1918:. The chemical potential may be thought of as a force which, when imbalanced, pushes an exchange of particles, either with the surroundings, or between phases inside the system. In cases where there are a mixture of chemicals and phases, this is a useful concept. For example, if a container holds liquid water and water vapor, there will be a chemical potential (which is negative) for the liquid which pushes the water molecules into the vapor (evaporation) and a chemical potential for the vapor, pushing vapor molecules into the liquid (condensation). Only when these "forces" equilibrate, and the chemical potential of each phase is equal, is equilibrium obtained.
32:
102:
1672:
3155:. In cases where there are a mixture of chemicals and phases, this is a useful concept. For example, if a container holds water and water vapor, there will be a chemical potential (which is negative) for the liquid, pushing water molecules into the vapor (evaporation) and a chemical potential for the vapor, pushing vapor molecules into the liquid (condensation). Only when these "forces" equilibrate is equilibrium obtained.
1902:, and their product is the energy lost by the system due to work. Here, pressure is the driving force, volume is the associated displacement, and the two form a pair of conjugate variables. In a similar way, temperature differences drive changes in entropy, and their product is the energy transferred by heat transfer. The thermodynamic force is always an
1913:
The theory of thermodynamic potentials is not complete until one considers the number of particles in a system as a variable on par with the other extensive quantities such as volume and entropy. The number of particles is, like volume and entropy, the displacement variable in a conjugate pair. The
1829:
Just as a small increment of energy in a mechanical system is the product of a force times a small displacement, so an increment in the energy of a thermodynamic system can be expressed as the sum of the products of certain generalized "forces" which, when unbalanced, cause certain generalized
1811:
In the above description, the product of two conjugate variables yields an energy. In other words, the conjugate pairs are conjugate with respect to energy. In general, conjugate pairs can be defined with respect to any thermodynamic state function. Conjugate pairs with respect to
3032:
2331:
1773:, a small increment of energy is the product of a force times a small displacement. A similar situation exists in thermodynamics. An increment in the energy of a thermodynamic system can be expressed as the sum of the products of certain
2836:
1910:, yielding an extensive energy. The intensive (force) variable is the derivative of the (extensive) internal energy with respect to the extensive (displacement) variable, with all other extensive variables held constant.
2563:
in the internal energy of the system is given by the sum of the flow of energy across the boundaries of the system due to the corresponding conjugate pair. These concepts will be expanded upon in the following sections.
2907:
1816:
are often used, in which the product of the conjugate pairs yields an entropy. Such conjugate pairs are particularly useful in the analysis of irreversible processes, as exemplified in the derivation of the
2533:
Here, the temperature, pressure, and chemical potential are the generalized forces, which drive the generalized changes in entropy, volume, and particle number respectively. These parameters all affect the
1391:
2922:
3131:. Temperature is the driving force, entropy is the associated displacement, and the two form a pair of conjugate variables. The temperature/entropy pair of conjugate variables is the only
2759:
2145:
3109:
3069:
2723:
2238:
2098:
2561:
1900:
2689:
1226:
1171:
1116:
923:
876:
791:
744:
656:
609:
2461:
827:
695:
1061:
2508:
2178:
2624:
1855:
560:
2528:
2481:
2434:
2414:
2394:
2374:
2354:
2230:
2204:
2046:
2016:
1983:
1953:
1875:
899:
852:
767:
720:
632:
585:
2767:
1701:
1830:"displacements" to occur, with their product being the energy transferred as a result. These forces and their associated displacements are called
1402:
1290:
2851:
524:
1797:
of the internal energy with respect to the extensive (displacement) variable, while all other extensive variables are held constant.
1380:
2587:, namely idealized, "infinitely slow" processes. Time-dependent thermodynamic processes far away from equilibrium are studied by
1413:
1781:, and the product of the two is the energy transferred as a result. These forces and their associated displacements are called
2912:
In the case of pure compression (i.e. no shearing forces), the stress tensor is simply the negative of the pressure times the
3322:
983:
2567:
While dealing with processes in which systems exchange matter or energy, classical thermodynamics is not concerned with the
1694:
1281:
950:
517:
333:
2638:. Pressure is the driving force, volume is the associated displacement, and the two form a pair of conjugate variables.
3174:
3027:{\displaystyle \delta w=V\,(-p\delta _{ij})\,\mathrm {d} \varepsilon _{ij}=-\sum _{k}pV\,\mathrm {d} \varepsilon _{kk}}
1465:
1439:
960:
414:
2729:
component of the strain tensor, then the mechanical work done as the result of a stress-induced infinitesimal strain
3341:
3303:
3270:
366:
75:
53:
46:
1518:
989:
388:
2732:
1687:
2588:
2114:
1618:
150:
3074:
1513:
1593:
1366:
343:
1818:
978:
181:
171:
186:
176:
3044:
2698:
2326:{\displaystyle \mathrm {d} U=T\,\mathrm {d} S-p\,\mathrm {d} V+\sum _{i}\mu _{i}\,\mathrm {d} N_{i}\,,}
1470:
1434:
212:
146:
1508:
1263:
1011:
457:
270:
260:
2072:
2541:
1880:
40:
2664:
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2025:
1805:
1778:
1755:
1675:
1503:
1300:
1181:
1126:
1071:
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942:
478:
467:
133:
3253:. Translated by J.B. Sykes; W.H. Reid. With A. M. Kosevich and L. P. Pitaevskii (3rd ed.).
