529:
5739:
750:
3753:
In the original work by
Clapeyron, the following argument is advanced. Clapeyron considered a Carnot process of saturated water vapor with horizontal isobars. As the pressure is a function of temperature alone, the isobars are also isotherms. If the process involves an infinitesimal amount of water,
86:
confirmed the relation experimentally in 1849–50, and it was historically important as a very early successful application of theoretical thermodynamics. Its relevance to meteorology and climatology is the increase of the water-holding capacity of the atmosphere by about 7% for every 1 °C
2453:
3332:
3467:
3580:
5363:
215:
2634:
633:
2327:
4604:
3924:
920:
839:
4130:
3201:
2334:
1126:
1011:
1308:
2826:
1747:
3205:
4875:
474:
5336:
2900:
3744:
2150:
2077:
5932:
5264:
1413:
3336:
5844:
1681:
5088:
3472:
1502:
4244:
1224:
1175:
131:
79:
decades later. Kelvin said of Carnot's argument that "nothing in the whole range of
Natural Philosophy is more remarkable than the establishment of general laws by such a process of reasoning."
6032:
5368:
5150:
2550:
1548:
1455:
2694:
2018:
5010:
One of the uses of this equation is to determine if a phase transition will occur in a given situation. Consider the question of how much pressure is needed to melt ice at a temperature
1791:
253:
2227:
1621:
4498:
850:
769:
4166:
3960:
3132:
5184:
3840:
516:
4046:
1864:
5002:
with temperature under typical atmospheric conditions, and hence the water-holding capacity of the atmosphere increases by about 7% for every 1 °C rise in temperature.
3985:
3802:
3777:
3043:
2997:
2545:
2518:
1064:
952:
5033:
1582:
1237:
5734:{\displaystyle {\begin{aligned}{\frac {\mathrm {d} ^{2}P}{\mathrm {d} T^{2}}}&={\frac {1}{v_{2}-v_{1}}}\left\\{}&+{\frac {1}{v_{2}-v_{1}}}\left,\end{aligned}}}
4778:
1927:
355:
328:
3845:
3067:
2765:
1955:
1686:
1039:
3087:
1975:
1059:
5766:
5348:(area ~ 1 cm). This shows that ice skating cannot be simply explained by pressure-caused melting point depression, and in fact the mechanism is quite complex.
4902:
4733:
4696:
4637:
4429:
4051:
4798:
1823:
6719:
4457:
4013:
405:
5274:
2834:
4996:
4950:
4926:
4663:
4395:
4363:
4343:
4319:
4284:
4264:
3127:
3107:
2944:
2920:
2760:
2736:
2716:
2218:
2194:
2170:
1904:
1884:
1355:
1328:
940:
745:{\displaystyle \mathrm {d} s=\left({\frac {\partial s}{\partial v}}\right)_{T}\,\mathrm {d} v+\left({\frac {\partial s}{\partial T}}\right)_{v}\,\mathrm {d} T.}
628:
605:
578:
394:
301:
277:
3607:
2088:
2029:
762:
composed of two contiguous phases, condensed matter and ideal gas, of a single substance, in mutual thermodynamic equilibrium, at constant temperature and
6578:
Krafcik, Matthew; Sánchez
Velasco, Eduardo (2014). "Beyond Clausius–Clapeyron: Determining the second derivative of a first-order phase transition line".
6305:
4998:(for which the unit is degree Celsius). Therefore, the August–Roche–Magnus equation implies that saturation water vapor pressure changes approximately
1360:
942:
is the pressure. Since pressure and temperature are constant, the derivative of pressure with respect to temperature does not change. Therefore, the
5038:
1459:
6509:
2448:{\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {s_{\beta }-s_{\alpha }}{v_{\beta }-v_{\alpha }}}={\frac {\Delta s}{\Delta v}},}
6725:
and was not known in this form. Clausius determined the form 30 years later and added his name to the eponymous
Clausius–Clapeyron relation.
4188:
5356:
While the
Clausius–Clapeyron relation gives the slope of the coexistence curve, it does not provide any information about its curvature or
5853:
17:
3327:{\displaystyle \int _{P_{1}}^{P_{2}}{\frac {\mathrm {d} P}{P}}\cong {\frac {L}{R}}\int _{T_{1}}^{T_{2}}{\frac {\mathrm {d} T}{T^{2}}},}
5195:
4964:
1509:
1417:
6037:
5775:
2651:
1980:
1628:
520:
The equation expresses this in a more convenient form just in terms of the latent heat, for moderate temperatures and pressures.
71:
6096:"An Account of Carnot's Theory of the Motive Power of Heat; with Numerical Results deduced from Regnault's Experiments on Steam"
4185:
For transitions between a gas and a condensed phase with the approximations described above, the expression may be rewritten as
1587:
6669:
6650:
6627:
6288:
6203:
6134:
6465:"The Relationship between Relative Humidity and the Dewpoint Temperature in Moist Air: A Simple Conversion and Applications"
1180:
1131:
4405:, for instance (1 bar, 373 K) for water, determines the rest of the curve. Conversely, the relationship between
6761:
6756:
5097:
3807:
1227:
3462:{\displaystyle \ln P{\Big |}_{P=P_{1}}^{P_{2}}\cong -{\frac {L}{R}}\cdot \left.{\frac {1}{T}}\right|_{T=T_{1}}^{T_{2}},}
480:
6335:
6345:
4490:
6751:
5979:"Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen"
62:
3575:{\displaystyle \ln {\frac {P_{2}}{P_{1}}}\cong -{\frac {L}{R}}\left({\frac {1}{T_{2}}}-{\frac {1}{T_{1}}}\right).}
537:
6151:
5035:
below 0 °C. Note that water is unusual in that its change in volume upon melting is negative. We can assume
2486:
1758:
220:
6444:
5981:[On the motive power of heat and the laws which can be deduced therefrom regarding the theory of heat].
1226:
are respectively the change in specific entropy and specific volume. Given that a phase change is an internally
540:. The Clausius–Clapeyron relation can be used to find the relationship between pressure and temperature along
66:
5847:
210:{\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {L}{T\,\Delta v}}={\frac {\Delta s}{\Delta v}},}
6313:
2629:{\displaystyle \Delta v=v_{\text{g}}\left(1-{\frac {v_{\text{c}}}{v_{\text{g}}}}\right)\approx v_{\text{g}}}
6220:
5340:
To provide a rough example of how much pressure this is, to melt ice at −7 °C (the temperature many
5157:
1231:
75:, which was published in 1824 but largely ignored until it was rediscovered by Clausius, Clapeyron, and
5935:
3585:
4965:
discussion of the accuracy of different approximating formulae for saturation vapour pressure of water
4640:
4322:
4018:
3589:
1828:
1334:
of the system. Given constant pressure and temperature (during a phase change) and the definition of
6746:
6741:
6505:
6464:
3965:
3782:
3757:
3002:
2956:
2322:{\displaystyle -(s_{\beta }-s_{\alpha })\,\mathrm {d} T+(v_{\beta }-v_{\alpha })\,\mathrm {d} P=0.}
83:
2523:
2496:
5947:
4398:
4287:
2739:
374:
42:
6247:
5013:
1558:
6280:
5769:
4790:
4782:
4599:{\displaystyle {\frac {\mathrm {d} e_{s}}{\mathrm {d} T}}={\frac {L_{v}(T)e_{s}}{R_{v}T^{2}}},}
915:{\displaystyle \mathrm {d} s=\left({\frac {\partial P}{\partial T}}\right)_{v}\,\mathrm {d} v,}
834:{\displaystyle \mathrm {d} s=\left({\frac {\partial s}{\partial v}}\right)_{T}\,\mathrm {d} v.}
4747:
1977:, are in contact and at equilibrium with each other. Their chemical potentials are related by
1909:
337:
310:
6638:
5957:
4489:. The Clausius–Clapeyron equation for water vapor under typical atmospheric conditions (near
3196:{\displaystyle {\frac {\mathrm {d} P}{P}}\cong {\frac {L}{R}}{\frac {\mathrm {d} T}{T^{2}}},}
3052:
1940:
1024:
4135:
3929:
3072:
1960:
1044:
6587:
6479:
6431:
6419:
5990:
5744:
4999:
4880:
4711:
4699:
4674:
4615:
4408:
4366:
3597:
2923:
1553:
6079:
This relationship was deduced by Carnot, but it is called the
Clausius-Clapeyron equation.
