:
4690:
4666:
4664:
4663:
4658:
4656:
4655:
4639:
4637:
4636:
4631:
4619:
4617:
4616:
4611:
4609:
4607:
4606:
4601:
4592:
4562:
4560:
4559:
4554:
4546:
4545:
4542:
4527:
4526:
4523:
4508:
4507:
4504:
4483:
4481:
4480:
4475:
4467:
4466:
4463:
4443:
4441:
4440:
4435:
4424:
4423:
4420:
4400:
4398:
4397:
4392:
4381:
4376:
4375:
4374:
4371:
4361:
4335:
4333:
4332:
4327:
4319:
4318:
4315:
4300:
4299:
4296:
4281:
4280:
4277:
4256:
4254:
4253:
4248:
4240:
4239:
4236:
4216:
4214:
4213:
4208:
4197:
4196:
4193:
4161:
4159:
4158:
4153:
4142:
4140:
4139:
4136:
4130:
4122:
4104:
4102:
4101:
4096:
4088:
4087:
4084:
4069:
4067:
4066:
4063:
4057:
4049:
4041:
4040:
4037:
4012:
4010:
4009:
4004:
4002:
4001:
3998:
3980:
3978:
3977:
3972:
3970:
3969:
3968:
3967:
3962:
3959:
3941:
3939:
3938:
3933:
3931:
3927:
3926:
3925:
3924:
3923:
3918:
3915:
3892:
3890:
3889:
3884:
3882:
3880:
3879:
3876:
3870:
3869:
3868:
3855:
3847:
3846:
3845:
3844:
3839:
3836:
3818:
3816:
3815:
3810:
3802:
3801:
3800:
3799:
3794:
3791:
3773:
3771:
3770:
3765:
3751:
3749:
3748:
3743:
3741:
3740:
3721:
3719:
3718:
3713:
3711:
3710:
3709:
3708:
3703:
3700:
3686:
3684:
3683:
3674:
3673:
3672:
3659:
3654:
3652:
3651:
3648:
3642:
3641:
3640:
3627:
3622:
3621:
3620:
3619:
3614:
3611:
3590:
3588:
3587:
3582:
3580:
3579:
3567:
3566:
3563:
3551:(at temperature
3546:
3544:
3543:
3538:
3530:
3529:
3510:
3508:
3507:
3502:
3500:
3499:
3481:
3477:
3476:
3475:
3451:
3449:
3448:
3443:
3441:
3440:
3421:
3419:
3418:
3413:
3411:
3410:
3409:
3408:
3403:
3400:
3382:
3380:
3379:
3374:
3359:
3357:
3356:
3351:
3349:
3347:
3346:
3337:
3336:
3335:
3322:
3317:
3316:
3315:
3314:
3309:
3306:
3288:
3286:
3285:
3280:
3278:
3276:
3275:
3272:
3266:
3265:
3264:
3251:
3246:
3242:
3241:
3240:
3239:
3238:
3233:
3230:
3204:
3202:
3201:
3196:
3194:
3193:
3192:
3191:
3186:
3183:
3169:
3167:
3166:
3157:
3156:
3155:
3142:
3137:
3135:
3134:
3131:
3125:
3124:
3123:
3110:
3105:
3104:
3103:
3102:
3097:
3094:
3070:
3068:
3067:
3062:
3060:
3059:
3047:
3046:
3043:
3030:
3028:
3027:
3022:
3020:
3018:
3017:
3008:
3007:
3006:
2993:
2988:
2987:
2986:
2985:
2980:
2977:
2959:
2957:
2956:
2951:
2949:
2948:
2932:
2930:
2929:
2924:
2922:
2921:
2918:
2902:
2900:
2899:
2894:
2892:
2891:
2890:
2889:
2884:
2881:
2863:
2861:
2860:
2855:
2840:
2838:
2837:
2832:
2830:
2829:
2798:
2796:
2795:
2790:
2782:
2781:
2778:
2766:
2765:
2762:
2750:
2749:
2746:
2710:
2708:
2707:
2702:
2691:
2690:
2687:
2668:
2666:
2665:
2660:
2624:
2622:
2621:
2616:
2614:
2609:
2601:
2596:
2567:
2565:
2564:
2559:
2557:
2556:
2553:
2540:
2538:
2537:
2532:
2517:
2515:
2514:
2509:
2488:
2486:
2485:
2480:
2462:
2460:
2459:
2454:
2446:
2444:
2443:
2440:
2434:
2426:
2399:
2397:
2396:
2391:
2383:
2378:
2370:
2322:
2320:
2319:
2314:
2312:
2311:
2308:
2289:
2287:
2286:
2281:
2279:
2277:
2276:
2273:
2267:
2259:
2254:
2253:
2250:
2228:
2226:
2225:
2220:
2218:
2213:
2205:
2186:to the system.
