3367:
combustion is highly exothermic, which releases heat. The overall change in internal energy does not reveal the mode of energy transfer and quantifies only the net work and heat. The difference between initial and final states of the system's internal energy does not account for the extent of the energy interactions transpired. Therefore, internal energy is a state function (i.e. exact differential), while heat and work are path functions (i.e. inexact differentials) because integration must account for the path taken.
28:
1598:
3204:. The third stage is similar to the first stage, except the heat is lost by contact with a cold reservoir, while the fourth cycle is like the second except work is done onto the system by the surroundings to compress the gas. Because the overall changes in heat and work are different over different parts of the cycle, there is a nonzero net change in the heat and work, indicating that the differentials
2162:
requires path independence in order to express the values of a given vector field in terms of the partial derivatives of another function that is the multivariate analogue of the antiderivative. This is because there can be no unique representation of an antiderivative for inexact differentials since
1668:
is always path independent since the integral acts to invert the differential operator. Consequently, a quantity with an inexact differential cannot be expressed as a function of only the variables within the differential. I.e., its value cannot be inferred just by looking at the initial and final
3007:
3366:
barrier for combustion is transferred as heat into the system, resulting in changes to the system's internal energy. In a process, the energy input to start a fire may comprise both work and heat, such as when one rubs tinder (work) and experiences friction (heat) to start a fire. The ensuing
3937:
2844:
2416:. This example captures the essential idea behind the inexact differential in one dimension. Note that if we allowed ourselves to change directions, then we could take a step forward and then backward at any point in time in going from
3811:
1317:
1962:
3431:
3692:
3525:
1780:
2840:
3130:
is the differential change in work. Based on the constants of the thermodynamic system, we are able to parameterize the average energy in several different ways. E.g., in the first stage of the
3062:
1896:
1838:
2667:
2549:
3843:
2154:
3615:
2046:
2004:
2489:
2345:
2611:
2770:
3202:
3167:
3964:
1152:
1097:
1042:
2705:
849:
802:
717:
670:
582:
535:
4047:
3834:
3263:, meaning its change can be inferred just by comparing two different states of the system (independently of its transition path), which we can therefore indicate with
3248:
3225:
3128:
3105:
2092:
1714:
1627:
753:
621:
2725:
2260:
987:
2414:
1901:
1328:
2368:
486:
3082:
2790:
2745:
2572:
2509:
2454:
2434:
2388:
2313:
2293:
2120:
2066:
1216:
825:
778:
693:
646:
558:
511:
3697:
3002:{\displaystyle \Delta g|_{\gamma }=\int _{\gamma }|\mathrm {d} x|=\int _{A}^{B}\mathrm {d} x+\int _{B}^{A}(-\mathrm {d} x)=2\int _{A}^{B}\mathrm {d} x=2AB}
450:
4121:
3386:
1306:
3471:
1339:
1716:
is a differential for which the integral over some two paths with the same end points is different. Specifically, there exist integrable paths
2795:
909:
1620:
1207:
443:
321:
876:
3975:
259:
1391:
1365:
886:
340:
3024:
4031:
3624:
292:
1444:
2456:
and in-so-doing increase the overall distance covered to an arbitrarily large number while keeping the net displacement constant.
915:
314:
2125:
4154:
2164:
1613:
1719:
1544:
76:
2163:
their variation is inconsistent along different paths. This stipulation of path independence is a necessary addendum to the
1439:
3169:
enters the gas. During the second stage, the gas is allowed to freely expand, outputting some differential amount of work
1519:
1292:
269:
2347:(without changing directions) your net displacement and total distance covered are both equal to the length of said line
1649:
904:
107:
97:
3991:
3018:
112:
102:
2218:
Within statistical mechanics, inexact differentials are often denoted with a bar through the differential operator,
3981:
1843:
1785:
1434:
1396:
1360:
72:
2616:
4002:
2518:
1189:
937:
383:
196:
186:
3134:
a gas is heated by a reservoir, giving us an isothermal expansion of that gas. Some differential amount of heat
3554:
2009:
1967:
4149:
1656:
to express changes in path dependent quantities such as heat and work, but is defined more generally within
1601:
1429:
1226:
929:
868:
404:
393:
59:
2195:
1534:
1454:
1251:
335:
89:
64:
2222:. In LaTeX the command "\rlap{\textrm{d}}{\bar{\phantom{w}}}" is an approximation or simply "\dj" for a
2167:
because in one-dimensional calculus there is only one path in between two points defined by a function.
1469:
1046:
359:
205:
54:
2462:
2390:(without changing directions) then your net displacement is zero while your total distance covered is
2318:
2048:
respectively to make explicit the path dependence of the change of the quantity we are considering as
2581:
1549:
1474:
1464:
264:
138:
126:
3932:{\displaystyle x\,\delta u=2xy\,\mathrm {d} x+x^{2}\,\mathrm {d} y=\mathrm {\mathrm {d} } (x^{2}y)}
3694:. Thus, along the first path we get a value of 2. However, along the second path we get a value of
2750:
1494:
1489:
1256:
278:
244:
239:
152:
3172:
3137:
4127:
4041:
3996:
3986:
3942:
3376:
2099:
1674:
1665:
1583:
1246:
1194:
1107:
1052:
997:
426:
410:
297:
249:
234:
224:
33:
27:
3439:
is an inexact differential, because its effect on the state of the system can be compensated by
2672:
1241:
831:
784:
699:
652:
564:
517:
4027:
3816:
3363:
3230:
3207:
3110:
3087:
2239:
2095:
2074:
1696:
1661:
1578:
1539:
1529:
1101:
899:
735:
727:
600:
229:
219:
161:
3375:
It is sometimes possible to convert an inexact differential into an exact one by means of an
2710:
2245:
1669:
states of a given system. Inexact differentials are primarily used in calculations involving
2159:
1499:
1484:
1424:
1419:
1236:
1231:
957:
881:
349:
214:
1449:
1297:
951:
592:
415:
176:
143:
3806:{\textstyle \int _{0}^{1}2y\,\mathrm {d} x|_{y=0}+\int _{0}^{1}x\,\mathrm {d} y|_{x=1}=1}
2393:
2238:
Within mathematics, inexact differentials are usually just referred more generally to as
4115:
4062:
3362:
A fire requires heat, fuel, and an oxidizing agent. The energy required to overcome the
2350:
468:
3259:
3067:
2775:
2730:
2557:
2552:
2494:
2439:
2419:
2373:
2298:
2278:
2105:
2051:
1682:
1653:
1504:
1274:
810:
763:
678:
631:
543:
496:
374:
254:
191:
181:
49:
19:
4133:
4143:
2227:
2219:
2191:
1678:
1573:
891:
460:
421:
133:
3131:
1524:
1509:
1459:
942:
3446:
1657:
1479:
287:
1568:
1514:
1957:{\displaystyle \int _{\gamma _{1}}\delta u\not =\int _{\gamma _{2}}\delta u}
166:
3379:. The most common example of this in thermodynamics is the definition of
1282:
1199:
991:
399:
171:
3380:
3339:, such a change of state does not uniquely identify the amount of work
388:
2190:
is used, a convention which originated in the 19th century work of
3999:
for solving non-exact differential equations by making them exact
3452:
when the exchange occurs at reversible conditions (therefore the
3009:
exactly the results we expected from the verbal argument before.