1758:
are expressed in terms of conjugate pairs. The product of two quantities that are conjugate has
905:
858:
773:
726:
638:
591:
3254:
3200:
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3038:
1608:
1325:
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163:
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57:
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809:
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433:
279:
128:
1031:
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200:
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8:
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226:
20:
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1837:
1563:
542:
3223:
3148:
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2399:
2379:
2359:
2339:
2215:
2189:
2157:
2031:
2001:
1968:
1938:
1921:
The most commonly considered conjugate thermodynamic variables are (with corresponding
1915:
1907:
1903:
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107:
101:
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1652:
1613:
1603:
1175:
973:
801:
303:
293:
235:
3227:
3215:
2595:, allowing systems near and far away from equilibrium to be studied, respectively.
1573:
1558:
1498:
1493:
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1305:
955:
423:
288:
3246:
3152:
2913:
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2183:
1763:
1759:
1751:
1719:
1523:
1371:
1025:
666:
489:
250:
217:
2831:{\displaystyle \delta w=V\sum _{ij}\sigma _{ij}\,\mathrm {d} \varepsilon _{ij}}
1793:, yielding an extensive energy transfer. The intensive (force) variable is the
1715:
1578:
1348:
448:
328:
265:
255:
123:
93:
3335:
3295:
2658:
2108:
2101:
2019:
1647:
965:
534:
495:
207:
3219:
1877:
acts as a generalized force: Pressure differences force a change in volume
1813:
1598:
1583:
1533:
1016:
3120:
1932:
1731:
1553:
361:
2657:, and changes in volume are generalized to the volume multiplied by the
3242:
3179:
2568:
1794:
1642:
1588:
2902:{\displaystyle \delta w=V\sigma _{ij}\,\mathrm {d} \varepsilon _{ij}}
2845:
for the tensors, in which repeated indices are assumed to be summed:
2642:
2630:
acts as a generalized force – pressure differences force a change in
240:
16:
Pair of values which express a thermodynamic system's internal energy
3289:
2627:
1995:
1739:
1356:
1273:
1065:
473:
245:
3171:: analogous conjugate variable pairs found in classical mechanics.
3071:) is the fractional change in volume so that the above reduces to
3124:
2641:
The above holds true only for non-viscous fluids. In the case of
2232:
of particles, a small change in the internal energy is given by:
1962:
1735:
462:
3258:
1956:
1743:
3294:. Revised by Kenneth S. Pitzer and Leo Brewer (2nd ed.).
3251:
Theory of
Elasticity (Course of Theoretical Physics Volume 7)
2634:, and their product is the energy lost by the system due to
2591:. This can be done through linear or non-linear analysis of
3135:
term; the other terms are essentially all various forms of
3128:
438:
3142:
2598:
3201:"Use of Legendre transforms in chemical thermodynamics"
1922:
1804:
can be used as a tool to recall and derive some of the
3315:
3077:
3047:
2925:
2854:
2770:
2735:
2701:
2667:
2609:
2544:
2516:
2489:
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2442:
2422:
2402:
2382:
2362:
2342:
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2218:
2192:
2166:
2117:
2075:
2034:
2004:
1971:
1941:
1883:
1863:
1840:
1184:
1129:
1074:
1034:
908:
887:
861:
840:
812:
776:
755:
729:
708:
677:
641:
620:
594:
573:
545:
3312:
3103:
3063:
3026:
2901:
2830:
2753:
2717:
2683:
2618:
2583:. A central notion for this connection is that of
2555:
2522:
2502:
2475:
2455:
2428:
2408:
2388:
2368:
2348:
2325:
2224:
2198:
2172:
2139:
2092:
2040:
2010:
1977:
1947:
1894:
1869:
1849:
1220:
1165:
1110:
1055:
917:
893:
870:
846:
821:
785:
761:
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714:
689:
650:
626:
603:
579:
554:
3317:(2nd ed.). New York: John Wiley & Sons.
3235:
3127:, and their product is the energy transferred by
2653:solids, the pressure force is generalized to the
3333:
1914:generalized force component of this pair is the
19:For a more general mathematical discussion, see
3192:
3290:Lewis, Gilbert Newton; Randall, Merle (1961).
3241:
1695:
3151:is like a force which pushes an increase in
2754:{\displaystyle \mathrm {\varepsilon } _{ij}}
3114:
2571:at which such processes take place, termed
2538:of a thermodynamic system. A small change
1702:
1688:
100:
3005:
2964:
2938:
2880:
2809:
2319:
2303:
2271:
2256:
2140:{\displaystyle V\times \varepsilon _{ij}}
2089:
76:Learn how and when to remove this message
3104:{\displaystyle \delta w=-p\mathrm {d} V}
2661:. These then form a conjugate pair. If
39:This article includes a list of general
3198:
3143:Chemical potential/particle number pair
2599:Pressure/volume and stress/strain pairs
1785:. The thermodynamic force is always an
3334:
1777:that, when unbalanced, cause certain
2695:component of the stress tensor, and
25:
13:
3283:
3175:Intensive and extensive properties
3094:
3007:
2966:
2882:
2811:
2579:is usually used synonymously with
2546:
2305:
2273:
2258:
2243:
2212:For a system with different types
1906:and the displacement is always an
1885:
1789:and the displacement is always an
1726:is expressed in terms of pairs of
909:
862:
777:
730:
642:
595:
415:Intensive and extensive properties
45:it lacks sufficient corresponding
14:
3353:
3064:{\displaystyle \varepsilon _{kk}}
2718:{\displaystyle \varepsilon _{ij}}
2463:is the chemical potential of the
1671:
1670:
990:Table of thermodynamic equations
30:
2530:-type particles in the system.