5978:
1799:
8:
6696:
4434:
3990:
30:"Clapeyron equation" and "Clapeyron's equation" redirect here. For a state equation, see
6591:
6483:
6435:
5994:
6547:"Friction on Ice: How Temperature, Pressure, and Speed Control the Slipperiness of Ice"
6527:
4981:
4935:
4911:
4648:
4380:
4348:
4328:
4304:
4269:
4249:
3112:
3092:
2929:
2905:
2745:
2721:
2701:
2203:
2179:
2155:
1889:
1869:
1340:
1313:
943:
925:
613:
590:
563:
379:
369:
The
Clausius–Clapeyron equation applies to vaporization of liquids where vapor follows
286:
262:
6665:
6646:
6623:
6341:
6284:
6199:
6130:
6027:
5357:
4460:
4370:
1335:
6388:
6062:
4963:
approximation, though this attribution is historically inaccurate. But see also the
1121:{\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {\Delta s}{\Delta v}},}
1006:{\displaystyle \mathrm {d} s={\frac {\mathrm {d} P}{\mathrm {d} T}}\,\mathrm {d} v,}
6595:
6558:
6487:
6460:
6439:
6398:
6107:
6006:
5998:
5952:
4486:
2470:
947:
844:
755:
558:
541:
358:
116:) diagram, for any phase change the line separating the two phases is known as the
54:
50:
4786:
1303:{\displaystyle \mathrm {d} u=\delta q+\delta w=T\,\mathrm {d} s-P\,\mathrm {d} v,}
6513:
6058:
5344:
rinks are set at) would require balancing a small car (mass ~ 1000 kg) on a
4929:
2490:
2197:
2082:
1331:
585:
554:
331:
58:
3919:{\displaystyle W={\frac {\mathrm {d} p}{\mathrm {d} T}}\,\mathrm {d} T(V''-V'),}
2821:{\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {L}{T\,\Delta v}},}
1742:{\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {L}{T\,\Delta v}}.}
6563:
6546:
6545:
Liefferink, Rinse W.; Hsia, Feng-Chun; Weber, Bart; Bonn, Daniel (2021-02-08).
6011:
5360:. The second derivative of the coexistence curve of phases 1 and 2 is given by
46:
6111:
4125:{\displaystyle {\frac {\mathrm {d} p}{\mathrm {d} T}}={\frac {L}{T(v''-v')}},}
6735:
6002:
4870:{\displaystyle e_{s}(T)=6.1094\exp \left({\frac {17.625T}{T+243.04}}\right),}
4482:
2645:
1014:
759:
533:
370:
31:
2926:
of the substance. Instead of the specific, corresponding molar values (i.e.
469:{\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {PL}{T^{2}R}},}
5331:{\displaystyle {\frac {\Delta P}{\Delta T}}=-13.5~{\text{MPa}}/{\text{K}}.}
4736:
4374:
4291:
2895:{\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {PL}{T^{2}R}}}
2637:
1506:
Given constant pressure and temperature (during a phase change), we obtain
6491:
6091:
5341:
4971:
4703:
4666:
4478:
4474:
3109:
varies between any two such points, as a function of temperature. But if
608:
581:
304:
280:
105:
76:
6504:
IPCC, Climate Change 2007: Working Group I: The
Physical Science Basis,
1929:. Instead of the specific, corresponding molar values may also be used.
1013:
and the total derivative of pressure with respect to temperature may be
6368:
Clapeyron, E (1834). "Mémoire sur la puissance motrice de la chaleur".
4471:
4298:
2221:
396:
and liquid volume is neglected as being much smaller than vapor volume
6599:
6127:
Elements of classical thermodynamics: for advanced students of physics
3739:{\displaystyle P_{\text{vap}}(T)\cong 1~{\text{bar}}\cdot \exp \left.}
2145:{\displaystyle \mathrm {d} \mu =M(-s\,\mathrm {d} T+v\,\mathrm {d} P)}
2072:{\displaystyle \mathrm {d} \mu _{\alpha }=\mathrm {d} \mu _{\beta }.}
45:, specifies the temperature dependence of pressure, most importantly
6095:
2457:
from which the derivation of the
Clapeyron equation continues as in
6403:
5927:{\displaystyle \kappa _{T}=-(1/v)(\mathrm {d} v/\mathrm {d} P)_{T}}
4975:
2641:
1625:
Substituting this result into the pressure derivative given above (
1018:
763:
125:
101:
6194:
Wark, Kenneth (1988) . "Generalized
Thermodynamic Relationships".
4401:. If the latent heat is known, then knowledge of one point on the
6420:"Improved Magnus Form Approximation of Saturation Vapor Pressure"
5345:
4905:
4402:
3596:
requiring specific-volume data. For instance, for water near its
3046:
2173:
2023:
1794:
256:
117:
6390:
Improved Magnus' Form Approximation of Saturation Vapor Pressure
5259:{\displaystyle \Delta v=-9.05\times 10^{-5}~\mathrm {m^{3}/kg} }
3600:, with a molar enthalpy of vaporization of 40.7 kJ/mol and
1408:{\displaystyle \mathrm {d} h=T\,\mathrm {d} s+v\,\mathrm {d} P,}
5187:
2478:
5839:{\displaystyle \alpha =(1/v)(\mathrm {d} v/\mathrm {d} T)_{P}}
2464:
528:
2762:
is the temperature. Substituting into the Clapeyron equation
2482:
1676:{\displaystyle \mathrm {d} P/\mathrm {d} T=\Delta s/\Delta v}
121:
5083:{\displaystyle \Delta P={\frac {L}{T\,\Delta v}}\,\Delta T,}
400:. It is often used to calculate vapor pressure of a liquid.