3426:{\displaystyle \mathrm {d} S={\frac {\delta Q_{\text{rev}}}{T}}}
2223:
1670:
364:
3687:{\textstyle \int _{0}^{1}2y\,\mathrm {\mathrm {d} } x|_{y=1}=2}
3520:{\displaystyle \delta u=2y\,\mathrm {d} x+x\,\mathrm {d} y.}
3456:
subscript), it produces an exact differential: the entropy
2613:, which does not have an antiderivative. The path taken is
1652:
whose integral is path dependent. It is most often used in
1775:{\displaystyle \gamma _{1},\gamma _{2}:\to \mathbb {R} }
3527:
This must be inexact by considering going to the point
2835:{\displaystyle \Delta f=\int _{\gamma }\mathrm {d} x=0}
3700:
3627:
3557:
3945:
3846:
3819:
3474:
3389:
3233:
3210:
3175:
3140:
3113:
3090:
3070:
3027:
2847:
2798:
2778:
2753:
2733:
2713:
2675:
2619:
2584:
2560:
2521:
2497:
2465:
2442:
2422:
2396:
2376:
2353:
2321:
2301:
2281:
2248:
2128:
2108:
2077:
2054:
2012:
1970:
1904:
1846:
1788:
1722:
1699:
1110:
1055:
1000:
960:
834:
813:
787:
766:
738:
702:
681:
655:
634:
603:
567:
546:
520:
499:
471:
3347:
transferred, but only the change in internal energy
4064:Vorlesungen über die mechanische Theorie der Wärme
3958:
3931:
3828:
3805:
3686:
3609:
3539:, then that corresponds to first integrating over
3519:
3425:
3242:
3219:
3196:
3161:
3122:
3099:
3076:
3056:
3001:
2834:
2784:
2764:
2739:
2719:
2699:
2661:
2605:
2566:
2543:
2503:
2483:
2459:Reworking the above with differentials and taking
2448:
2428:
2408:
2382:
2362:
2339:
2307:
2287:
2254:
2160:fundamental theorem of calculus for line integrals
2148:
2114:
2086:
2060:
2040:
1998:
1956:
1890:
1832:
1774:
1708:
1146:
1091:
1036:
981:
843:
819:
796:
772:
747:
711:
687:
664:
640:
615:
576:
552:
529:
505:
480:
3017:Inexact differentials show up explicitly in the
2792:and negative afterward, yielding the integrals,
4084:Fundamentals of Statistical and Thermal Physics
3057:{\displaystyle \mathrm {d} U=\delta Q-\delta W}
1891:{\displaystyle \gamma _{1}(1)=\gamma _{2}(1)}
1833:{\displaystyle \gamma _{1}(0)=\gamma _{2}(0)}
1621:
8:
4046:: CS1 maint: multiple names: authors list (
2662:{\displaystyle \gamma :\to {\overline {AB}}}
3836:an exact differential by multiplying it by
2544:{\displaystyle \mathrm {d} f=\mathrm {d} x}
2149:{\displaystyle \mathrm {d} f\neq \delta u}
1628:
1614:
1177:
329:
148:
26:
15:
4069:Lectures on the Mechanical Theory of Heat
3949:
3944:
3917:
3904:
3903:
3892:
3891:
3885:
3870:
3869:
3850:
3845:
3818:
3785:
3780:
3771:
3770:
3761:
3756:
3737:
3732:
3723:
3722:
3710:
3705:
3699:
3666:
3661:
3651:
3650:
3649:
3637:
3632:
3626:
3589:
3584:
3576:
3567:
3562:
3556:
3506:
3505:
3491:
3490:
3473:
3411:
3401:
3390:
3388:
3232:
3209:
3174:
3139:
3112:
3089:
3069:
3028:
3026:
2979:
2973:
2968:
2947:
2935:
2930:
2915:
2909:
2904:
2892:
2884:
2879:
2873:
2860:
2855:
2846:
2818:
2812:
2797:
2777:
2754:
2752:
2732:
2712:
2674:
2644:
2618:
2598:
2590:
2585:
2583:
2559:
2533:
2522:
2520:
2496:
2466:
2464:
2441:
2421:
2395:
2375:
2352:
2322:
2320:
2300:
2280:
2247:
2129:
2127:
2107:
2076:
2053:
2030:
2025:
2020:
2011:
1988:
1983:
1978:
1969:
1964:In this case, we denote the integrals as
1940:
1935:
1914:
1909:
1903:
1873:
1851:
1845:
1815:
1793:
1787:
1768:
1767:
1740:
1727:
1721:
1698:
1109:
1054:
999:
959:
833:
812:
786:
765:
737:
701:
680:
654:
633:
602:
566:
545:
519:
498:
470:
3610:{\textstyle \int _{0}^{1}x\,dy|_{x=0}=0}
3468:Consider the inexact differential form,
3445:. However, when divided by the absolute
2747:and strictly decreasing afterward. Then
2071:More generally, an inexact differential
4014:
3107:is the differential change in heat and
2041:{\displaystyle \Delta u|_{\gamma _{2}}}
1999:{\displaystyle \Delta u|_{\gamma _{1}}}
1374:
1351:
1305:
1265:
1215:
1180:
373:
348:
277:
204:
151:
18:
4039:
7:
4099:Principles of Mathematical Analysis
3976:Closed and exact differential forms
2180:Instead of the differential symbol
3905:
3893:
3871:
3772:
3724:
3652:
3507:
3492:
3391:
3029:
2980:
2948:
2916:
2885:
2848:
2819:
2799:
2755:
2591:
2534:
2523:
2130:
2013:
1971:
835:
788:
703:
656:
568:
521:
341:Intensive and extensive properties
14:
2206:(work) are path-dependent, while
1664:. In contrast, an integral of an
2484:{\displaystyle {\overline {AB}}}
2340:{\displaystyle {\overline {AB}}}
2242:which are often written just as
1597:
1596:
916:Table of thermodynamic equations
4128:Exact and Inexact Differentials
4122:Exact and Inexact Differentials
4024:The World of Physical Chemistry
3250:must be inexact differentials.