1466:Maxwell's thermodynamic surface
2961:
2939:
2589:non-equilibrium thermodynamics
2093:{\displaystyle \sigma _{ij}\,}
1808:based on conjugate variables.
1200:
1188:
1145:
1133:
1090:
1078:
1050:
1038:
1:
3185:
3123:differences drive changes in
2556:{\displaystyle \mathrm {d} U}
1895:{\displaystyle \mathrm {d} V}
1857:conjugate pair. The pressure
1367:Mechanical equivalent of heat
2684:{\displaystyle \sigma _{ij}}
2603:As an example, consider the
2575:. For this reason, the term
1834:. For example, consider the
1819:Onsager reciprocal relations
979:Onsager reciprocal relations
7:
3313:Callen, Herbert B. (1998).
3158:
1824:
1779:generalized "displacements"
1471:Entropy as energy dispersal
1282:"Perpetual motion" machines
1221:{\displaystyle G(T,p)=H-TS}
1166:{\displaystyle A(T,V)=U-TS}
1111:{\displaystyle H(S,p)=U+pV}
10:
3358:
2581:equilibrium thermodynamics
2206: (particles or mole)
918:{\displaystyle \partial T}
871:{\displaystyle \partial V}
786:{\displaystyle \partial p}
739:{\displaystyle \partial V}
651:{\displaystyle \partial T}
604:{\displaystyle \partial S}
18:
1392:An Inquiry Concerning the
3342:Thermodynamic properties
3115:Temperature/entropy pair
2456:{\displaystyle \mu _{i}}
1806:thermodynamic potentials
1756:thermodynamic potentials
1405:Heterogeneous Substances
822:{\displaystyle \alpha =}
690:{\displaystyle \beta =-}
3220:10.1351/pac200173081349
3199:Alberty, R. A. (2001).
2483:-th particle type, and
1992:Mechanical parameters:
60:more precise citations.
3165:Generalized coordinate
3105:
3065:
3041:of the strain tensor (
3028:
2903:
2832:
2755:
2719:
2685:
2620:
2593:irreversible processes
2557:
2524:
2504:
2477:
2457:
2430:
2410:
2390:
2370:
2350:
2327:
2226:
2200:
2174:
2141:
2094:
2042:
2012:
1979:
1949:
1896:
1871:
1851:
1222:
1167:
1112:
1057:
1056:{\displaystyle U(S,V)}
919:
895:
872:
848:
823:
787:
763:
740:
716:
691:
652:
628:
605:
581:
556:
535:Specific heat capacity
139:Quantum thermodynamics
3263:Butterworth-Heinemann
3106:
3066:
3029:
2904:
2833:
2756:
2720:
2686:
2621:
2585:quasistatic processes
2558:
2525:
2505:
2503:{\displaystyle N_{i}}
2478:
2458:
2431:
2411:
2391:
2371:
2351:
2328:
2227:
2201:
2175:
2154:Material parameters:
2142:
2095:
2043:
2013:
1980:
1950:
1897:
1872:
1852:
1403:On the Equilibrium of
1223:
1168:
1121:Helmholtz free energy
1113:
1058:
920:
896:
873:
849:
824:
788:
764:
741:
717:
692:
653:
629:
606:
582:
557:
3298:: McGraw-Hill Book.