6445:
10.1175/1520-0450(1996)035<0601:IMFAOS>2.0.CO;2
6394:
6248:"PvT Surface for a Substance which Contracts Upon Freezing"
3409:
65:. However, this relation was in fact originally derived by
2474:
1497:{\displaystyle \mathrm {d} s={\frac {\mathrm {d} h}{T}}.}
6639:"4. Water-Air systems § 4.8 Clausius–Clapeyron Equation"
6544:
6274:
3592:
and temperature to the latent heat of the phase change
1932:
283:(the amount of energy absorbed in the transformation),
6662:
Thermodynamics and an Introduction to Thermostatistics
6577:
5741:
where subscripts 1 and 2 denote the different phases,
4239:{\displaystyle \ln P=-{\frac {L}{R}}{\frac {1}{T}}+c,}
6699:
5856:
5778:
5747:
5366:
5277:
5198:
5160:
5100:
5041:
5016:
4984:
4938:
4914:
4883:
4801:
4750:
4714:
4677:
4651:
4618:
4501:
4437:
4411:
4383:
4351:
4331:
4307:
4272:
4252:
4191:
4180:
4138:
4054:
4021:
3993:
3968:
3932:
3848:
3810:
3785:
3760:
3610:
3475:
3339:
3208:
3135:
3115:
3095:
3075:
3055:
3005:
2959:
2932:
2908:
2837:
2768:
2748:
2724:
2704:
2654:
2553:
2526:
2499:
2337:
2230:
2206:
2182:
2158:
2091:
2032:
1983:
1963:
1943:
1912:
1892:
1872:
1831:
1802:
1761:
1689:
1631:
1590:
1561:
1512:
1462:
1420:
1363:
1343:
1316:
1240:
1219:{\displaystyle \Delta v\equiv v_{\beta }-v_{\alpha }}
1183:
1170:{\displaystyle \Delta s\equiv s_{\beta }-s_{\alpha }}
1134:
1067:
1047:
1027:
955:
928:
853:
772:
636:
616:
593:
566:
483:
408:
382:
340:
313:
289:
265:
223:
134:
3584:
These last equations are useful because they relate
2458:
6361:
6129:(Repr ed.). Cambridge: Univ. Pr. p. 116.
5145:{\displaystyle L=3.34\times 10^{5}~\mathrm {J/kg} }
6713:
5926:
5838:
5760:
5733:
5330:
5258:
5178:
5144:
5082:
5027:
4990:
4944:
4920:
4896:
4869:
4772:
4727:
4690:
4657:
4631:
4598:
4451:
4423:
4389:
4357:
4337:
4313:
4278:
4258:
4238:
4160:
4124:
4040:
4007:
3979:
3954:
3918:
3834:
3796:
3771:
3738:
3574:
3461:
3326:
3195:
3121:
3101:
3081:
3061:
3037:
2991:
2938:
2914:
2894:
2820:
2754:
2730:
2710:
2688:
2628:
2539:
2512:
2485:), and occurs at temperatures much lower than the
2447:
2321:
2212:
2188:
2164:
2144:
2071:
2012:
1969:
1949:
1921:
1898:
1878:
1858:
1817:
1785:
1741:
1675:
1615:
1576:
1542:
1496:
1449:
1407:
1349:
1322:
1302:
1218:
1169:
1120:
1053:
1033:
1005:
934:
914:
833:
744:
622:
599:
572:
548:
510:
468:
388:
349:
322:
295:
271:
247:
209:
6334:Masterton, William L.; Hurley, Cecile N. (2008).
5266:(change in specific volume from solid to liquid),
3779:, and an infinitesimal difference in temperature
3352:
6733:
6417:
6386:
6198:(5th ed.). New York, NY: McGraw-Hill, Inc.
6028:"Mémoire sur la puissance motrice de la chaleur"
3987:in the liquid phase and vapor phases. The ratio
2644:is also low, the gas may be approximated by the
27:Relation between vapour pressure and temperature
6472:Bulletin of the American Meteorological Society
6418:Alduchov, Oleg A.; Eskridge, Robert E. (1996).
6387:Alduchov, Oleg; Eskridge, Robert (1997-11-01),
6340:(6th ed.). Cengage Learning. p. 230.
6333:
1543:{\displaystyle \Delta s={\frac {\Delta h}{T}}.}
1450:{\displaystyle \mathrm {d} h=T\,\mathrm {d} s,}
6636:
4952:is an absolute temperature, e.g. in kelvins).
4795:provides a solution under that approximation:
4744:The temperature dependence of the latent heat
4466:
364:
6450:Equation 25 provides these coefficients.
6408:Equation 21 provides these coefficients.
6275:Çengel, Yunus A.; Boles, Michael A. (1998) .
6270:
6268:
6245:
6218:
2689:{\displaystyle v_{\text{g}}={\frac {RT}{P}},}
2013:{\displaystyle \mu _{\alpha }=\mu _{\beta }.}
6306:"Clapeyron and Clausius–Clapeyron Equations"
4345:is a constant. For a liquid–gas transition,
2520:greatly exceeds that of the condensed phase
754:The Clausius–Clapeyron relation describes a
57:of a single constituent. It is named after
6297:
6100:Transactions of the Edinburgh Royal Society
2465:Ideal gas approximation at low temperatures
1786:{\displaystyle \mathrm {d} P/\mathrm {d} T}
248:{\displaystyle \mathrm {d} P/\mathrm {d} T}
95:
6283:(3rd ed.). Boston, MA.: McGraw-Hill.
6265:
6189:
6187:
6185:
6183:
6181:
6179:
6177:
6175:
6173:
4970:Under typical atmospheric conditions, the
4780:can be neglected in this application. The
2902:for low temperatures and pressures, where
946:of specific entropy may be changed into a
6617:
6562:
6443:
6402:
6367:
6337:Chemistry : principles and reactions
6239:
6149:
6025:
6010:
5170:
5070:
5060:
3962:is the difference between the volumes of
3879:
3820:
3748:
3667:
2950:= 8.31 J/(mol⋅K)) may also be used.
2805:
2304:
2263:
2130:
2115:
1846:
1726:
1435:
1393:
1378:
1288:
1273:
991:
900:
819:
730:
683:
171:
6506:"FAQ 3.2 How is Precipitation Changing?"
6459:
6277:Thermodynamics – An Engineering Approach
6047:: 153–190. ark:/12148/bpt6k4336791/f157.
6019:
5976:
4015:is the efficiency of the Carnot engine,
1616:{\displaystyle \Delta s={\frac {L}{T}}.}
527:
6637:Iribarne, J. V.; Godson, W. L. (2013).
6622:(3rd ed.). Butterworth–Heinemann.
6312:. University of Arizona. Archived from
6303:
6170:
6124:
6090:
6057:
5970:
4932:(whereas everywhere else on this page,
72:Reflections on the Motive Power of Fire
14:
6734:
6659:
6525:
6051:
6084:
4463:is used to estimate the latent heat.
4048:. Substituting and rearranging gives
1230:, and that our system is closed, the
6193:
6069:. California Institute of Technology
5351:
1933:Derivation from Gibbs–Duhem relation
1906:, and the change in specific volume
334:change of the phase transition, and
5179:{\displaystyle T=273\,\mathrm {K} }
3835:{\displaystyle Q=L\,\mathrm {d} x,}
255:is the slope of the tangent to the
120:. The Clapeyron relation gives the
87:(1.8 °F) rise in temperature.
24:
6618:Yau, M. K.; Rogers, R. R. (1989).
6304:Salzman, William R. (2001-08-21).
5907:
5894:
5819:
5806:
5699:
5689:
5560:
5550:
5392:
5376:
5289:
5281:
5252:
5249:
5235:
5199:
5172:
5152:(latent heat of fusion for water),
5138:
5135:
5127:
5071:
5061:
5042:
5018:
4955:This is also sometimes called the
4523:
4506:
4181:Chemistry and chemical engineering
4069:
4059:
4023:
3970:
3881:
3869:
3859:
3822:
3787:
3762:
3301:
3242:
3170:
3140:
2852:
2842:
2806:
2783:
2773:
2554:
2433:
2425:
2352:
2342:
2306:
2265:
2132:
2117:
2093:
2052:
2034:
1913:
1847:
1776:
1763:
1727:
1704:
1694:
1667:
1656:
1646:
1633:
1591:
1568:
1525:
1513:
1478:
1464:
1437:
1422:
1395:
1380:
1365:
1290:
1275:
1242:
1184:
1135:
1106:
1098:
1082:
1072:
993:
981:
971:
957:
902:
881:
873:
855:
821:
800:
792:
774:
732:
711:
703:
685:
664:
656:
638:
536:. The dotted green line gives the
511:{\displaystyle V={\frac {RT}{P}}.}
423:
413:
341:
314:
238:
225:
195:
187:
172:
149:
139:
25:
6773:
6370:Journal de l ́École Polytechnique
6221:"The Clausius-Clapeyron Equation"
6150:Koziol, Andrea; Perkins, Dexter.