2185:
2183:
2182:
2177:
2165:
2163:
2162:
2157:
2155:
2154:
2132:
2130:
2129:
2124:
2116:
2112:
2111:
2109:
2108:
2099:
2098:
2097:
2084:
2081:
2076:
2050:
2049:
2046:
2021:
2019:
2018:
2013:
2011:
2007:
2006:
2005:
1987:
1986:
1974:
1973:
1949:
1947:
1946:
1941:
1933:
1932:
1929:
1899:
1897:
1896:
1891:
1889:
1888:
1885:
1872:
1870:
1869:
1864:
1849:
1847:
1846:
1841:
1820:
1818:
1817:
1812:
1793:
1791:
1790:
1785:
1783:
1782:
1779:
1757:
1755:
1754:
1749:
1738:
1736:
1735:
1732:
1726:
1718:
1710:
1709:
1706:
1644:
1637:
1630:
1614:
1613:
1321:Key publications
1302:
1301:("living force")
1251:Brownian ratchet
1246:Entropy and life
1241:Entropy and time
1192:
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:
163:
44:
30:
21:
5316:
5315:
5311:
5310:
5309:
5307:
5306:
5305:
5286:
5285:
5276:
5274:
5267:
5258:
5256:
5253:
5249:
5240:
5238:
5228:
5225:
5220:
5182:
5180:Further reading
5176:
5175:
5166:
5162:
5156:Thermal Physics
5153:
5149:
5140:
5136:
5123:
5119:
5114:
5087:
5049:
5037:
5012:
4996:
4995:
4967:
4955:
4933:
4921:
4914:
4913:
4882:
4870:
4851:
4839:
4832:
4831:
4807:
4802:
4801:
4772:
4760:
4735:
4702:
4685:
4679:
4678:
4647:
4642:
4641:
4622:
4621:
4596:
4580:
4579:
4576:
4537:
4518:
4499:
4488:
4487:
4458:
4447:
4446:
4415:
4404:
4403:
4366:
4362:
4352:
4351:
4310:
4291:
4272:
4261:
4260:
4231:
4220:
4219:
4188:
4177:
4176:
4131:
4123:
4113:
4112:
4108:In particular,
4079:
4058:
4050:
4032:
4018:
4017:
3993:
3988:
3987:
3984:
3957:
3952:
3944:
3943:
3913:
3908:
3904:
3900:
3895:
3894:
3871:
3860:
3856:
3834:
3829:
3821:
3820:
3789:
3784:
3776:
3775:
3753:
3752:
3732:
3724:
3723:
3698:
3693:
3675:
3664:
3660:
3643:
3632:
3628:
3609:
3604:
3593:
3592:
3571:
3558:
3553:
3552:
3521:
3513:
3512:
3491:
3467:
3463:
3459:
3454:
3453:
3432:
3427:
3426:
3398:
3393:
3385:
3384:
3362:
3361:
3338:
3327:
3323:
3304:
3299:
3291:
3290:
3267:
3256:
3252:
3228:
3223:
3219:
3215:
3210:
3209:
3181:
3176:
3158:
3147:
3143:
3126:
3115:
3111:
3092:
3087:
3076:
3075:
3051:
3038:
3033:
3032:
3009:
2998:
2994:
2975:
2970:
2962:
2961:
2940:
2935:
2934:
2913:
2908:
2907:
2879:
2874:
2866:
2865:
2843:
2842:
2821:
2813:
2812:
2773:
2757:
2741:
2733:
2732:
2721:
2682:
2671:
2670:
2642:
2641:
2602:
2581:
2580:
2548:
2543:
2542:
2520:
2519:
2491:
2490:
2468:
2467:
2435:
2427:
2417:
2416:
2371:
2361:
2360:
2345:Rudolf Clausius
2337:
2303:
2295:
2294:
2268:
2260:
2245:
2237:
2236:
2206:
2199:
2198:
2168:
2167:
2146:
2138:
2137:
2100:
2089:
2085:
2061:
2057:
2041:
2027:
2026:
1997:
1978:
1965:
1964:
1960:
1955:
1954:
1924:
1913:
1912:
1880:
1875:
1874:
1852:
1851:
1823:
1822:
1800:
1799:
1774:
1763:
1762:
1727:
1719:
1701:
1687:
1686:
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:
1120:
1119:
1118:
1065:
1064:
1063:
1010:
1009:
1008:
970:
969:
968:
966:Internal energy
961:
946:
936:
935:
910:
885:
875:
874:
873:
844:
843:
823:
822:
797:
796:
776:
775:
748:
747:
712:
711:
691:
690:
665:
664:
644:
643:
613:
612:
607:Compressibility
577:
576:
556:
555:
530:
529:
509:
508:
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:
5314:
5312:
5304:
5303:
5298:
5288:
5287:
5284:
5283:
5265:
5247:
5224:
5223:External links
5221:
5219:
5218:
5211:
5204:
5197:
5190:
5181:
5178:
5177:
5174:
5173:
5160:
5147:
5134:
5116:
5115:
5113:
5110:
5109:
5108:
5103:
5098:
5093:
5086:
5083:
5075:
5074:
5058:
5053:
5046:
5041:
5034:
5031:
5028:
5021:
5016:
5011:
5006:
5003:
4976:
4971:
4964:
4959:
4952:
4949:
4942:
4937:
4930:
4925:
4901:
4898:
4891:
4886:
4879:
4874:
4867:
4860:
4855:
4848:
4843:
4814:
4810:
4798:
4797:
4781:
4776:
4769:
4764:
4757:
4754:
4751:
4744:
4739:
4734:
4729:
4726:
4723:
4720:
4717:
4711:
4706:
4700:
4694:
4689:
4654:
4650:
4629:
4605:
4600:
4595:
4590:
4587:
4575:
4572:
4564:
4563:
4552:
4549:
4540:
4536:
4533:
4530:
4521:
4517:
4514:
4511:
4502:
4498:
4495:
4485:
4473:
4470:
4461:
4457:
4454:
4444:
4433:
4430:
4427:
4418:
4414:
4411:
4401:
4390:
4387:
4384:
4379:
4369:
4365:
4359:
4337:
4336:
4325:
4322:
4313:
4309:
4306:
4303:
4294:
4290:
4287:
4284:
4275:
4271:
4268:
4258:
4246:
4243:
4234:
4230:
4227:
4217:
4206:
4203:
4200:
4191:
4187:
4184:
4163:
4162:
4151:
4148:
4145:
4134:
4129:
4126:
4120:
4106:
4105:
4094:
4091:
4082:
4078:
4075:
4072:
4061:
4056:
4053:
4047:
4044:
4035:
4031:
4028:
4025:
3996:
3966:
3955:
3951:
3930:
3922:
3911:
3907:
3903:
3874:
3867:
3863:
3859:
3853:
3850:
3843:
3832:
3828:
3808:
3805:
3798:
3787:
3783:
3763:
3760:
3739:
3735:
3731:
3707:
3696:
3692:
3689:
3682:
3678:
3671:
3667:
3663:
3657:
3646:
3639:
3635:
3631:
3625:
3618:
3607:
3603:
3600:
3578:
3574:
3570:
3561:
3536:
3533:
3528:
3524:
3520:
3498:
3494:
3490:
3487:
3484:
3480:
3474:
3470:
3466:
3462:
3439:
3435:
3407:
3396:
3392:
3372:
3369:
3345:
3341:
3334:
3330:
3326:
3320:
3313:
3302:
3298:
3270:
3263:
3259:
3255:
3249:
3245:
3237:
3226:
3222:
3218:
3206:
3205:
3190:
3179:
3175:
3172:
3165:
3161:
3154:
3150:
3146:
3140:
3129:
3122:
3118:
3114:
3108:
3101:
3090:
3086:
3083:
3058:
3054:
3050:
3041:
3016:
3012:
3005:
3001:
2997:
2991:
2984:
2973:
2969:
2947:
2943:
2916:
2888:
2877:
2873:
2853:
2850:
2828:
2824:
2820:
2788:
2785:
2776:
2772:
2769:
2760:
2756:
2753:
2744:
2740:
2720:
2717:
2700:
2697:
2694:
2685:
2681:
2678:
2658:
2655:
2652:
2649:
2626:
2625:
2612:
2608:
2605:
2599:
2595:
2591:
2588:
2551:
2530:
2527:
2507:
2504:
2501:
2498:
2478:
2475:
2464:
2463:
2452:
2449:
2438:
2433:
2430:
2424:
2401:
2400:
2389:
2386:
2381:
2377:
2374:
2368:
2336:
2333:
2306:
2302:
2291:
2290:
2271:
2266:
2263:
2257:
2248:
2244:
2216:
2212:
2209:
2191:state function
2175:
2153:
2149:
2145:
2134:
2133:
2122:
2119:
2115:
2107:
2103:
2096:
2092:
2088:
2080:
2075:
2072:
2069:
2065:
2060:
2056:
2053:
2044:
2040:
2037:
2034:
2010:
2004:
2000:
1996:
1993:
1990:
1985:
1981:
1977:
1972:
1968:
1963:
1939:
1936:
1927:
1923:
1920:
1900:is the common
1883:
1862:
1859:
1839:
1836:
1833:
1830:
1810:
1807:
1777:
1773:
1770:
1759:
1758:
1747:
1744:
1741:
1730:
1725:
1722:
1716:
1713:
1704:
1700:
1697:
1694:
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:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5313:
5302:
5299:
5297:
5294:
5293:
5291:
5273:
5272:
5266:
5252:
5248:
5236:
5232:
5227:
5226:
5222:
5216:
5212:
5209:
5205:
5202:
5198:
5195:
5191:
5188:
5184:
5183:
5179:
5170:
5164:
5161:
5157:
5151:
5148:
5144:
5138:
5135:
5132:
5128:
5127:
5121:
5118:
5111:
5107:
5104:
5102:
5099:
5097:
5094:
5092:
5089:
5088:
5084:
5082:
5080:
5056:
5051:
5044:
5039:
5032:
5029:
5026:
5019:
5014:
5009:
5004:
5001:
4994:
4993:
4992:
4974:
4969:
4962:
4957:
4950:
4947:
4940:
4935:
4928:
4923:
4899:
4896:
4889:
4884:
4877:
4872:
4865:
4858:
4853:
4846:
4841:
4828:
4812:
4808:
4779:
4774:
4767:
4762:
4755:
4752:
4749:
4742:
4737:
4732:
4727:
4724:
4718:
4715:
4709:
4704:
4698:
4692:
4687:
4677:
4676:
4675:
4672:
4670:
4652:
4648:
4627:
4603:
4598:
4593:
4588:
4585:
4573:
4571:
4569:
4550:
4547:
4538:
4534:
4531:
4528:
4519:
4515:
4512:
4509:
4500:
4496:
4493:
4486:
4471:
4468:
4459:
4455:
4452:
4445:
4431:
4428:
4425:
4416:
4412:
4409:
4402:
4388:
4385:
4382:
4377:
4367:
4363:
4357:
4350:
4349:
4348:
4345:
4342:
4323:
4320:
4311:
4307:
4304:
4301:
4292:
4288:
4285:
4282:
4273:
4269:
4266:
4259:
4244:
4241:
4232:
4228:
4225:
4218:
4204:
4201:
4198:
4189:
4185:
4182:
4175:
4174:
4173:
4171:
4166:
4149:
4146:
4143:
4132:
4127:
4124:
4118:
4111:
4110:
4109:
4092:
4089:
4080:
4076:
4073:
4070:
4059:
4054:
4051:
4045:
4042:
4033:
4029:
4026:
4023:
4016:
4015:
4014:
3994:
3982:
3964:
3953:
3949:
3928:
3920:
3909:
3905:
3901:
3872:
3865:
3861:
3857:
3851:
3848:
3841:
3830:
3826:
3806:
3803:
3796:
3785:
3781:
3761:
3758:
3737:
3733:
3729:
3705:
3694:
3690:
3687:
3680:
3676:
3669:
3665:
3661:
3655:
3644:
3637:
3633:
3629:
3623:
3616:
3605:
3601:
3598:
3576:
3572:
3568:
3559:
3550:
3534:
3531:
3526:
3522:
3518:
3496:
3492:
3488:
3485:
3482:
3478:
3472:
3468:
3464:
3460:
3437:
3433:
3423:
3405:
3394:
3390:
3370:
3367:
3343:
3339:
3332:
3328:
3324:
3318:
3311:
3300:
3296:
3268:
3261:
3257:
3253:
3247:
3243:
3235:
3224:
3220:
3216:
3188:
3177:
3173:
3170:
3163:
3159:
3152:
3148:
3144:
3138:
3127:
3120:
3116:
3112:
3106:
3099:
3088:
3084:
3081:
3074:
3073:
3072:
3056:
3052:
3048:
3039:
3014:
3010:
3003:
2999:
2995:
2989:
2982:
2971:
2967:
2945:
2941:
2914:
2906:
2886:
2875:
2871:
2851:
2848:
2826:
2822:
2818:
2810:
2805:
2801:
2800:
2786:
2783:
2774:
2770:
2767:
2758:
2754:
2751:
2742:
2738:
2728:
2726:
2718:
2716:
2712:
2695:
2692:
2683:
2656:
2653:
2650:
2639:
2635:
2631:
2610:
2606:
2603:
2597:
2593:
2589:
2579:
2578:
2577:
2575:
2571:
2549:
2528:
2525:
2505:
2502:
2499:
2496:
2476:
2473:
2450:
2447:
2436:
2431:
2428:
2422:
2415:
2414:
2413:
2410:
2406:
2387:
2384:
2379:
2375:
2372:
2366:
2359:
2358:
2357:
2355:
2349:
2346:
2342:
2334:
2332:
2329:
2324:
2304:
2300:
2269:
2264:
2261:
2255:
2246:
2242:
2235:
2234:
2233:
2230:
2214:
2210:
2207:
2196:
2192:
2187:
2173:
2151:
2147:
2143:
2120:
2117:
2113:
2105:
2101:
2094:
2090:
2086:
2078:
2073:
2070:
2067:
2063:
2058:
2054:
2051:
2042:
2038:
2035:
2032:
2025:
2024:
2023:
2008:
2002:
1998:
1994:
1991:
1988:
1983:
1979:
1975:
1970:
1966:
1961:
1951:
1937:
1934:
1925:
1921:
1918:
1909:
1907:
1903:
1881:
1860:
1857:
1837:
1834:
1831:
1828:
1808:
1805:
1797:
1794:is the total
1775:
1771:
1768:
1745:
1742:
1739:
1728:
1723:
1720:
1714:
1711:
1702:
1698:
1695:
1692:
1685:
1684:
1683:
1681:
1677:
1673:
1669:
1665:
1661:
1657:
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:
1387:
1381:
1378:
1376:
1373:
1372:
1370:
1369:
1364:
1358:
1357:
1350:
1347:
1346:
1339:
1336:
1335:
1328:
1327:
1326:
1325:
1322:
1318:
1313:
1310:
1308:
1305:
1303:
1299:
1295:
1294:
1290:
1287:
1286:
1284:
1283:
1278:
1272:
1269:
1267:
1264:
1262:
1259:
1257:
1254:
1252:
1249:
1247:
1244:
1242:
1239:
1238:
1236:
1235:
1232:
1228:
1223:
1220:
1219:
1215:
1212:
1210:
1207:
1205:
1202:
1201:
1199:
1198:
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:
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390:
386:
380:
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361:
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348:
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313:
310:
308:
305:
303:
300:
299:
298:
297:
294:
290:
285:
282:
280:
277:
275:
274:Reversibility
272:
270:
267:
265:
262:
260:
257:
255:
252:
250:
247:
245:
242:
240:
237:
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199:
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182:
179:
177:
174:
173:
172:
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168:
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159:
156:
154:
151:
149:
148:Closed system
146:
145:
142:
137:
136:
128:
125:
123:
120:
118:
115:
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110:
109:
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100:
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67:
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62:
61:
54:
53:
49:
43:
39:
38:
35:
31:
19:
5275:. Retrieved
5270:
5257:. Retrieved
5239:. Retrieved
5235:the original
5214:
5207:
5200:
5193:
5186:
5168:
5163:
5155:
5150:
5142:
5137:
5125:
5120:
5076:
4829:
4799:
4673:
4577:
4565:
4338:
4167:
4164:
4107:
3983:
3424:
3207:
2806:
2802:
2730:
2729:
2722:
2713:
2637:
2633:
2627:
2573:
2465:
2408:
2404:
2402:
2353:
2350:
2338:
2325:
2292:
2231:
2188:
2135:
1952:
1910:
1760:
1659:
1655:
1653:
1449:Carathéodory
1380:Heat engines
1352:
1341:
1330:
1312:Motive power
1297:
957:Free entropy
928:
895:
428:
427: /
417:
416: /
408:introduction
401:
400: /
339:
302:Heat engines
89: /
5277:December 1,
4013:, one gets
2570:temperature
1902:temperature
1668:heat engine
1271:Synergetics
952:Free energy
398:Temperature
259:Quasistatic
254:Isenthalpic
211:Instruments
201:Equilibrium
153:Open system
87:Equilibrium
69:Statistical
5290:Categories
5259:October 5,
5241:October 4,
5112:References
4341:reversible
1583:Nucleation
1427:Scientists
1231:Philosophy
944:Potentials
307:Heat pumps
264:Polytropic
249:Isentropic
239:Isothermal
5033:−
5027:≤
5002:η
4951:−
4948:≤
4897:≤
4725:η
4722:→
4586:η
4532:∮
4513:∮
4494:∮
4453:∮
4410:∮
4364:δ
4358:∮
4321:≥
4305:∮
4286:∮
4267:∮
4226:∮
4199:≥
4183:∮
4144:≤
4125:δ
4119:∮
4074:∮
4071:≤
4052:δ
4046:∮
4027:∮
4024:−
3858:δ
3852:−
3804:≤
3759:≤
3730:δ
3662:δ
3656:≤
3630:δ
3599:−
3569:≤
3549:reservoir
3532:≤
3519:δ
3489:δ
3486:−
3465:δ
3368:≥
3325:δ
3254:δ
3145:δ
3139:≤
3113:δ
3082:−
3049:≥
2996:δ
2905:reservoir
2849:≥
2819:δ
2809:reservoir
2784:≥
2680:Δ
2648:Δ
2604:δ
2598:∮
2587:Δ
2526:δ
2497:δ
2474:δ
2448:≤
2429:δ
2423:∮
2373:δ
2367:∮
2262:δ
2256:≥
2208:δ
2193:known as
2144:δ
2118:≤
2087:δ
2064:∑
2055:∮
2036:∮
2033:−
1992:…
1935:≥
1919:∮
1858:δ
1829:δ
1806:δ
1769:∮
1740:≤
1721:δ
1715:∮
1696:∮
1693:−
1672:heat pump
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
5085:See also
4620:, 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
2568:is the
2403:with 𝛿
2335:History
2195:entropy
1796: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
4800:where
4344:cyclic
4339:For a
2466:where
2136:where
1761:where
1666:(e.g.
1529:Planck
1519:Nernst
1494:Kelvin
1454:Carnot
744:
609:
477:
419:Volume
334:Note:
293:Cycles
122:Second
112:Zeroth
5254:(PDF)
4505:Total
4278:Total
3960:Total
3401:Total
2882:Total
2747:Total
2719:Proof
1577:Other
1544:Stahl
1499:Lewis
1489:Joule
1479:Gibbs
1474:Duhem
167:State
127:Third
117:First
5279:2011
5261:2010
5243:2010
4640:and
4137:surr
4064:surr
3999:surr
3877:cold
3649:Cold
3564:Cold
2693:>
2688:surr
2554:surr
2503:>
2441:surr
2293:for
2274:surr
1886:surr
1835:>
1733:surr
1654:The
1549:Tait
379:Heat
374:Work
104:Laws
5129:at
4543:Sys
4524:Res
4464:Sys
4421:Res
4372:rev
4316:Sys
4297:Res
4237:Sys
4194:Res
4085:Sys
4038:Res
3916:Sys
3837:Res
3792:Sys
3701:Sys
3612:Res
3307:Sys
3273:Hot
3231:Res
3184:Sys
3132:Hot
3095:Res
3044:Hot
2978:Sys
2919:Hot
2779:Res
2763:Sys
2309:sys
2251:sys
2229:.
2047:Res
1930:Res
1780:Res
1707:Res
1670:or
1392:Art
338:in
5292::
4551:0.
4324:0.
4093:0.
2121:0.