3075:
3045:
2923:
2852:
2768:
2733:
2699:
2665:
2626:conjugate pair. The
2607:
2542:
2514:
2487:
2467:
2440:
2420:
2400:
2380:
2360:
2356:is internal energy,
2340:
2239:
2216:
2190:
2173:{\displaystyle \mu }
2164:
2115:
2073:
2032:
2002:
1969:
1939:
1929:Thermal parameters:
1881:
1861:
1838:
1802:thermodynamic square
1775:generalized "forces"
1416:Motive Power of Fire
1182:
1127:
1072:
1032:
984:Bridgman's equations
961:Fundamental relation
906:
885:
859:
838:
810:
774:
753:
727:
706:
675:
639:
618:
592:
571:
543:
2057:or, more generally,
1832:conjugate variables
1783:conjugate variables
1728:conjugate variables
1394:Source ... Friction
1326:Loschmidt's paradox
518:Material properties
396:Conjugate variables
21:Conjugate variables
3149:chemical potential
3119:In a similar way,
3101:
3061:
3024:
2998:
2899:
2828:
2795:
2751:
2715:
2681:
2619:{\displaystyle pV}
2616:
2553:
2520:
2500:
2473:
2453:
2426:
2406:
2386:
2366:
2346:
2323:
2292:
2222:
2196:
2170:
2158:chemical potential
2137:
2090:
2038:
2008:
1975:
1945:
1916:chemical potential
1908:extensive variable
1904:intensive variable
1892:
1867:
1850:{\displaystyle pV}
1847:
1791:extensive variable
1787:intensive variable
1748:chemical potential
1658:Order and disorder
1414:Reflections on the
1321:Heat death paradox
1218:
1163:
1108:
1053:
915:
891:
868:
844:
819:
783:
759:
736:
712:
687:
648:
624:
601:
577:
555:{\displaystyle c=}
552:
525:Property databases
501:Reduced properties
485:Chemical potential
449:Functions of state
372:Thermal efficiency
108:Carnot heat engine
3324:978-0-471-86256-7
3169:generalized force
2989:
2843:Einstein notation
2783:
2523:{\displaystyle i}
2510:is the number of
2476:{\displaystyle i}
2429:{\displaystyle V}
2409:{\displaystyle p}
2389:{\displaystyle S}
2369:{\displaystyle T}
2349:{\displaystyle U}
2283:
2225:{\displaystyle i}
2199:{\displaystyle N}
2048: (m = J Pa)
2041:{\displaystyle V}
2011:{\displaystyle p}
1978:{\displaystyle S}
1948:{\displaystyle T}
1870:{\displaystyle p}
1771:mechanical system
1712:
1711:
1653:Self-organization
1478:
1477:
1176:Gibbs free energy
974:Maxwell relations
932:
931:
928:
927:
894:{\displaystyle V}
847:{\displaystyle 1}
802:Thermal expansion
796:
795:
762:{\displaystyle V}
715:{\displaystyle 1}
661:
660:
627:{\displaystyle N}
580:{\displaystyle T}
508:
507:
424:Process functions
410:Property diagrams
389:System properties
379:
378:
344:Endoreversibility
236:Equation of state
86:
85:
78:
3349:
3328:
3309:
3277:
3276:
3239:
3233:
3231:
3214:(8): 1349–1380.
3205:
3196:
3110:
3108:
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3102:
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3068:
3067:
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3031:
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3025:
3023:
3022:
3010:
2997:
2982:
2981:
2969:
2960:
2959:
2908:
2906:
2905:
2900:
2898:
2897:
2885:
2879:
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2829:
2827:
2826:
2814:
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2807:
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2760:
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2757:
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2750:
2749:
2741:
2724:
2722:
2721:
2716:
2714:
2713:
2690:
2688:
2687:
2682:
2680:
2679:
2625:
2623:
2622:
2617:
2562:
2560:
2559:
2554:
2549:
2529:
2527:
2526:
2521:
2509:
2507:
2506:
2501:
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2498:
2482:
2480:
2479:
2474:
2462:
2460:
2459:
2454:
2452:
2451:
2435:
2433:
2432:
2427:
2415:
2413:
2412:
2407:
2395:
2393:
2392:
2387:
2376:is temperature,
2375:
2373:
2372:
2367:
2355:
2353:
2352:
2347:
2332:
2330:
2329:
2324:
2318:
2317:
2308:
2302:
2301:
2291:
2276:
2261:
2246:
2231:
2229:
2228:
2223:
2205:
2203:
2202:
2197:
2179:
2177:
2176:
2171:
2146:
2144:
2143:
2138:
2136:
2135:
2099:
2097:
2096:
2091:
2088:
2087:
2047:
2045:
2044:
2039:
2017:
2015:
2014:
2009:
1984:
1982:
1981:
1976:
1954:
1952:
1951:
1946:
1901:
1899:
1898:
1893:
1888:
1876:
1874:
1873:
1868:
1856:
1854:
1853:
1848:
1704:
1697:
1690:
1674:
1673:
1381:Key publications
1362:
1361:("living force")
1311:Brownian ratchet
1306:Entropy and life
1301:Entropy and time
1252:
1251:
1227:
1225:
1224:
1219:
1172:
1170:
1169:
1164:
1117:
1115:
1114:
1109:
1062:
1060:
1059:
1054:
956:Clausius theorem
951:Carnot's theorem
924:
922:
921:
916:
900:
898:
897:
892:
877:
875:
874:
869:
853:
851:
850:
845:
832:
831:
828:
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825:
820:
792:
790:
789:
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768:
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743:
742:
737:
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633:
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559:
558:
553:
531:
530:
404:
403:
223:
222:
104:
90:
89:
81:
74:
70:
67:
61:
56:this article by
47:inline citations
34:
33:
26:
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3351:
3350:
3348:
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3346:
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3331:
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3306:
3286:
3284:Further reading
3281:
3280:
3273:
3247:Lifshitz, E. M.