6063:"Illustrations of Thermodynamics"
4491:standard temperature and pressure
2547:. Therefore, one may approximate
6033:Journal de l'École polytechnique
361:change of the phase transition.
6687:
6610:
6571:
6538:
6519:
6498:
6453:
6411:
6380:
6327:
6067:The Feynman Lectures on Physics
4397:is the specific latent heat of
4175:
4041:{\displaystyle \mathrm {d} T/T}
1859:{\displaystyle L/(T\,\Delta v)}
1751:This result (also known as the
1552:Substituting the definition of
549:Derivation from state postulate
128:to this curve. Mathematically,
6424:Journal of Applied Meteorology
6212:
6143:
6118:
5915:
5890:
5887:
5873:
5827:
5802:
5799:
5785:
5677:
5625:
5543:
5497:
4818:
4812:
4767:
4761:
4555:
4549:
4485:), motivating interest in its
4377:; for a solid–gas transition,
4113:
4091:
3910:
3888:
3842:and the corresponding work is
3627:
3621:
3032:
3006:
2986:
2960:
2301:
2275:
2260:
2234:
2139:
2106:
1853:
1840:
1812:
1806:
584:substance to be a function of
523:
13:
1:
6620:Short Course in Cloud Physics
5963:
5848:thermal expansion coefficient
3980:{\displaystyle \mathrm {d} x}
3797:{\displaystyle \mathrm {d} T}
3772:{\displaystyle \mathrm {d} x}
3129:is approximated as constant,
3038:{\displaystyle (P_{2},T_{2})}
2992:{\displaystyle (P_{1},T_{1})}
90:
3045:be any two points along the
2540:{\displaystyle v_{\text{c}}}
2513:{\displaystyle v_{\text{g}}}
2473:of a substance is between a
1866:of the specific latent heat
7:
6580:American Journal of Physics
6152:"Teaching Phase Equilibria"
6125:Pippard, Alfred B. (1981).
5941:
4467:Meteorology and climatology
2830:Clausius–Clapeyron equation
1232:first law of thermodynamics
538:anomalous behavior of water
365:Clausius–Clapeyron equation
63:Benoît Paul Émile Clapeyron
39:Clausius–Clapeyron relation
18:Clausius-Clapeyron relation
10:
6778:
6762:Engineering thermodynamics
6757:Atmospheric thermodynamics
6645:. Springer. pp. 60–.
6643:Atmospheric Thermodynamics
6564:10.1103/PhysRevX.11.011025
6254:. Georgia State University
5936:isothermal compressibility
5028:{\displaystyle {\Delta T}}
5005:
2081:One may therefore use the
1577:{\displaystyle L=\Delta h}
29:
6112:10.1017/S0080456800022481
6026:Clapeyron, M. C. (1834).
5186:(absolute temperature in
4641:saturation vapor pressure
3590:saturation vapor pressure
6680:
6279:. McGraw-Hill Series in
6003:10.1002/andp.18501550403
4773:{\displaystyle L_{v}(T)}
1922:{\displaystyle \Delta v}
350:{\displaystyle \Delta s}
323:{\displaystyle \Delta v}
96:Exact Clapeyron equation
6752:Thermodynamic equations
6310:Chemical Thermodynamics
4172:during the transition.
3804:, the heat absorbed is
3604:= 8.31 J/(mol⋅K),
3062:{\displaystyle \alpha }
2489:of that substance, the
2477:and a condensed phase (
2022:Furthermore, along the
1950:{\displaystyle \alpha }
1034:{\displaystyle \alpha }
82:Kelvin and his brother
43:chemical thermodynamics
6721:was simply called the
6715:
6693:In the original work,
6660:Callen, H. B. (1985).
6281:Mechanical Engineering
6246:Carl Rod Nave (2006).
5928:
5840:
5772:at constant pressure,
5770:specific heat capacity
5762:
5735:
5332:
5260:
5180:
5146:
5084:
5029:
4992:
4946:
4922:
4898:
4871:
4774:
4729:
4692:
4659:
4633:
4600:
4477:drives many important
4453:
4425:
4391:
4359:
4339:
4315:
4280:
4260:
4240:
4168:denotes the change in
4162:
4161:{\displaystyle v''-v'}
4126:
4042:
4009:
3981:
3956:
3955:{\displaystyle V''-V'}
3920:
3836:
3798:
3773:
3749:Clapeyron's derivation
3740:
3576:
3463:
3328:
3197:
3123:
3103:
3083:
3082:{\displaystyle \beta }
3063:
3039:
2993:
2940:
2916:
2896:
2822:
2756:
2732:
2712:
2690:
2630:
2541:
2514:
2449:
2323:
2214:
2190:
2166:
2146:
2073:
2014:
1971:
1970:{\displaystyle \beta }
1951:
1923:
1900:
1880:
1860:
1819:
1787:
1743:
1677:
1617:
1578:
1544:
1498:
1451:
1409:
1351:
1324:
1304:
1220:
1171:
1122:
1055:
1054:{\displaystyle \beta }
1035:
1021:from an initial phase
1007:
936:
916:
843:Using the appropriate
835:
746:
624:
601:
574:
545:
512:
470:
390:
351:
324:
297:
273:
249:
211:
6716:
6526:Zorina, Yana (2000).
6492:10.1175/BAMS-86-2-225
6219:Clausius; Clapeyron.
6158:. Carleton University
5977:Clausius, R. (1850).