2606:{\displaystyle |\mathrm {d} x|}
2165:fundamental theorem of calculus
1392:Maxwell's thermodynamic surface
3926:
3910:
3781:
3733:
3662:
3585:
2955:
2941:
2893:
2880:
2856:
2727:is strictly increasing before
2694:
2682:
2641:
2638:
2626:
2599:
2586:
2370:. If you then return to point
2021:
1979:
1885:
1879:
1863:
1857:
1827:
1821:
1805:
1799:
1764:
1761:
1749:
1126:
1114:
1071:
1059:
1016:
1004:
976:
964:
1:
3281:. Since we can go from state
2765:{\displaystyle \mathrm {d} x}
2551:, an exact differential with
1293:Mechanical equivalent of heat
3978:for a higher-level treatment
3197:{\displaystyle \delta W=PdV}
3162:{\displaystyle \delta Q=TdS}
2654:
2476:
2332:
905:Onsager reciprocal relations
4026:. Oxford University Press.
4022:Laidler, Keith, J. (1993).
3992:Exact differential equation
3959:{\displaystyle x\,\delta u}
3343:done to the system or heat
3019:first law of thermodynamics
3013:First law of thermodynamics
2275:When you walk from a point
2226:character, which needs the
1397:Entropy as energy dispersal
1208:"Perpetual motion" machines
1147:{\displaystyle G(T,p)=H-TS}
1092:{\displaystyle A(T,V)=U-TS}
1037:{\displaystyle H(S,p)=U+pV}
4171:
3982:Differential (mathematics)
3966:is an exact differential.
3617:and then integrating over
3460:is also a state function.
2700:{\displaystyle t\in (0,1)}
2669:where there exists a time
2210:(internal energy) is not.
2102:, i.e., for all functions
844:{\displaystyle \partial T}
797:{\displaystyle \partial V}
712:{\displaystyle \partial p}
665:{\displaystyle \partial V}
577:{\displaystyle \partial T}
530:{\displaystyle \partial S}
4061:Neumann, Carl G. (1875).
4003:Conservative vector field
3299:either by providing heat
2574:. On the other hand, the
1318:An Inquiry Concerning the
4136:– from Wolfram MathWorld
4118:– from Wolfram MathWorld
4071:]. Leipzig: Teubner.
3829:{\displaystyle \delta u}
3243:{\displaystyle \delta W}
3220:{\displaystyle \delta Q}
3123:{\displaystyle \delta W}
3100:{\displaystyle \delta Q}
2087:{\displaystyle \delta u}
1709:{\displaystyle \delta u}
1693:An inexact differential
1331:Heterogeneous Substances
748:{\displaystyle \alpha =}
616:{\displaystyle \beta =-}
4124:– University of Arizona
4082:Reif, Fredrick (1965).
3531:. If we first increase
2720:{\displaystyle \gamma }
2255:{\displaystyle \omega }
4155:Multivariable calculus
4097:Rudin, Walter (2013).
3960:
3933:
3830:
3807:
3688:
3611:
3521:
3427:
3244:
3221:
3198:
3163:
3124:
3101:
3078:
3058:
3003:
2836:
2786:
2766:
2741:
2721:
2701:
2663:
2607:
2568:
2545:
2505:
2485:
2450:
2430:
2410:
2384:
2364:
2341:
2309:
2289:
2256:
2196:Carl Gottfried Neumann
2150:
2116:
2088:
2062:
2042:
2000:
1958:
1892:
1834:
1776:
1710:
1646:imperfect differential
1148:
1093:
1038:
983:
982:{\displaystyle U(S,V)}
845:
821:
798:
774:
749:
713:
689:
666:
642:
617:
578:
554:
531:
507:
482:
461:Specific heat capacity
65:Quantum thermodynamics
4130:– University of Texas
3961:
3934:
3831:
3808:
3689:
3612:
3522:
3428:
3245:
3222:
3199:
3164:
3125:
3102:
3079:
3059:
3004:
2837:
2787:
2767:
2742:
2722:
2702:
2664:
2608:
2569:
2546:
2506:
2486:
2451:
2431:
2411:
2385:
2365:
2342:
2310:
2290:
2257:
2214:Statistical Mechanics
2151:
2117:
2089:
2063:
2043:
2001:
1959:
1893:
1835:
1777:
1711:
1329:On the Equilibrium of
1149:
1094:
1047:Helmholtz free energy
1039:
984:
846:
822:
799:
775:
750:
714:
690:
667:
643:
618:
579:
555:
532:
508:
483:
4116:Inexact Differential
3943:
3844:
3817:
3698:
3625:
3555:
3472:
3387:
3231:
3208:
3173:
3138:
3111:
3088:
3068:
3025:
2845:
2796:
2776:
2751:
2731:
2711:
2673:
2617:
2582:
2558:
2519:
2495:
2463:
2440:
2420:
2394:
2374:
2351:
2319:
2299:
2279:
2246:
2126:
2106:
2075:
2052:
2010:
1968:
1902:
1844:
1786:
1720:
1697:
1642:inexact differential
1342:Motive Power of Fire
1108:
1053:
998:
958:
910:Bridgman's equations
887:Fundamental relation
832:
811:
785:
764:
736:
700:
679:
653:
632:
601:
565:
544:
518:
497:
469:
3766:
3715:
3642:
3572:
3547:. Integrating over
3371:Integrating factors
2978:
2940:
2914:
2772:is positive before
2409:{\displaystyle 2AB}
1320:Source ... Friction
1252:Loschmidt's paradox
444:Material properties
322:Conjugate variables