3240:
3236:
3208:Pure Appl. Chem
3203:
3197:
3193:
3188:
3161:
3153:particle number
3145:
3117:
3093:
3076:
3073:
3072:
3052:
3048:
3046:
3043:
3042:
3015:
3011:
3006:
2993:
2974:
2970:
2965:
2952:
2948:
2924:
2921:
2920:
2890:
2886:
2881:
2871:
2867:
2853:
2850:
2849:
2819:
2815:
2810:
2800:
2796:
2787:
2769:
2766:
2765:
2742:
2737:
2736:
2734:
2731:
2730:
2706:
2702:
2700:
2697:
2696:
2672:
2668:
2666:
2663:
2662:
2636:mechanical work
2608:
2605:
2604:
2601:
2545:
2543:
2540:
2539:
2536:internal energy
2515:
2512:
2511:
2494:
2490:
2488:
2485:
2484:
2468:
2465:
2464:
2447:
2443:
2441:
2438:
2437:
2421:
2418:
2417:
2401:
2398:
2397:
2381:
2378:
2377:
2361:
2358:
2357:
2341:
2338:
2337:
2313:
2309:
2304:
2297:
2293:
2287:
2272:
2257:
2242:
2240:
2237:
2236:
2217:
2214:
2213:
2191:
2188:
2187:
2184:particle number
2165:
2162:
2161:
2128:
2124:
2116:
2113:
2112:
2080:
2076:
2074:
2071:
2070:
2033:
2030:
2029:
2003:
2000:
1999:
1970:
1967:
1966:
1940:
1937:
1936:
1884:
1882:
1879:
1878:
1862:
1859:
1858:
1839:
1836:
1835:
1827:
1760:units of energy
1754:. In fact, all
1752:particle number
1720:internal energy
1708:
1663:
1662:
1638:
1630:
1629:
1628:
1488:
1480:
1479:
1458:
1444:
1419:
1415:
1408:
1404:
1397:
1393:
1360:
1353:
1335:
1316:Maxwell's demon
1278:
1249:
1248:
1232:
1231:
1230:
1183:
1180:
1179:
1178:
1128:
1125:
1124:
1123:
1073:
1070:
1069:
1068:
1033:
1030:
1029:
1028:
1026:Internal energy
1021:
1006:
996:
995:
970:
945:
935:
934:
933:
907:
904:
903:
886:
883:
882:
860:
857:
856:
839:
836:
835:
811:
808:
807:
775:
772:
771:
754:
751:
750:
728:
725:
724:
707:
704:
703:
676:
673:
672:
667:Compressibility
640:
637:
636:
619:
616:
615:
593:
590:
589:
572:
569:
568:
544:
541:
540:
520:
510:
509:
490:Particle number
443:
402:
391:
381:
380:
339:Irreversibility
251:State of matter
218:Isolated system
203:
193:
192:
191:
166:
156:
155:
151:Non-equilibrium
143:
118:
110:
82:
71:
65:
62:
52:Please help to
51:
35:
31:
24:
17:
12:
11:
5:
3355:
3345:
3344:
3330:
3329:
3323:
3310:
3304:
3292:Thermodynamics
3285:
3282:
3279:
3278:
3271:
3234:
3190:
3189:
3187:
3184:
3183:
3182:
3177:
3172:
3160:
3157:
3144:
3141:
3116:
3113:
3111:as it should.
3100:
3096:
3092:
3089:
3086:
3083:
3080:
3058:
3055:
3051:
3035:
3034:
3021:
3018:
3014:
3009:
3004:
3001:
2996:
2992:
2988:
2985:
2980:
2977:
2973:
2968:
2963:
2958:
2955:
2951:
2947:
2944:
2941:
2937:
2934:
2931:
2928:
2910:
2909:
2896:
2893:
2889:
2884:
2877:
2874:
2870:
2866:
2863:
2860:
2857:
2839:
2838:
2825:
2822:
2818:
2813:
2806:
2803:
2799:
2793:
2790:
2786:
2782:
2779:
2776:
2773:
2748:
2745:
2740:
2712:
2709:
2705:
2678:
2675:
2671:
2643:viscous fluids
2615:
2612:
2600:
2597:
2577:thermodynamics
2552:
2548:
2519:
2497:
2493:
2472:
2450:
2446:
2425:
2405:
2385:
2365:
2345:
2334:
2333:
2322:
2316:
2312:
2307:
2300:
2296:
2290:
2286:
2282:
2279:
2275:
2270:
2267:
2264:
2260:
2255:
2252:
2249:
2245:
2221:
2210:
2209:
2208:
2207:
2195:
2181:
2169:
2151:
2150:
2149:
2148:
2134:
2131:
2127:
2123:
2120:
2107:Volume ×
2105:
2086:
2083:
2079:
2061:
2060:
2059:
2058:
2052:
2051:
2050:
2049:
2037:
2023:
2007:
1989:
1988:
1987:
1986:
1974:
1960:
1944:
1891:
1887:
1866:
1846:
1843:
1826:
1823:
1716:thermodynamics
1710:
1709:
1707:
1706:
1699:
1692:
1684:
1681:
1680:
1679:
1678:
1665:
1664:
1661:
1660:
1655:
1650:
1645:
1639:
1636:
1635:
1632:
1631:
1627:
1626:
1621:
1616:
1611:
1606:
1601:
1596:
1591:
1586:
1581:
1576:
1571:
1566:
1561:
1556:
1551:
1546:
1541:
1536:
1531:
1526:
1521:
1516:
1511:
1506:
1501:
1496:
1490:
1489:
1486:
1485:
1482:
1481:
1476:
1475:
1474:
1473:
1468:
1460:
1459:
1457:
1456:
1453:
1449:
1446:
1445:
1443:
1442:
1437:
1435:Thermodynamics
1431:
1428:
1427:
1423:
1422:
1421:
1420:
1411:
1409:
1400:
1398:
1389:
1384:
1383:
1377:
1376:
1375:
1374:
1369:
1364:
1352:
1351:
1349:Caloric theory
1345:
1342:
1341:
1337:
1336:
1334:
1333:
1328:
1323:
1318:
1313:
1308:
1303:
1297:
1294:
1293:
1287:
1286:
1285:
1284:
1277:
1276:
1271:
1266:
1260:
1257:
1256:
1250:
1247:
1246:
1243:
1239:
1238:
1237:
1234:
1233:
1229:
1228:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1173:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1118:
1107:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1077:
1063:
1052:
1049:
1046:
1043:
1040:
1037:
1022:
1020:
1019:
1014:
1008:
1007:
1002:
1001:
998:
997:
994:
993:
986:
981:
976:
969:
968:
963:
958:
953:
947:
946:
941:
940:
937:
936:
930:
929:
926:
925:
914:
911:
901:
890:
879:
878:
867:
864:
854:
843:
829:
818:
815:
805:
798:
797:
794:
793:
782:
779:
769:
758:
747:
746:
735:
732:
722:
711:
697:
686:
683:
680:
670:
663:
662:
659:
658:
647:
644:
634:
623:
612:
611:
600:
597:
587:
576:
562:
551:
548:
538:
529:
528:
527:
521:
516:
515:
512:
511:
506:
505:
504:
503:
498:
493:
482:
471:
452:
451:
445:
444:
442:
441:
436:
430:
427:
426:
420:
419:
418:
417:
412:
393:
392:
387:
386:
383:
382:
377:
376:
375:
374:
369:
364:
356:
355:
349:
348:
347:
346:
341:
336:
331:
329:Free expansion
326:
321:
316:
311:
306:
301:
296:
291:
283:
282:
276:
275:
274:
273:
268:
266:Control volume
263:
258:
256:Phase (matter)
253:
248:
243:
238:
230:
229:
221:
220:
215:
210:
204:
199:
198:
195:
194:
190:
189:
184:
179:
174:
168:
167:
162:
161:
158:
157:
154:
153:
142:
141:
136:
131:
126:
120:
119:
116:
115:
112:
111:
106:The classical
105:
97:
96:
94:Thermodynamics
84:
83:
38:
36:
29:
15:
9:
6:
4:
3:
2:
3354:
3343:
3340:
3339:
3337:
3326:
3320:
3316:
3311:
3307:
3305:9780071138093
3301:
3297:
3296:New York City
3293:
3288:
3287:
3274:
3272:9780750626330
3268:
3264:
3260:
3256:
3252:
3248:
3244:
3243:Landau, L. D.
3238:
3229:
3225:
3221:
3217:
3213:
3209:
3202:
3195:
3191:
3181:
3178:
3176:
3173:
3170:
3166:
3163:
3162:
3156:
3154:
3150:
3140:
3138:
3134:
3130:
3126:
3122:
3112:
3098:
3090:
3087:
3084:
3081:
3078:
3056:
3053:
3049:
3040:
3019:
3016:
3012:
3002:
2999:
2994:
2990:
2986:
2983:
2978:
2975:
2971:
2956:
2953:
2949:
2945:
2942:
2935:
2932:
2929:
2926:
2919:
2918:
2917:
2915:
2894:
2891:
2887:
2875:
2872:
2868:
2864:
2861:
2858:
2855:
2848:
2847:
2846:
2844:
2823:
2820:
2816:
2804:
2801:
2797:
2791:
2788:
2784:
2780:
2777:
2774:
2771:
2764:
2763:
2762:
2746:
2743:
2738:
2728:
2710:
2707:
2703:
2694:
2676:
2673:
2669:
2660:
2659:strain tensor
2656:
2655:stress tensor
2652:
2648:
2644:
2639:
2637:
2633:
2629:
2613:
2610:
2596:
2594:
2590:
2586:
2582:
2578:
2574:
2570:
2565:
2550:
2537:
2531:
2517:
2495:
2491:
2470:
2448:
2444:
2423:
2416:is pressure,
2403:
2383:
2363:
2343:
2320:
2314:
2310:
2298:
2294:
2288:
2284:
2280:
2277:
2268:
2265:
2262:
2253:
2250:
2247:
2235:
2234:
2233:
2219:
2193:
2185:
2182:
2167:
2159:
2156:
2155:
2153:
2152:
2132:
2129:
2125:
2121:
2118:
2110:
2106:
2103:
2084:
2081:
2077:
2068:
2065:
2064:
2063:
2062:
2056:
2055:
2054:
2053:
2035:
2027:
2024:
2021:
2005:
1997:
1994:
1993:
1991:
1990:
1972:
1964:
1961:
1958:
1942:
1934:
1931:
1930:
1928:
1927:
1926:
1924:
1919:
1917:
1911:
1909:
1905:
1889:
1864:
1844:
1841:
1833:
1822:
1820:
1815:
1809:
1807:
1803:
1798:
1796:
1792:
1788:
1784:
1780:
1776:
1772:
1767:
1765:
1762:or sometimes
1761:
1757:
1753:
1749:
1745:
1741:
1737:
1733:
1729:
1725:
1721:
1717:
1705:
1700:
1698:
1693:
1691:
1686:
1685:
1683:
1682:
1677:
1669:
1668:
1667:
1666:
1659:
1656:
1654:
1651:
1649:
1648:Self-assembly
1646:
1644:
1641:
1640:
1634:
1633:
1625:
1622:
1620:
1619:van der Waals
1617:
1615:
1612:
1610:
1607:
1605:
1602:
1600:
1597:
1595:
1592:
1590:
1587:
1585:
1582:
1580:
1577:
1575:
1572:
1570:
1567:
1565:
1562:
1560:
1557:
1555:
1552:
1550:
1547:
1545:
1544:von Helmholtz
1542:
1540:
1537:
1535:
1532:
1530:
1527:
1525:
1522:
1520:
1517:
1515:
1512:
1510:
1507:
1505:
1502:
1500:
1497:
1495:
1492:
1491:
1484:
1483:
1472:
1469:
1467:
1464:
1463:
1462:
1461:
1454:
1451:
1450:
1448:
1447:
1441:
1438:
1436:
1433:
1432:
1430:
1429:
1425:
1424:
1418:
1417:
1410:
1407:
1406:
1399:
1396:
1395:
1388:
1387:
1386:
1385:
1382:
1379:
1378:
1373:
1370:
1368:
1365:
1363:
1359:
1355:
1354:
1350:
1347:
1346:
1344:
1343:
1339:
1338:
1332:
1329:
1327:
1324:
1322:
1319:
1317:
1314:
1312:
1309:
1307:
1304:
1302:
1299:
1298:
1296:
1295:
1292:
1289:
1288:
1283:
1280:
1279:
1275:
1272:
1270:
1267:
1265:
1262:
1261:
1259:
1258:
1254:
1253:
1244:
1241:
1240:
1236:
1235:
1215:
1212:
1209:
1206:
1203:
1197:
1194:
1191:
1185:
1177:
1174:
1160:
1157:
1154:
1151:
1148:
1142:
1139:
1136:
1130:
1122:
1119:
1105:
1102:
1099:
1096:
1093:
1087:
1084:
1081:
1075:
1067:
1064:
1047:
1044:
1041:
1035:
1027:
1024:
1023:
1018:
1015:
1013:
1010:
1009:
1005:
1000:
999:
992:
991:
987:
985:
982:
980:
977:
975:
972:
971:
967:
966:Ideal gas law
964:
962:
959:
957:
954:
952:
949:
948:
944:
939:
938:
912:
902:
888:
881:
880:
865:
855:
841:
834:
833:
830:
816:
813:
806:
803:
800:
799:
780:
770:
756:
749:
748:
733:
723:
709:
702:
701:
698:
684:
681:
678:
671:
668:
665:
664:
645:
635:
621:
614:
613:
598:
588:
574:
567:
566:
563:
549:
546:
539:
536:
533:
532:
526:
523:
522:
519:
514:
513:
502:
499:
497:
496:Vapor quality
494:
492:
491:
486:
483:
481:
480:
475:
472:
469:
465:
464:
459:
456:
455:
454:
453:
450:
447:
446:
440:
437:
435:
432:
431:
429:
428:
425:
422:
421:
416:
413:
411:
408:
407:
406:
405:
401:
397:
390:
385:
384:
373:
370:
368:
365:
363:
360:
359:
358:
357:
354:
351:
350:
345:
342:
340:
337:
335:
334:Reversibility
332:
330:
327:
325:
322:
320:
317:
315:
312:
310:
307:
305:
302:
300:
297:
295:
292:
290:
287:
286:
285:
284:
281:
278:
277:
272:
269:
267:
264:
262:
259:
257:
254:
252:
249:
247:
244:
242:
239:
237:
234:
233:
232:
231:
228:
225:
224:
219:
216:
214:
211:
209:
208:Closed system
206:
205:
202:
197:
196:
188:
185:
183:
180:
178:
175:
173:
170:
169:
165:
160:
159:
152:
148:
145:
144:
140:
137:
135:
132:
130:
127:
125:
122:
121:
114:
113:
109:
103:
99:
98:
95:
92:
91:
88:
80:
77:
69:
59:
55:
49:
48:
42:
37:
28:
27:
22:
3314:
3291:
3250:
3237:
3211:
3207:
3194:
3146:
3136:
3132:
3118:
3036:
2911:
2840:
2726:
2692:
2640:
2602:
2580:
2576:
2566:
2532:
2396:is entropy,
2335:
2211:
1985: (J K)
1920:
1912:
1831:
1828:
1810:
1799:
1782:
1768:
1727:
1713:
1509:Carathéodory
1440:Heat engines
1412:
1401:
1390:
1372:Motive power
1357:
1017:Free entropy
988:
488:
487: /
477:
476: /
468:introduction
461:
460: /
399:
395:
362:Heat engines
149: /
87:
72:
63:
44:
3121:temperature
2914:unit tensor
2436:is volume,
1933:Temperature
1732:temperature
1331:Synergetics
1012:Free energy
458:Temperature
319:Quasistatic
314:Isenthalpic
271:Instruments
261:Equilibrium
213:Open system
147:Equilibrium
129:Statistical
58:introducing
3255:Waltham MA
3186:References
3180:Bond graph
2841:or, using
2147:(m = J Pa)
1795:derivative
1643:Nucleation
1487:Scientists
1291:Philosophy
1004:Potentials
367:Heat pumps
324:Polytropic
309:Isentropic
299:Isothermal
41:references
3088:−
3079:δ
3050:ε
3013:ε
2991:∑
2987:−
2972:ε
2950:δ
2943:−
2927:δ
2888:ε
2869:σ
2856:δ
2817:ε
2798:σ
2785:∑
2772:δ
2739:ε
2704:ε
2670:σ
2445:μ
2295:μ
2285:∑
2266:−
2168:μ
2126:ε
2122:×
2078:σ
1624:Waterston
1574:von Mayer
1529:de Donder
1519:Clapeyron
1499:Boltzmann
1494:Bernoulli
1455:Education
1426:Timelines
1210:−
1155:−
943:Equations
910:∂
863:∂
814:α
778:∂
731:∂
685:−
679:β
643:∂
596:∂
304:Adiabatic
294:Isochoric
280:Processes
241:Ideal gas
124:Classical
66:July 2019
3336:Category
3249:(1986).