5281:.
5263:.
5245:.
5073:.
5057:1
5052:T
5045:2
5040:T
5030:1
5020:1
5015:Q
5010:W
5005:=
4975:1
4970:T
4963:2
4958:T
4941:1
4936:Q
4929:2
4924:Q
4900:0
4890:2
4885:T
4878:2
4873:Q
4866:+
4859:1
4854:T
4847:1
4842:Q
4813:2
4809:Q
4796:,
4780:1
4775:Q
4768:2
4763:Q
4756:+
4753:1
4750:=
4743:1
4738:Q
4733:W
4728:=
4719:W
4716:=
4710:2
4705:Q
4699:+
4693:1
4688:Q
4653:1
4649:Q
4628:W
4604:1
4599:Q
4594:W
4589:=
4548:=
4539:S
4535:d
4529:+
4520:S
4516:d
4510:=
4501:S
4497:d
4472:0
4469:=
4460:S
4456:d
4432:,
4429:0
4426:=
4417:S
4413:d
4389:,
4386:0
4383:=
4378:T
4368:Q
4312:S
4308:d
4302:+
4293:S
4289:d
4283:=
4274:S
4270:d
4245:0
4242:=
4233:S
4229:d
4205:,
4202:0
4190:S
4186:d
4150:,
4147:0
4133:T
4128:Q
4090:=
4081:S
4077:d
4060:T
4055:Q
4043:=
4034:S
4030:d
3995:T
3965:2
3954:S
3950:d
3929:|
3921:2
3910:S
3906:d
3902:|
3873:T
3866:2
3862:Q
3849:=
3842:2
3831:S
3827:d
3807:0
3797:2
3786:S
3782:d
3762:0
3738:2
3734:Q
3706:2
3695:S
3691:d
3688:=
3681:2
3677:T
3670:2
3666:Q
3645:T
3638:2
3634:Q
3624:=
3617:2
3606:S
3602:d
3577:2
3573:T
3560:T
3535:0
3527:2
3523:Q
3511:(
3497:2
3493:Q
3483:=
3479:|
3473:2
3469:Q
3461:|
3438:2
3434:T
3406:1
3395:S
3391:d
3371:0
3360:(
3344:1
3340:T
3333:1
3329:Q
3319:=
3312:1
3301:S
3297:d
3269:T
3262:1
3258:Q
3248:=
3244:|
3236:1
3225:S
3221:d
3217:|
3189:1
3178:S
3174:d
3171:=
3164:1
3160:T
3153:1
3149:Q
3128:T
3121:1
3117:Q
3107:=
3100:1
3089:S
3085:d
3057:1
3053:T
3040:T
3015:1
3011:T
3004:1
3000:Q
2990:=
2983:1
2972:S
2968:d
2946:1
2942:T
2915:T
2887:1
2876:S
2872:d
2852:0
2841:(
2827:1
2823:Q
2787:0
2775:S
2771:d
2768:+
2759:S
2755:d
2752:=
2743:S
2739:d
2699:)
2696:0
2684:S
2677:(
2657:0
2654:=
2651:S
2638:W
2634:Q
2611:T
2607:Q
2594:=
2590:S
2574:S
2550:T
2529:Q
2506:0
2500:Q
2477:Q
2451:0
2437:T
2432:Q
2409:T
2405:Q
2388:0
2385:=
2380:T
2376:Q
2354:Q
2305:S
2301:d
2270:T
2265:Q
2247:S
2243:d
2215:T
2211:Q
2174:n
2152:n
2148:Q
2114:)
2106:n
2102:T
2095:n
2091:Q
2079:N
2074:1
2071:=
2068:n
2059:(
2052:=
2043:S
2039:d
2009:)
2003:N
1999:T
1995:,
1989:,
1984:2
1980:T
1976:,
1971:1
1967:T
1962:(
1938:0
1926:S
1922:d
1882:T
1861:Q
1838:0
1832:Q
1809:Q
1776:S
1772:d
1746:,
1743:0
1729:T
1724:Q
1712:=
1703:S
1699:d
1643:e
1636:t
1629:v
1156:S
1153:T
1147:H
1144:=
1141:)
1138:p
1135:,
1132:T
1129:(
1126:G
1101:S
1098:T
1092:U
1089:=
1086:)
1083:V
1080:,
1077:T
1074:(
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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