4134:Exact Differential
3997:Integrating factor
3987:Exact differential
3956:
3929:
3826:
3803:
3752:
3701:
3684:
3628:
3607:
3558:
3551:first contributes
3535:and then increase
3517:
3423:
3377:integrating factor
3240:
3217:
3194:
3159:
3120:
3097:
3074:
3054:
2999:
2964:
2926:
2900:
2832:
2782:
2762:
2737:
2717:
2697:
2659:
2603:
2564:
2541:
2501:
2481:
2446:
2426:
2406:
2380:
2363:{\displaystyle AB}
2360:
2337:
2305:
2285:
2252:
2240:differential forms
2198:, indicating that
2146:
2112:
2100:exact differential
2084:
2058:
2038:
1996:
1954:
1888:
1830:
1772:
1706:
1666:exact differential
1584:Order and disorder
1340:Reflections on the
1247:Heat death paradox
1144:
1089:
1034:
979:
841:
817:
794:
770:
745:
709:
685:
662:
638:
613:
574:
550:
527:
503:
481:{\displaystyle c=}
478:
451:Property databases
427:Reduced properties
411:Chemical potential
375:Functions of state
298:Thermal efficiency
34:Carnot heat engine
3421:
3414:
3364:activation energy
3077:{\displaystyle U}
2785:{\displaystyle t}
2740:{\displaystyle t}
2657:
2567:{\displaystyle x}
2504:{\displaystyle x}
2479:
2449:{\displaystyle B}
2429:{\displaystyle A}
2383:{\displaystyle A}
2335:
2308:{\displaystyle B}
2288:{\displaystyle A}
2115:{\displaystyle f}
2096:differential form
2061:{\displaystyle u}
1677:because they are
1662:differential form
1638:
1637:
1579:Self-organization
1404:
1403:
1102:Gibbs free energy
900:Maxwell relations
858:
857:
854:
853:
820:{\displaystyle V}
773:{\displaystyle 1}
728:Thermal expansion
722:
721:
688:{\displaystyle V}
641:{\displaystyle 1}
587:
586:
553:{\displaystyle N}
506:{\displaystyle T}
434:
433:
350:Process functions
336:Property diagrams
315:System properties
305:
304:
270:Endoreversibility
162:Equation of state
4162:
4103:
4102:
4094:
4088:
4087:
4079:
4073:
4072:
4058:
4052:
4051:
4045:
4037:
4019:
3965:
3963:
3962:
3957:
3938:
3936:
3935:
3930:
3922:
3921:
3909:
3908:
3896:
3890:
3889:
3874:
3839:
3835:
3833:
3832:
3827:
3812:
3810:
3809:
3804:
3796:
3795:
3784:
3775:
3765:
3760:
3748:
3747:
3736:
3727:
3714:
3709:
3693:
3691:
3690:
3685:
3677:
3676:
3665:
3656:
3655:
3641:
3636:
3620:
3616:
3614:
3613:
3608:
3600:
3599:
3588:
3571:
3566:
3550:
3546:
3542:
3538:
3534:
3530:
3526:
3524:
3523:
3518:
3510:
3495:
3459:
3444:
3438:
3432:
3430:
3429:
3424:
3422:
3417:
3416:
3415:
3412:
3402:
3394:
3353:
3346:
3342:
3338:
3318:
3298:
3289:
3280:
3271:
3256:
3253:Internal energy
3249:
3247:
3246:
3241:
3226:
3224:
3223:
3218:
3203:
3201:
3200:
3195:
3168:
3166:
3165:
3160:
3129:
3127:
3126:
3121:
3106:
3104:
3103:
3098:
3083:
3081:
3080:
3075:
3063:
3061:
3060:
3055:
3032:
3008:
3006:
3005:
3000:
2983:
2977:
2972:
2951:
2939:
2934:
2919:
2913:
2908:
2896:
2888:
2883:
2878:
2877:
2865:
2864:
2859:
2841:
2839:
2838:
2833:
2822:
2817:
2816:
2791:
2789:
2788:
2783:
2771:
2769:
2768:
2763:
2758:
2746:
2744:
2743:
2738:
2726:
2724:
2723:
2718:
2706:
2704:
2703:
2698:
2668:
2666:
2665:
2660:
2658:
2653:
2645:
2612:
2610:
2609:
2604:
2602:
2594:
2589:
2578:differential is
2573:
2571:
2570:
2565:
2550:
2548:
2547:
2542:
2537:
2526:
2515:differential is
2510:
2508:
2507:
2502:
2491:to be along the
2490:
2488:
2487:
2482:
2480:
2475:
2467:
2455:
2453:
2452:
2447:
2435:
2433:
2432:
2427:
2415:
2413:
2412:
2407:
2389:
2387:
2386:
2381:
2369:
2367:
2366:
2361:
2346:
2344:
2343:
2338:
2336:
2331:
2323:
2314:
2312:
2311:
2306:
2294:
2292:
2291:
2286:
2261:
2259:
2258:
2253:
2209:
2205:
2201:
2189:
2183:
2155:
2153:
2152:
2147:
2133:
2121:
2119:
2118:
2113:
2098:which is not an
2093:
2091:
2090:
2085:
2067:
2065:
2064:
2059:
2047:
2045:
2044:
2039:
2037:
2036:
2035:
2034:
2024:
2005:
2003:
2002:
1997:
1995:
1994:
1993:
1992:
1982:
1963:
1961:
1960:
1955:
1947:
1946:
1945:
1944:
1921:
1920:
1919:
1918:
1897:
1895:
1894:
1889:
1878:
1877:
1856:
1855:
1839:
1837:
1836:
1831:
1820:
1819:
1798:
1797:
1781:
1779:
1778:
1773:
1771:
1745:
1744:
1732:
1731:
1715:
1713:
1712:
1707:
1630:
1623:
1616:
1600:
1599:
1307:Key publications
1288:
1287:("living force")
1237:Brownian ratchet
1232:Entropy and life
1227:Entropy and time
1178:
1153:
1151:
1150:
1145:
1098:
1096:
1095:
1090:
1043:
1041:
1040:
1035:
988:
986:
985:
980:
882:Clausius theorem
877:Carnot's theorem
850:
848:
847:
842:
826:
824:
823:
818:
803:
801:
800:
795:
779:
777:
776:
771:
758:
757:
754:
752:
751:
746:
718:
716:
715:
710:
694:
692:
691:
686:
671:
669:
668:
663:
647:
645:
644:
639:
626:
625:
622:
620:
619:
614:
583:
581:
580:
575:
559:
557:
556:
551:
536:
534:
533:
528:
512:
510:
509:
504:
491:
490:
487:
485:
484:
479:
457:
456:
330:
149:
30:
16:
4170:
4169:
4165:
4164:
4163:
4161:
4160:
4159:
4140:
4139:
4112:
4107:
4106:
4096:
4095:
4091:
4081:
4080:
4076:
4060:
4059:
4055:
4038:
4034:
4021:
4020:
4016:
4011:
3972:
3941:
3940:
3913:
3881:
3842:
3841:
3837:
3815:
3814:
3779:
3731:
3696:
3695:
3660:
3623:
3622:
3618:
3583:
3553:
3552:
3548:
3544:
3540:
3536:
3532:
3528:
3470:
3469:
3466:
3457:
3455:
3440:
3434:
3407:
3403:
3385:
3384:
3373:
3360:
3348:
3344:
3340:
3337:
3330:
3320:
3317:
3310:
3300:
3297:
3291:
3288:
3282:
3279:
3273:
3270:
3264:
3254:
3229:
3228:
3206:
3205:
3171:
3170:
3136:
3135:
3109:
3108:
3086:
3085:
3084:is the energy,
3066:
3065:
3023:
3022:
3015:
2869:
2854:
2843:
2842:
2808:
2794:
2793:
2774:
2773:
2749:
2748:
2729:
2728:
2709:
2708:
2671:
2670:
2646:
2615:
2614:
2580:
2579:
2556:
2555:
2517:
2516:
2493:
2492:
2468:
2461:
2460:
2438:
2437:
2418:
2417:
2392:
2391:
2372:
2371:
2349:
2348:
2324:
2317:
2316:
2297:
2296:
2277:
2276:
2273:
2268:
2244:
2243:
2236:
2216:
2207:
2203:
2199:
2185:
2181:
2178:
2173:
2124:
2123:
2104:
2103:
2073:
2072:
2050:
2049:
2026:
2019:
2008:
2007:
1984:
1977:
1966:
1965:
1936:
1931:
1910:
1905:
1900:
1899:
1869:
1847:
1842:
1841:
1811:
1789:
1784:
1783:
1736:
1723:
1718:
1717:
1695:
1694:
1691:
1683:state functions
1634:
1589:
1588:
1564:
1556:
1555:
1554:
1414:
1406:
1405:
1384:
1370:
1345:
1341:
1334:
1330:
1323:
1319:
1286:
1279:
1261:
1242:Maxwell's demon
1204:
1175:
1174:
1158:
1157:
1156:
1106:
1105:
1104:
1051:
1050:
1049:
996:
995:
994:
956:
955:
954:
952:Internal energy
947:
932:
922:
921:
896:
871:
861:
860:
859:
830:
829:
809:
808:
783:
782:
762:
761:
734:
733:
698:
697:
677:
676:
651:
650:
630:
629:
599:
598:
593:Compressibility
563:
562:
542:
541:
516:
515:
495:
494:
467:
466:
446:
436:
435:
416:Particle number
369:
328:
317:
307:
306:
265:Irreversibility
177:State of matter
144:Isolated system
129:
119:
118:
117:
92:
82:
81:
77:Non-equilibrium
69:
44:
36:
12:
11:
5:
4168:
4166:
4158:
4157:
4152:
4150:Thermodynamics
4142:
4141:
4138:
4137:
4131:
4125:
4119:
4111:
4110:External links
4108:
4105:
4104:
4101:. McGraw Hill.
4089:
4086:. McGraw Hill.
4074:
4053:
4032:
4013:
4012:
4010:
4007:
4006:
4005:
4000:
3994:
3989:
3984:
3979:
3971:
3968:
3955:
3952:
3948:
3928:
3925:
3920:
3916:
3912:
3907:
3902:
3899:
3895:
3888:
3884:
3880:
3877:
3873:
3868:
3865:
3862:
3859:
3856:
3853:
3849:
3825:
3822:
3813:. We can make
3802:
3799:
3794:
3791:
3788:
3783:
3778:
3774:
3769:
3764:
3759:
3755:
3751:
3746:
3743:
3740:
3735:
3730:
3726:
3721:
3718:
3713:
3708:
3704:
3683:
3680:
3675:
3672:
3669:
3664:
3659:
3654:
3648:
3645:
3640:
3635:
3631:
3606:
3603:
3598:
3595:
3592:
3587:
3582:
3579:
3575:
3570:
3565:
3561:
3543:and then over
3516:
3513:
3509:
3504:
3501:
3498:
3494:
3489:
3486:
3483:
3480:
3477:
3465:
3462:
3453:
3433:In this case,
3420:
3410:
3406:
3400:
3397:
3393:
3372:
3369:
3359:
3356:
3335:
3328:
3315:
3308:
3295:
3286:
3277:
3268:
3260:state function
3239:
3236:
3216:
3213:
3193:
3190:
3187:
3184:
3181:
3178:
3158:
3155:
3152:
3149:
3146:
3143:
3119:
3116:
3096:
3093:
3073:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3031:
3014:
3011:
2998:
2995:
2992:
2989:
2986:
2982:
2976:
2971:
2967:
2963:
2960:
2957:
2954:
2950:
2946:
2943:
2938:
2933:
2929:
2925:
2922:
2918:
2912:
2907:
2903:
2899:
2895:
2891:
2887:
2882:
2876:
2872:
2868:
2863:
2858:
2853:
2850:
2831:
2828:
2825:
2821:
2815:
2811:
2807:
2804:
2801:
2781:
2761:
2757:
2736:
2716:
2696:
2693:
2690:
2687:
2684:
2681:
2678:
2656:
2652:
2649:
2643:
2640:
2637:
2634:
2631:
2628:
2625:
2622:
2601:
2597:
2593:
2588:
2576:total distance
2563:
2553:antiderivative
2540:
2536:
2532:
2529:
2525:
2500:
2478:
2474:
2471:
2445:
2425:
2405:
2402:
2399:
2379:
2359:
2356:
2334:
2330:
2327:
2304:
2284:
2272:
2271:Total distance
2269:
2267:
2264:
2251:
2235:
2232:
2215:
2212:
2194:mathematician
2177:
2176:Thermodynamics
2174:
2172:
2169:
2145:
2142:
2139:
2136:
2132:
2111:
2083:
2080:
2057:
2033:
2029:
2023:
2018:
2015:
1991:
1987:
1981:
1976:
1973:
1953:
1950:
1943:
1939:
1934:
1930:
1927:
1924:
1917:
1913:
1908:
1887:
1884:
1881:
1876:
1872:
1868:
1865:
1862:
1859:
1854:
1850:
1829:
1826:
1823:
1818:
1814:
1810:
1807:
1804:
1801:
1796:
1792:
1770:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1743:
1739:
1735:
1730:
1726:
1705:
1702:
1690:
1687:
1679:path functions
1654:thermodynamics
1636:
1635:
1633:
1632:
1625:
1618:
1610:
1607:
1606:
1605:
1604:
1591:
1590:
1587:
1586:
1581:
1576:
1571:
1565:
1562:
1561:
1558:
1557:
1553:
1552:
1547:
1542:
1537:
1532:
1527:
1522:
1517:
1512:
1507:
1502:
1497:
1492:
1487:
1482:
1477:
1472:
1467:
1462:
1457:
1452:
1447:
1442:
1437:
1432:
1427:
1422:
1416:
1415:
1412:
1411:
1408:
1407:
1402:
1401:
1400:
1399:
1394:
1386:
1385:
1383:
1382:
1379:
1375:
1372:
1371:
1369:
1368:
1363:
1361:Thermodynamics
1357:
1354:
1353:
1349:
1348:
1347:
1346:
1337:
1335:
1326:
1324:
1315:
1310:
1309:
1303:
1302:
1301:
1300:
1295:
1290:
1278:
1277:
1275:Caloric theory
1271:
1268:
1267:
1263:
1262:
1260:
1259:
1254:
1249:
1244:
1239:
1234:
1229:
1223:
1220:
1219:
1213:
1212:
1211:
1210:
1203:
1202:
1197:
1192:
1186:
1183:
1182:
1176:
1173:
1172:
1169:
1165:
1164:
1163:
1160:
1159:
1155:
1154:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1113:
1099:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1044:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
989:
978:
975:
972:
969:
966:
963:
948:
946:
945:
940:
934:
933:
928:
927:
924:
923:
920:
919:
912:
907:
902:
895:
894:
889:
884:
879:
873:
872:
867:
866:
863:
862:
856:
855:
852:
851:
840:
837:
827:
816:
805:
804:
793:
790:
780:
769:
755:
744:
741:
731:
724:
723:
720:
719:
708:
705:
695:
684:
673:
672:
661:
658:
648:
637:
623:
612:
609:
606:
596:
589:
588:
585:
584:
573:
570:
560:
549:
538:
537:
526:
523:
513:
502:
488:
477:
474:
464:
455:
454:
453:
447:
442:
441:
438:
437:
432:
431:
430:
429:
424:
419:
408:
397:
378:
377:
371:
370:
368:
367:
362:
356:
353:
352:
346:
345:
344:
343:
338:
319:
318:
313:
312:
309:
308:
303:
302:
301:
300:
295:
290:
282:
281:
275:
274:
273:
272:
267:
262:
257:
255:Free expansion
252:
247:
242:
237:
232:
227:
222:
217:
209:
208:
202:
201:
200:
199:
194:
192:Control volume
189:
184:
182:Phase (matter)
179:
174:
169:
164:
156:
155:
147:
146:
141:
136:
130:
125:
124:
121:
120:
116:
115:
110:
105:
100:
94:
93:
88:
87:
84:
83:
80:
79:
68:
67:
62:
57:
52:
46:
45:
42:
41:
38:
37:
32:The classical
31:
23:
22:
20:Thermodynamics
13:
10:
9:
6:
4:
3:
2:
4167:
4156:
4153:
4151:
4148:
4147:
4145:
4135:
4132:
4129:
4126:
4123:
4120:
4117:
4114:
4113:
4109:
4100:
4093:
4090:
4085:
4078:
4075:
4070:
4066:
4065:
4057:
4054:
4049:
4043:
4035:
4033:0-19-855919-4
4029:
4025:
4018:
4015:
4008:
4004:
4001:
3998:
3995:
3993:
3990:
3988:
3985:
3983:
3980:
3977:
3974:
3973:
3969:
3967:
3953:
3950:
3946:
3923:
3918:
3914:
3900:
3897:
3886:
3882:
3878:
3875:
3866:
3863:
3860:
3857:
3854:
3851:
3847:
3823:
3820:
3800:
3797:
3792:
3789:
3786:
3776:
3767:
3762:
3757:
3753:
3749:
3744:
3741:
3738:
3728:
3719:
3716:
3711:
3706:
3702:
3681:
3678:
3673:
3670:
3667:
3657:
3646:
3643:
3638:
3633:
3629:
3604:
3601:
3596:
3593:
3590:
3580:
3577:
3573:
3568:
3563:
3559:
3514:
3511:
3502:
3499:
3496:
3487:
3484:
3481:
3478:
3475:
3463:
3461:
3451:
3448:
3443:
3437:
3418:
3408:
3404:
3398:
3395:
3382:
3378:
3370:
3368:
3365:
3358:Heat and work
3357:
3355:
3352:
3334:
3327:
3323:
3314:
3307:
3303:
3294:
3285:
3276:
3267:
3262:
3261:
3251:
3237:
3234:
3214:
3211:
3191:
3188:
3185:
3182:
3179:
3176:
3156:
3153:
3150:
3147:
3144:
3141:
3133:
3117:
3114:
3094:
3091:
3071:
3051:
3048:
3045:
3042:
3039:
3036:
3033:
3020:
3012:
3010:
2996:
2993:
2990:
2987:
2984:
2974:
2969:
2965:
2961:
2958:
2952:
2944:
2936:
2931:
2927:
2923:
2920:
2910:
2905:
2901:
2897:
2889:
2874:
2870:
2866:
2861:
2851:
2829:
2826:
2823:
2813:
2809:
2805:
2802:
2779:
2759:
2734:
2714:
2691:
2688:
2685:
2679:
2676:
2650:
2647:
2635:
2632:
2629:
2623:
2620:
2595:
2577:
2561:
2554:
2538:
2530:
2527:
2514:
2498:
2472:
2469:
2457:
2443:
2423:
2403:
2400:
2397:
2377:
2357:
2354:
2328:
2325:
2315:along a line
2302:
2282:
2270:
2265:
2263:
2249:
2241:
2233:
2231:
2229:
2225:
2221:
2213:
2211:
2197:
2193:
2188:
2184:, the symbol
2175:
2170:
2168:
2166:
2161:
2156:
2143:
2140:
2137:
2134:
2109:
2101:
2097:
2081:
2078:
2069:
2055:
2031:
2027:
2016:
1989:
1985:
1974:
1951:
1948:
1941:
1937:
1932:
1928:
1925:
1922:
1915:
1911:
1906:
1882:
1874:
1870:
1866:
1860:
1852:
1848:
1824:
1816:
1812:
1808:
1802:
1794:
1790:
1758:
1755:
1752:
1746:
1741:
1737:
1733:
1728:
1724:
1703:
1700:
1688:
1686:
1684:
1680:
1676:
1672:
1667:
1663:
1660:as a type of
1659:
1655:
1651:
1647:
1643:
1631:
1626:
1624:
1619:
1617:
1612:
1611:
1609:
1608:
1603:
1595:
1594:
1593:
1592:
1585:
1582:
1580:
1577:
1575:
1574:Self-assembly
1572:
1570:
1567:
1566:
1560:
1559:
1551:
1548:
1546:
1545:van der Waals
1543:
1541:
1538:
1536:
1533:
1531:
1528:
1526:
1523:
1521:
1518:
1516:
1513:
1511:
1508:
1506:
1503:
1501:
1498:
1496:
1493:
1491:
1488:
1486:
1483:
1481:
1478:
1476:
1473:
1471:
1470:von Helmholtz
1468:
1466:
1463:
1461:
1458:
1456:
1453:
1451:
1448:
1446:
1443:
1441:
1438:
1436:
1433:
1431:
1428:
1426:
1423:
1421:
1418:
1417:
1410:
1409:
1398:
1395:
1393:
1390:
1389:
1388:
1387:
1380:
1377:
1376:
1373:
1367:
1364:
1362:
1359:
1358:
1356:
1355:
1350:
1344:
1343:
1336:
1333:
1332:
1325:
1322:
1321:
1314:
1313:
1312:
1311:
1308:
1304:
1299:
1296:
1294:
1291:
1289:
1285:
1281:
1280:
1276:
1273:
1272:
1270:
1269:
1264:
1258:
1255:
1253:
1250:
1248:
1245:
1243:
1240:
1238:
1235:
1233:
1230:
1228:
1225:
1224:
1222:
1221:
1218:
1214:
1209:
1206:
1205:
1201:
1198:
1196:
1193:
1191:
1188:
1187:
1185:
1184:
1179:
1170:
1167:
1166:
1162:
1161:
1141:
1138:
1135:
1132:
1129:
1123:
1120:
1117:
1111:
1103:
1100:
1086:
1083:
1080:
1077:
1074:
1068:
1065:
1062:
1056:
1048:
1045:
1031:
1028:
1025:
1022:
1019:
1013:
1010:
1007:
1001:
993:
990:
973:
970:
967:
961:
953:
950:
949:
944:
941:
939:
936:
935:
931:
926:
925:
918:
917:
913:
911:
908:
906:
903:
901:
898:
897:
893:
892:Ideal gas law
890:
888:
885:
883:
880:
878:
875:
874:
870:
865:
864:
838:
828:
814:
807:
806:
791:
781:
767:
760:
759:
756:
742:
739:
732:
729:
726:
725:
706:
696:
682:
675:
674:
659:
649:
635:
628:
627:
624:
610:
607:
604:
597:
594:
591:
590:
571:
561:
547:
540:
539:
524:
514:
500:
493:
492:
489:
475:
472:
465:
462:
459:
458:
452:
449:
448:
445:
440:
439:
428:
425:
423:
422:Vapor quality
420:
418:
417:
412:
409:
407:
406:
401:
398:
395:
391:
390:
385:
382:
381:
380:
379:
376:
372:
366:
363:
361:
358:
357:
355:
354:
351:
347:
342:
339:
337:
334:
333:
332:
331:
327:
323:
316:
311:
310:
299:
296:
294:
291:
289:
286:
285:
284:
283:
280:
276:
271:
268:
266:
263:
261:
260:Reversibility
258:
256:
253:
251:
248:
246:
243:
241:
238:
236:
233:
231:
228:
226:
223:
221:
218:
216:
213:
212:
211:
210:
207:
203:
198:
195:
193:
190:
188:
185:
183:
180:
178:
175:
173:
170:
168:
165:
163:
160:
159:
158:
157:
154:
150:
145:
142:
140:
137:
135:
134:Closed system
132:
131:
128:
123:
122:
114:
111:
109:
106:
104:
101:
99:
96:
95:
91:
86:
85:
78:
74:
71:
70:
66:
63:
61:
58:
56:
53:
51:
48:
47:
40:
39:
35:
29:
25:
24:
21:
17:
4098:
4092:
4083:
4077:
4068:
4063:
4056:
4023:
4017:
3621:contributes
3467:
3449:
3441:
3435:
3374:
3361:
3350:
3332:
3325:
3321:
3312:
3305:
3301:
3292:
3283:
3274:
3265:
3258:
3252:
3132:Carnot cycle
3016:
2575:
2513:net distance
2512:
2458:
2274:
2237:
2217:
2186:
2179:
2157:
2070:
1692:
1650:differential
1645:
1641:
1639:
1435:Carathéodory
1366:Heat engines
1338:
1327:
1316:
1298:Motive power
1283:
943:Free entropy
914:
414:
413: /
403:
402: /
394:introduction
387:
386: /
325:
288:Heat engines
75: /
3840:, yielding
3447:temperature
2511:-axis, the
2295:to a point
2234:Mathematics
2228:T1 encoding
2202:(heat) and
1658:mathematics
1257:Synergetics
938:Free energy
384:Temperature
245:Quasistatic
240:Isenthalpic
197:Instruments
187:Equilibrium
139:Open system
73:Equilibrium
55:Statistical
4144:Categories
4009:References
2707:such that
1782:such that
1689:Definition
1569:Nucleation
1413:Scientists
1217:Philosophy