5929:
5841:
5763:
5761:{\displaystyle c_{p}}
5736:
5333:
5261:
5181:
5147:
5085:
5030:
4993:
4947:
4923:
4899:
4897:{\displaystyle e_{s}}
4872:
4775:
4730:
4728:{\displaystyle R_{v}}
4693:
4691:{\displaystyle L_{v}}
4660:
4634:
4632:{\displaystyle e_{s}}
4601:
4454:
4426:
4424:{\displaystyle \ln P}
4392:
4360:
4340:
4316:
4288:specific gas constant
4281:
4261:
4241:
4163:
4127:
4043:
4010:
3982:
3957:
3921:
3837:
3799:
3774:
3741:
3577:
3464:
3329:
3198:
3124:
3104:
3084:
3064:
3040:
2994:
2941:
2917:
2897:
2823:
2757:
2740:specific gas constant
2733:
2713:
2691:
2631:
2542:
2515:
2450:
2324:
2215:
2191:
2167:
2147:
2074:
2015:
1972:
1952:
1924:
1901:
1881:
1861:
1820:
1788:
1744:
1678:
1618:
1579:
1545:
1499:
1452:
1410:
1352:
1325:
1305:
1221:
1172:
1123:
1056:
1036:
1008:
937:
917:
836:
747:
625:
602:
575:
531:
513:
471:
391:
375:specific gas constant
352:
325:
298:
274:
250:
212:
49:, at a discontinuous
6697:
6532:The Physics Factbook
5948:Van 't Hoff equation
5854:
5776:
5745:
5364:
5275:
5196:
5158:
5098:
5090:and substituting in
5039:
5014:
4982:
4936:
4912:
4881:
4799:
4748:
4712:
4700:specific latent heat
4675:
4649:
4616:
4499:
4481:phenomena (notably,
4435:
4409:
4381:
4367:specific latent heat
4349:
4329:
4323:absolute temperature
4305:
4270:
4250:
4189:
4136:
4052:
4019:
3991:
3966:
3930:
3846:
3808:
3783:
3758:
3608:
3598:normal boiling point
3473:
3337:
3206:
3133:
3113:
3093:
3073:
3053:
3003:
2957:
2930:
2924:specific latent heat
2906:
2835:
2766:
2746:
2722:
2702:
2652:
2551:
2524:
2497:
2487:critical temperature
2459:the previous section
2335:
2331:Rearrangement gives
2228:
2204:
2180:
2156:
2089:
2030:
1981:
1961:
1941:
1937:Suppose two phases,
1910:
1890:
1870:
1829:
1818:{\displaystyle P(T)}
1800:
1759:
1755:) equates the slope
1687:
1629:
1588:
1559:
1554:specific latent heat
1510:
1460:
1418:
1361:
1341:
1314:
1238:
1181:
1132:
1065:
1045:
1025:
953:
926:
851:
770:
634:
614:
591:
564:
481:
406:
380:
338:
311:
287:
263:
221:
132:
6714:{\displaystyle 1/T}
6592:2014AmJPh..82..301K
6484:2005BAMS...86..225L
6436:1996JApMe..35..601A
6227:. Purdue University
6225:Bodner Research Web
5995:1850AnP...155..500C
4452:{\displaystyle 1/T}
4008:{\displaystyle W/Q}
3455:
3387:
3296:
3237:
3049:between two phases
6711:
6512:2018-11-02 at the
6012:2027/uc1.$ b242250
5983:Annalen der Physik
5924:
5836:
5758:
5731:
5729:
5328:
5256:
5176:
5142:
5080:
5025:
4988:
4978:depends weakly on
4942:
4918:
4894:
4867:
4770:
4725:
4688:
4655:
4629:
4596:
4459:is linear, and so
4449:
4421:
4387:
4355:
4335:
4311:
4276:
4256:
4236:
4158:
4122:
4038:
4005:
3977:
3952:
3916:
3832:
3794:
3769:
3736:
3572:
3459:
3407:
3349:
3324:
3268:
3209:
3193:
3119:
3099:
3079:
3059:
3035:
2989:
2936:
2912:
2892:
2828:we can obtain the
2818:
2752:
2728:
2708:
2686:
2626:
2537:
2510:
2445:
2319:
2210:
2186:
2162:
2142:
2069:
2010:
1967:
1947:
1919:
1896:
1886:, the temperature
1876:
1856:
1815:
1783:
1753:Clapeyron equation
1739:
1673:
1613:
1574:
1540:
1494:
1447:
1405:
1347:
1320:
1300:
1228:reversible process
1216:
1167:
1118:
1051:
1031:
1003:
944:partial derivative
932:
912:
831:
742:
620:
597:
570:
546:
508:
466:
386:
347:
320:
293:
269:
245:
207:
6671:978-0-471-86256-7
6652:978-94-010-2642-0
6629:978-0-7506-3215-7
6600:10.1119/1.4858403
6551:Physical Review X
6290:978-0-07-011927-7
6205:978-0-07-068286-3
6156:serc.carleton.edu
6136:978-0-521-09101-5
5958:Lee–Kesler method
5707:
5618:
5568:
5489:
5446:
5407:
5358:second derivative
5352:Second derivative
5323:
5313:
5309:
5296:
5232:
5125:
5068:
4991:{\displaystyle T}
4945:{\displaystyle T}
4921:{\displaystyle T}
4858:
4658:{\displaystyle T}
4591:
4531:
4461:linear regression
4403:coexistence curve
4390:{\displaystyle L}
4371:specific enthalpy
4358:{\displaystyle L}
4338:{\displaystyle c}
4314:{\displaystyle T}
4279:{\displaystyle R}
4266:is the pressure,
4259:{\displaystyle P}
4225:
4215:
4117:
4077:
3877:
3721:
3718:
3714:
3698:
3683:
3677:
3673:
3642:
3638:
3618:
3562:
3542:
3520:
3504:
3419:
3402:
3319:
3266:
3253:
3188:
3164:
3151:
3122:{\displaystyle L}
3102:{\displaystyle L}
3047:coexistence curve
2939:{\displaystyle L}
2915:{\displaystyle L}
2890:
2860:
2813:
2791:
2755:{\displaystyle T}
2731:{\displaystyle R}
2718:is the pressure,
2711:{\displaystyle P}
2681:
2662:
2623:
2606:
2603:
2593:
2570:
2534:
2507:
2493:of the gas phase
2440:
2417:
2360:
2213:{\displaystyle M}
2189:{\displaystyle v}
2165:{\displaystyle s}
2024:coexistence curve
1899:{\displaystyle T}
1879:{\displaystyle L}
1795:coexistence curve
1734:
1712:
1608:
1535:
1489:
1350:{\displaystyle h}
1336:specific enthalpy
1323:{\displaystyle u}
1113:
1090:
1041:to a final phase
989:
935:{\displaystyle P}
888:
807:
718:
671:
623:{\displaystyle T}
600:{\displaystyle v}
573:{\displaystyle s}
503:
461:
431:
389:{\displaystyle R}
296:{\displaystyle T}
272:{\displaystyle L}
257:coexistence curve
202:
179:
157:
118:coexistence curve
16:(Redirected from
6769:
6726:
6720:
6718:
6717:
6712:
6707:
6691:
6675:
6656:
6633:
6604:
6603:
6575:
6569:
6568:
6566:
6542:
6536:
6535:
6523:
6517:
6502:
6496:
6495:
6469:
6457:
6451:
6449:
6447:
6415:
6409:
6407:
6406:
6384:
6378:
6377:
6365:
6359:
6358:
6356:
6354:
6331:
6325:
6324:
6322:
6321:
6301:
6295:
6294:
6272:
6263:
6262:
6260:
6259:
6243:
6237:
6236:
6234:
6232:
6216:
6210:
6209:
6191:
6168:
6167:
6165:
6163:
6147:
6141:
6140:
6122:
6116:
6115:
6092:Thomson, William
6088:
6082:
6081:
6076:
6074:
6059:Feynman, Richard
6055:
6049:
6048:
6041:
6023:
6017:
6016:
6014:
5974:
5953:Antoine equation
5933:
5931:
5930:
5925:
5923:
5922:
5910:
5905:
5897:
5883:
5866:
5865:
5845:
5843:
5842:
5837:
5835:
5834:
5822:
5817:
5809:
5795:
5767:
5765:
5764:
5759:
5757:
5756:
5740:
5738:
5737:
5732:
5730:
5723:
5719:
5718:
5717:
5712:
5708:
5706:
5702:
5696:
5692:
5686:
5676:
5675:
5663:
5662:
5650:
5649:
5637:
5636:
5619:
5617:
5616:
5615:
5603:
5602:
5589:
5580:
5574:
5570:
5569:
5567:
5563:
5557:
5553:
5547:
5542:
5541:
5532:
5531:
5519:
5518:
5509:
5508:
5490:
5485:
5484:
5483:
5468:
5467:
5454:
5447:
5445:
5444:
5443:
5431:
5430:
5417:
5408:
5406:
5405:
5404:
5395:
5389:
5385:
5384:
5379:
5372:
5337:
5335:
5334:
5329:
5324:
5321:
5319:
5314:
5311:
5307:
5297:
5295:
5287:
5279:
5265:
5263:
5262:
5257:
5255:
5248:
5243:
5242:
5230:
5229:
5228:
5185:
5183:
5182:
5177:
5175:
5151:
5149:
5148:
5143:
5141:
5134:
5123:
5122:
5121:
5089:
5087:
5086:
5081:
5069:
5067:
5052:
5034:
5032:
5031:
5026:
5024:
4997:
4995:
4994:
4989:
4951:
4949:
4948:
4943:
4927:
4925:
4924:
4919:
4903:
4901:
4900:
4895:
4893:
4892:
4876:
4874:
4873:
4868:
4863:
4859:
4857:
4846:
4838:
4811:
4810:
4779:
4777:
4776:
4771:
4760:
4759:
4734:
4732:
4731:
4726:
4724:
4723:
4697:
4695:
4694:
4689:
4687:
4686:
4664:
4662:
4661:
4656:
4638:
4636:
4635:
4630:
4628:
4627:
4605:
4603:
4602:
4597:
4592:
4590:
4589:
4588:
4579:
4578:
4568:
4567:
4566:
4548:
4547:
4537:
4532:
4530:
4526:
4520:
4519:
4518:
4509:
4503:
4458:
4456:
4455:
4450:
4445:
4430:
4428:
4427:
4422:
4396:
4394:
4393:
4388:
4364:
4362:
4361:
4356:
4344:
4342:
4341:
4336:
4320:
4318:
4317:
4312:
4296:
4285:
4283:
4282:
4277:
4265:
4263:
4262:
4257:
4245:
4243:
4242:
4237:
4226:
4218:
4216:
4208:
4167:
4165:
4164:
4159:
4157:
4146:
4132:where lowercase
4131:
4129:
4128:
4123:
4118:
4116:
4112:
4101:
4083:
4078:
4076:
4072:
4066:
4062:
4056:
4047:
4045:
4044:
4039:
4034:
4026:
4014:
4012:
4011:
4006:
4001:
3986:
3984:
3983:
3978:
3973:
3961:
3959:
3958:
3953:
3951:
3940:
3925:
3923:
3922:
3917:
3909:
3898:
3884:
3878:
3876:
3872:
3866:
3862:
3856:
3841:
3839:
3838:
3833:
3825:
3803:
3801:
3800:
3795:
3790:
3778:
3776:
3775:
3770:
3765:
3745:
3743:
3742:
3737:
3732:
3728:
3727:
3723:
3722:
3720:
3719:
3716:
3712:
3704:
3699:
3691:
3684:
3679:
3678:
3675:
3671:
3662:
3643:
3640:
3636:
3620:
3619:
3616:
3603:
3581:
3579:
3578:
3573:
3568:
3564:
3563:
3561:
3560:
3548:
3543:
3541:
3540:
3528:
3521:
3513:
3505:
3503:
3502:
3493:
3492:
3483:
3468:
3466:
3465:
3460:
3454:
3453:
3452:
3442:
3441:
3440:
3424:
3420:
3412:
3403:
3395:
3386:
3385:
3384:
3374:
3373:
3372:
3356:
3355:
3333:
3331:
3330:
3325:
3320:
3318:
3317:
3308:
3304:
3298:
3295:
3294:
3293:
3283:
3282:
3281:
3267:
3259:
3254:
3249:
3245:
3239:
3236:
3235:
3234:
3224:
3223:
3222:
3202:
3200:
3199:
3194:
3189:
3187:
3186:
3177:
3173:
3167:
3165:
3157:
3152:
3147:
3143:
3137:
3128:
3126:
3125:
3120:
3108:
3106:
3105:
3100:
3088:
3086:
3085:
3080:
3068:
3066:
3065:
3060:
3044:
3042:
3041:
3036:
3031:
3030:
3018:
3017:
2998:
2996:
2995:
2990:
2985:
2984:
2972:
2971:
2949:
2945:
2943:
2942:
2937:
2921:
2919:
2918:
2913:
2901:
2899:
2898:
2893:
2891:
2889:
2885:
2884:
2874:
2866:
2861:
2859:
2855:
2849:
2845:
2839:
2827:
2825:
2824:
2819:
2814:
2812:
2797:
2792:
2790:
2786:
2780:
2776:
2770:
2761:
2759:
2758:
2753:
2737:
2735:
2734:
2729:
2717:
2715:
2714:
2709:
2695:
2693:
2692:
2687:
2682:
2677:
2669:
2664:
2663:
2660:
2635:
2633:
2632:
2627:
2625:
2624:
2621:
2612:
2608:
2607:
2605:
2604:
2601:
2595:
2594:
2591:
2585:
2572:
2571:
2568:
2546:
2544:
2543:
2538:
2536:
2535:
2532:
2519:
2517:
2516:
2511:
2509:
2508:
2505:
2471:phase transition
2454:
2452:
2451:
2446:
2441:
2439:
2431:
2423:
2418:
2416:
2415:
2414:
2402:
2401:
2391:
2390:
2389:
2377:
2376:
2366:
2361:
2359:
2355:
2349:
2345:
2339:
2328:
2326:
2325:
2320:
2309:
2300:
2299:
2287:
2286:
2268:
2259:
2258:
2246:
2245:
2219:
2217:
2216:
2211:
2195:
2193:
2192:
2187:
2172:is the specific
2171:
2169:
2168:
2163:
2151:
2149:
2148:
2143:
2135:
2120:
2096:
2078:
2076:
2075:
2070:
2065:
2064:
2055:
2047:
2046:
2037:
2019:
2017:
2016:
2011:
2006:
2005:
1993:
1992:
1976:
1974:
1973:
1968:
1956:
1954:
1953:
1948:
1928:
1926:
1925:
1920:
1905:
1903:
1902:
1897:
1885:
1883:
1882:
1877:
1865:
1863:
1862:
1857:
1839:
1825:to the function
1824:
1822:
1821:
1816:
1792:
1790:
1789:
1784:
1779:
1774:
1766:
1748:
1746:
1745:
1740:
1735:
1733:
1718:
1713:
1711:
1707:
1701:
1697:
1691:
1682:
1680:
1679:
1674:
1666:
1649:
1644:
1636:
1622:
1620:
1619:
1614:
1609:
1601:
1583:
1581:
1580:
1575:
1549:
1547:
1546:
1541:
1536:
1531:
1523:
1503:
1501:
1500:
1495:
1490:
1485:
1481:
1475:
1467:
1456:
1454:
1453:
1448:
1440:
1425:
1414:
1412:
1411:
1406:
1398:
1383:
1368:
1356:
1354:
1353:
1348:
1329:
1327:
1326:
1321:
1309:
1307:
1306:
1301:
1293:
1278:
1245:
1225:
1223:
1222:
1217:
1215:
1214:
1202:
1201:
1176:
1174:
1173:
1168:
1166:
1165:
1153:
1152:
1127:
1125:
1124:
1119:
1114:
1112:
1104:
1096:
1091:
1089:
1085:
1079:
1075:
1069:
1060:
1058:
1057:
1052:
1040:
1038:
1037:
1032:
1012:
1010:
1009:
1004:
996:
990:
988:
984:
978:
974:
968:
960:
948:total derivative
941:
939:
938:
933:
921:
919:
918:
913:
905:
899:
898:
893:
889:
887:
879:
871:
858:
845:Maxwell relation
840:
838:
837:
832:
824:
818:
817:
812:
808:
806:
798:
790:
777:
756:Phase transition
751:
749:
748:
743:
735:
729:
728:
723:
719:
717:
709:
701:
688:
682:
681:
676:
672:
670:
662:
654:
641:
629:
627:
626:
621:
606:
604:
603:
598:
579:
577:
576:
571:
559:specific entropy
542:phase boundaries
517:
515:
514:
509:
504:
499:
491:
475:
473:
472:
467:
462:
460:
456:
455:
445:
437:
432:
430:
426:
420:
416:
410:
395:
393:
392:
387:
359:specific entropy
356:
354:
353:
348:
329:
327:
326:
321:
302:
300:
299:
294:
279:is the specific
278:
276:
275:
270:
254:
252:
251:
246:
241:
236:
228:
216:
214:
213:
208:
203:
201:
193:
185:
180:
178:
163:
158:
156:
152:
146:
142:
136:
55:phases of matter
51:phase transition
21:
6777:
6776:
6772:
6771:
6770:
6768:
6767:
6766:
6747:1850 in science
6742:1849 in science
6732:
6731:
6730:
6729:
6723:Carnot function
6703:
6698:
6695:
6694:
6692:
6688:
6683:
6678:
6672:
6653:
6630:
6613:
6608:
6607:
6576:
6572:
6543:
6539:
6528:"Mass of a Car"
6524:
6520:
6514:Wayback Machine
6503:
6499:
6467:
6461:Lawrence, M. G.