3232:p. 1353.
3228:98264934
3159:See also
2916:so that
2628:pressure
2573:kinetics
2018: (
1996:Pressure
1955: (
1925:units):
1825:Overview
1740:pressure
1730:such as
1676:Category
1614:Thompson
1524:Clausius
1504:Bridgman
1358:Vis viva
1340:Theories
1274:Gas laws
1066:Enthalpy
474:Pressure
289:Isobaric
246:Real gas
134:Chemical
117:Branches
3129:heating
3125:entropy
2725:is the
2691:is the
2651:elastic
2647:plastic
1963:Entropy
1814:entropy
1736:entropy
1599:Smeaton
1594:Rankine
1584:Onsager
1569:Maxwell
1564:Massieu
1269:Entropy
1264:General
1255:History
1245:Culture
1242:History
466: (
463:Entropy
400:italics
201:Systems
54:improve
3321:
3302:
3269:
3259:Oxford
3226:
2632:volume
2336:where
2109:Strain
2104:= J m)
2067:Stress
2026:Volume
2022:= J m)
1769:For a
1744:volume
1724:system
1718:, the
1589:Planck
1579:Nernst
1554:Kelvin
1514:Carnot
804:
669:
537:
479:Volume
394:Note:
353:Cycles
182:Second
172:Zeroth
43:, but
3224:S2CID
3204:(PDF)
3039:trace
1764:power
1746:, or
1722:of a
1637:Other
1604:Stahl
1559:Lewis
1549:Joule
1539:Gibbs
1534:Duhem
227:State
187:Third
177:First
3319:ISBN
3300:ISBN
3267:ISBN
3167:and
3147:The
3137:work
3133:heat
3037:The
2761:is:
2649:and
2569:rate
1800:The
1750:and
1742:and
1734:and
1609:Tait
439:Heat
434:Work
164:Laws
3216:doi
2180:(J)
1714:In
1452:Art
398:in
3338::
3265:.
3261::
3257:,
3245:;
3222:.
3212:73
3210:.
3206:.
3139:.
2727:ij
2693:ij
2645:,
2186::
2160::
2111::
2102:Pa
2069::
2028::
2020:Pa
1998::
1965::
1935::
1923:SI
1821:.
1766:.
1738:,
3327:.
3308:.
3275:.
3230:.
3218::
3099:V
3095:d
3091:p
3085:=
3082:w
3057:k
3054:k
3020:k
3017:k
3008:d
3003:V
3000:p
2995:k
2984:=
2979:j
2976:i
2967:d
2962:)
2957:j
2954:i
2946:p
2940:(
2936:V
2933:=
2930:w
2895:j
2892:i
2883:d
2876:j
2873:i
2865:V
2862:=
2859:w
2824:j
2821:i
2812:d
2805:j
2802:i
2792:j
2789:i
2781:V
2778:=
2775:w
2747:j
2744:i
2711:j
2708:i
2677:j
2674:i
2614:V
2611:p
2551:U
2547:d
2518:i
2496:i
2492:N
2471:i
2449:i
2424:V
2404:p
2384:S
2364:T
2344:U
2321:,
2315:i
2311:N
2306:d
2299:i
2289:i
2281:+
2278:V
2274:d
2269:p
2263:S
2259:d
2254:T
2251:=
2248:U
2244:d
2220:i
2194:N
2133:j
2130:i
2119:V
2100:(
2085:j
2082:i
2036:V
2006:p
1973:S
1959:)
1957:K
1943:T
1890:V
1886:d
1865:p
1845:V
1842:p
1703:e
1696:t
1689:v
1216:S
1213:T
1207:H
1204:=
1201:)
1198:p
1195:,
1192:T
1189:(
1186:G
1161:S
1158:T
1152:U
1149:=
1146:)
1143:V
1140:,
1137:T
1134:(
1131:A
1106:V
1103:p
1100:+
1097:U
1094:=
1091:)
1088:p
1085:,
1082:S
1079:(
1076:H
1051:)
1048:V
1045:,
1042:S
1039:(
1036:U
913:T
889:V
866:V
842:1
817:=
781:p
757:V
734:V
710:1
682:=
646:T
622:N
599:S
575:T
550:=
547:c
470:)
79:)
73:(
68:)
64:(
50:.
23:.
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