930:Potentials
293:Heat pumps
250:Polytropic
235:Isentropic
225:Isothermal
4042:cite book
3951:δ
3939:. And so
3852:δ
3821:δ
3754:∫
3703:∫
3630:∫
3560:∫
3476:δ
3405:δ
3290:to state
3235:δ
3212:δ
3177:δ
3142:δ
3115:δ
3092:δ
3049:δ
3046:−
3040:δ
2966:∫
2945:−
2928:∫
2902:∫
2875:γ
2871:∫
2862:γ
2849:Δ
2814:γ
2810:∫
2800:Δ
2715:γ
2680:∈
2655:¯
2642:→
2621:γ
2477:¯
2333:¯
2250:ω
2141:δ
2138:≠
2079:δ
2028:γ
2014:Δ
1986:γ
1972:Δ
1949:δ
1938:γ
1933:∫
1923:δ
1912:γ
1907:∫
1871:γ
1849:γ
1813:γ
1791:γ
1765:→
1738:γ
1725:γ
1701:δ
1550:Waterston
1500:von Mayer
1455:de Donder
1445:Clapeyron
1425:Boltzmann
1420:Bernoulli
1381:Education
1352:Timelines
1136:−
1081:−
869:Equations
836:∂
789:∂
740:α
704:∂
657:∂
611:−
605:β
569:∂
522:∂
230:Adiabatic
220:Isochoric
206:Processes
167:Ideal gas
50:Classical
3970:See also
3319:or work
2266:Examples
2171:Notation
1929:≠
1602:Category
1540:Thompson
1450:Clausius
1430:Bridgman
1284:Vis viva
1266:Theories
1200:Gas laws
992:Enthalpy
400:Pressure
215:Isobaric
172:Real gas
60:Chemical
43:Branches
3464:Example
3381:entropy
1525:Smeaton
1520:Rankine
1510:Onsager
1495:Maxwell
1490:Massieu
1195:Entropy
1190:General
1181:History
1171:Culture
1168:History
392: (
389:Entropy
326:italics
127:Systems
4030:
3064:where
2192:German
2187:δ
1681:, not
1515:Planck
1505:Nernst
1480:Kelvin
1440:Carnot
730:
595:
463:
405:Volume
320:Note:
279:Cycles
108:Second
98:Zeroth
4067:[
3529:(1,1)
3257:is a
2094:is a
1648:is a
1563:Other
1530:Stahl
1485:Lewis
1475:Joule
1465:Gibbs
1460:Duhem
153:State
113:Third
103:First
4048:link
4028:ISBN
3272:and
3227:and
2224:dyet
2158:The
2006:and
1898:and
1675:work
1673:and
1671:heat
1535:Tait
365:Heat
360:Work
90:Laws
3454:rev
3450:and
3413:rev
2436:to
1644:or
1640:An
1378:Art
324:in
4146::
4044:}}
4040:{{
3442:δW
3436:δQ
3383::
3354:.
3331:−
3324:=
3311:−
3304:=
3021:,
2262:.
2230:.
2122:,
2068:.
1840:,
1685:.
4050:)
4036:.
3954:u
3947:x
3927:)
3924:y
3919:2
3915:x
3911:(
3906:d
3901:=
3898:y
3894:d
3887:2
3883:x
3879:+
3876:x
3872:d
3867:y
3864:x
3861:2
3858:=
3855:u
3848:x
3838:x
3824:u
3801:1
3798:=
3793:1
3790:=
3787:x
3782:|
3777:y
3773:d
3768:x
3763:1
3758:0
3750:+
3745:0
3742:=
3739:y
3734:|
3729:x
3725:d
3720:y
3717:2
3712:1
3707:0
3682:2
3679:=
3674:1
3671:=
3668:y
3663:|
3658:x
3653:d
3647:y
3644:2
3639:1
3634:0
3619:x
3605:0
3602:=
3597:0
3594:=
3591:x
3586:|
3581:y
3578:d
3574:x
3569:1
3564:0
3549:y
3545:x
3541:y
3537:x
3533:y
3515:.
3512:y
3508:d
3503:x
3500:+
3497:x
3493:d
3488:y
3485:2
3482:=
3479:u
3458:S
3419:T
3409:Q
3399:=
3396:S
3392:d
3351:U
3349:Δ
3345:Q
3341:W
3336:1
3333:U
3329:2
3326:U
3322:W
3316:1
3313:U
3309:2
3306:U
3302:Q
3296:2
3293:U
3287:1
3284:U
3278:2
3275:U
3269:1
3266:U
3255:U
3238:W
3215:Q
3192:V
3189:d
3186:P
3183:=
3180:W
3157:S
3154:d
3151:T
3148:=
3145:Q
3118:W
3095:Q
3072:U
3052:W
3043:Q
3037:=
3034:U
3030:d
2997:B
2994:A
2991:2
2988:=
2985:x
2981:d
2975:B
2970:A
2962:2
2959:=
2956:)
2953:x
2949:d
2942:(
2937:A
2932:B
2924:+
2921:x
2917:d
2911:B
2906:A
2898:=
2894:|
2890:x
2886:d
2881:|
2867:=
2857:|
2852:g
2830:0
2827:=
2824:x
2820:d
2806:=
2803:f
2780:t
2760:x
2756:d
2735:t
2695:)
2692:1
2689:,
2686:0
2683:(
2677:t
2651:B
2648:A
2639:]
2636:1
2633:,
2630:0
2627:[
2624::
2600:|
2596:x
2592:d
2587:|
2562:x
2539:x
2535:d
2531:=
2528:f
2524:d
2499:x
2473:B
2470:A
2444:B
2424:A
2404:B
2401:A
2398:2
2378:A
2358:B
2355:A
2329:B
2326:A
2303:B
2283:A
2220:đ
2208:U
2204:W
2200:Q
2182:d
2144:u
2135:f
2131:d
2110:f
2082:u
2056:u
2032:2
2022:|
2017:u
1990:1
1980:|
1975:u
1952:u
1942:2
1926:u
1916:1
1886:)
1883:1
1880:(
1875:2
1867:=
1864:)
1861:1
1858:(
1853:1
1828:)
1825:0
1822:(
1817:2
1809:=
1806:)
1803:0
1800:(
1795:1
1769:R
1762:]
1759:1
1756:,
1753:0
1750:[
1747::
1742:2
1734:,
1729:1
1704:u
1629:e
1622:t
1615:v
1142:S
1139:T
1133:H
1130:=
1127:)
1124:p
1121:,
1118:T
1115:(
1112:G
1087:S
1084:T
1078:U
1075:=
1072:)
1069:V
1066:,
1063:T
1060:(
1057:A
1032:V
1029:p
1026:+
1023:U
1020:=
1017:)
1014:p
1011:,
1008:S
1005:(
1002:H
977:)
974:V
971:,
968:S
965:(
962:U
839:T
815:V
792:V
768:1
743:=
707:p
683:V
660:V
636:1
608:=
572:T
548:N
525:S
501:T
476:=
473:c
396:)
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