6458:
6454:
6416:
6412:
6385:
6381:
6366:
6362:
6352:
6350:
6348:
6332:
6328:
6319:
6317:
6302:
6298:
6291:
6273:
6266:
6257:
6255:
6244:
6240:
6230:
6228:
6217:
6213:
6206:
6192:
6171:
6161:
6159:
6148:
6144:
6137:
6123:
6119:
6089:
6085:
6072:
6070:
6056:
6052:
6035:
6024:
6020:
5975:
5971:
5966:
5944:
5918:
5914:
5906:
5901:
5893:
5879:
5861:
5857:
5855:
5852:
5851:
5830:
5826:
5818:
5813:
5805:
5791:
5777:
5774:
5773:
5752:
5748:
5746:
5743:
5742:
5728:
5727:
5713:
5698:
5697:
5688:
5687:
5685:
5681:
5680:
5668:
5664:
5658:
5654:
5642:
5638:
5632:
5628:
5624:
5620:
5611:
5607:
5598:
5594:
5593:
5588:
5581:
5579:
5576:
5575:
5559:
5558:
5549:
5548:
5546:
5537:
5533:
5527:
5523:
5514:
5510:
5504:
5500:
5476:
5472:
5460:
5456:
5455:
5453:
5452:
5448:
5439:
5435:
5426:
5422:
5421:
5416:
5409:
5400:
5396:
5391:
5390:
5380:
5375:
5374:
5373:
5371:
5367:
5365:
5362:
5361:
5354:
5320:
5315:
5310:
5288:
5280:
5278:
5276:
5273:
5272:
5269:
5244:
5238:
5234:
5233:
5221:
5217:
5197:
5194:
5193:
5171:
5159:
5156:
5155:
5130:
5126:
5117:
5113:
5099:
5096:
5095:
5056:
5051:
5040:
5037:
5036:
5017:
5015:
5012:
5011:
5008:
4983:
4980:
4979:
4937:
4934:
4933:
4930:degrees Celsius
4913:
4910:
4909:
4888:
4884:
4882:
4879:
4878:
4847:
4839:
4837:
4833:
4806:
4802:
4800:
4797:
4796:
4755:
4751:
4749:
4746:
4745:
4742:
4739:of water vapor.
4719:
4715:
4713:
4710:
4709:
4682:
4678:
4676:
4673:
4672:
4650:
4647:
4646:
4623:
4619:
4617:
4614:
4613:
4584:
4580:
4574:
4570:
4569:
4562:
4558:
4543:
4539:
4538:
4536:
4522:
4521:
4514:
4510:
4505:
4504:
4502:
4500:
4497:
4496:
4469:
4441:
4436:
4433:
4432:
4410:
4407:
4406:
4382:
4379:
4378:
4350:
4347:
4346:
4330:
4327:
4326:
4306:
4303:
4302:
4297:divided by the
4294:
4271:
4268:
4267:
4251:
4248:
4247:
4217:
4207:
4190:
4187:
4186:
4183:
4178:
4170:specific volume
4150:
4139:
4137:
4134:
4133:
4105:
4094:
4087:
4082:
4068:
4067:
4058:
4057:
4055:
4053:
4050:
4049:
4030:
4022:
4020:
4017:
4016:
3997:
3992:
3989:
3988:
3969:
3967:
3964:
3963:
3944:
3933:
3931:
3928:
3927:
3902:
3891:
3880:
3868:
3867:
3858:
3857:
3855:
3847:
3844:
3843:
3821:
3809:
3806:
3805:
3786:
3784:
3781:
3780:
3761:
3759:
3756:
3755:
3751:
3715:
3708:
3703:
3690:
3689:
3685:
3674:
3663:
3661:
3657:
3653:
3639:
3615:
3611:
3609:
3606:
3605:
3601:
3556:
3552:
3547:
3536:
3532:
3527:
3526:
3522:
3512:
3498:
3494:
3488:
3484:
3482:
3474:
3471:
3470:
3448:
3444:
3443:
3436:
3432:
3425:
3411:
3408:
3394:
3380:
3376:
3375:
3368:
3364:
3357:
3351:
3350:
3338:
3335:
3334:
3313:
3309:
3300:
3299:
3297:
3289:
3285:
3284:
3277:
3273:
3272:
3258:
3241:
3240:
3238:
3230:
3226:
3225:
3218:
3214:
3213:
3207:
3204:
3203:
3182:
3178:
3169:
3168:
3166:
3156:
3139:
3138:
3136:
3134:
3131:
3130:
3114:
3111:
3110:
3094:
3091:
3090:
3074:
3071:
3070:
3054:
3051:
3050:
3026:
3022:
3013:
3009:
3004:
3001:
3000:
2980:
2976:
2967:
2963:
2958:
2955:
2954:
2947:
2931:
2928:
2927:
2907:
2904:
2903:
2880:
2876:
2875:
2867:
2865:
2851:
2850:
2841:
2840:
2838:
2836:
2833:
2832:
2801:
2796:
2782:
2781:
2772:
2771:
2769:
2767:
2764:
2763:
2747:
2744:
2743:
2723:
2720:
2719:
2703:
2700:
2699:
2670:
2668:
2659:
2655:
2653:
2650:
2649:
2620:
2616:
2600:
2596:
2590:
2586:
2584:
2577:
2573:
2567:
2563:
2552:
2549:
2548:
2531:
2527:
2525:
2522:
2521:
2504:
2500:
2498:
2495:
2494:
2491:specific volume
2467:
2432:
2424:
2422:
2410:
2406:
2397:
2393:
2392:
2385:
2381:
2372:
2368:
2367:
2365:
2351:
2350:
2341:
2340:
2338:
2336:
2333:
2332:
2305:
2295:
2291:
2282:
2278:
2264:
2254:
2250:
2241:
2237:
2229:
2226:
2225:
2205:
2202:
2201:
2198:specific volume
2181:
2178:
2177:
2157:
2154:
2153:
2131:
2116:
2092:
2090:
2087:
2086:
2060:
2056:
2051:
2042:
2038:
2033:
2031:
2028:
2027:
2001:
1997:
1988:
1984:
1982:
1979:
1978:
1962:
1959:
1958:
1942:
1939:
1938:
1935:
1911:
1908:
1907:
1891:
1888:
1887:
1871:
1868:
1867:
1835:
1830:
1827:
1826:
1801:
1798:
1797:
1775:
1770:
1762:
1760:
1757:
1756:
1722:
1717:
1703:
1702:
1693:
1692:
1690:
1688:
1685:
1684:
1662:
1645:
1640:
1632:
1630:
1627:
1626:
1600:
1589:
1586:
1585:
1560:
1557:
1556:
1524:
1522:
1511:
1508:
1507:
1477:
1476:
1474:
1463:
1461:
1458:
1457:
1436:
1421:
1419:
1416:
1415:
1394:
1379:
1364:
1362:
1359:
1358:
1342:
1339:
1338:
1332:internal energy
1315:
1312:
1311:
1289:
1274:
1241:
1239:
1236:
1235:
1210:
1206:
1197:
1193:
1182:
1179:
1178:
1161:
1157:
1148:
1144:
1133:
1130:
1129:
1105:
1097:
1095:
1081:
1080:
1071:
1070:
1068:
1066:
1063:
1062:
1046:
1043:
1042:
1026:
1023:
1022:
992:
980:
979:
970:
969:
967:
956:
954:
951:
950:
927:
924:
923:
901:
894:
880:
872:
870:
866:
865:
854:
852:
849:
848:
820:
813:
799:
791:
789:
785:
784:
773:
771:
768:
767:
731:
724:
710:
702:
700:
696:
695:
684:
677:
663:
655:
653:
649:
648:
637:
635:
632:
631:
615:
612:
611:
592:
589:
588:
586:specific volume
565:
562:
561:
555:state postulate
551:
526:
492:
490:
482:
479:
478:
451:
447:
446:
438:
436:
422:
421:
412:
411:
409:
407:
404:
403:
381:
378:
377:
367:
339:
336:
335:
332:specific volume
312:
309:
308:
288:
285:
284:
264:
261:
260:
237:
232:
224:
222:
219:
218:
194:
186:
184:
167:
162:
148:
147:
138:
137:
135:
133:
130:
129:
98:
93:
59:Rudolf Clausius
35:
28:
23:
22:
15:
12:
11:
5:
6775:
6765:
6764:
6759:
6754:
6749:
6744:
6728:
6727:
6710:
6706:
6702:
6685:
6684:
6682:
6679:
6677:
6676:
6670:
6657:
6651:
6634:
6628:
6614:
6612:
6609:
6606:
6605:
6586:(4): 301–305.
6570:
6537:
6518:
6497:
6478:(2): 225–233.
6452:
6430:(4): 601–609.
6410:
6404:10.2172/548871
6379:
6360:
6346:
6326:
6296:
6289:
6264:
6238:
6211:
6204:
6196:Thermodynamics
6169:
6142:
6135:
6117:
6106:(5): 541–574.
6083:
6050:
6018:
5989:(4): 500–524.
5968:
5967:
5965:
5962:
5961:
5960:
5955:
5950:
5943:
5940:
5921:
5917:
5913:
5909:
5904:
5900:
5896:
5892:
5889:
5886:
5882:
5878:
5875:
5872:
5869:
5864:
5860:
5833:
5829:
5825:
5821:
5816:
5812:
5808:
5804:
5801:
5798:
5794:
5790:
5787:
5784:
5781:
5755:
5751:
5726:
5722:
5716:
5711:
5705:
5701:
5695:
5691:
5684:
5679:
5674:
5671:
5667:
5661:
5657:
5653:
5648:
5645:
5641:
5635:
5631:
5627:
5623:
5614:
5610:
5606:
5601:
5597:
5592:
5587:
5584:
5582:
5578:
5577:
5573:
5566:
5562:
5556:
5552:
5545:
5540:
5536:
5530:
5526:
5522:
5517:
5513:
5507:
5503:
5499:
5496:
5493:
5488:
5482:
5479:
5475:
5471:
5466:
5463:
5459:
5451:
5442:
5438:
5434:
5429:
5425:
5420:
5415:
5412:
5410:
5403:
5399:
5394:
5388:
5383:
5378:
5370:
5369:
5353:
5350:
5327:
5318:
5306:
5303:
5300:
5294:
5291:
5286:
5283:
5268:
5267:
5254:
5251:
5247:
5241:
5237:
5227:
5224:
5220:
5216:
5213:
5210:
5207:
5204:
5201:
5191:
5174:
5169:
5166:
5163:
5153:
5140:
5137:
5133:
5129:
5120:
5116:
5112:
5109:
5106:
5103:
5092:
5079:
5076:
5073:
5066:
5063:
5059:
5055:
5050:
5047:
5044:
5023:
5020:
5007:
5004:
4987:
4941:
4917:
4891:
4887:
4866:
4862:
4856:
4853:
4850:
4845:
4842:
4836:
4832:
4829:
4826:
4823:
4820:
4817:
4814:
4809:
4805:
4769:
4766:
4763:
4758:
4754:
4741:
4740:
4722:
4718:
4707:
4685:
4681:
4670:
4654:
4644:
4626:
4622:
4610:
4595:
4587:
4583:
4577:
4573:
4565:
4561:
4557:
4554:
4551:
4546:
4542:
4535:
4529:
4525:
4517:
4513:
4508:
4468:
4465:
4448:
4444:
4440:
4420:
4417:
4414:
4386:
4354:
4334:
4310:
4275:
4255:
4235:
4232:
4229:
4224:
4221:
4214:
4211:
4206:
4203:
4200:
4197:
4194:
4182:
4179:
4177:
4174:
4156:
4153:
4149:
4145:
4142:
4121:
4115:
4111:
4108:
4104:
4100:
4097:
4093:
4090:
4086:
4081:
4075:
4071:
4065:
4061:
4037:
4033:
4029:
4025:
4004:
4000:
3996:
3976:
3972:
3950:
3947:
3943:
3939:
3936:
3915:
3912:
3908:
3905:
3901:
3897:
3894:
3890:
3887:
3883:
3875:
3871:
3865:
3861:
3854:
3851:
3831:
3828:
3824:
3819:
3816:
3813:
3793:
3789:
3768:
3764:
3750:
3747:
3735:
3731:
3726:
3711:
3707:
3702:
3697:
3694:
3688:
3682:
3670:
3666:
3660:
3656:
3652:
3649:
3646:
3635:
3632:
3629:
3626:
3623:
3614:
3571:
3567:
3559:
3555:
3551:
3546:
3539:
3535:
3531:
3525:
3519:
3516:
3511:
3508:
3501:
3497:
3491:
3487:
3481:
3478:
3458:
3451:
3447:
3439:
3435:
3431:
3428:
3423:
3418:
3415:
3410:
3406:
3401:
3398:
3393:
3390:
3383:
3379:
3371:
3367:
3363:
3360:
3354:
3348:
3345:
3342:
3323:
3316:
3312:
3307:
3303:
3292:
3288:
3280:
3276:
3271:
3265:
3262:
3257:
3252:
3248:
3244:
3233:
3229:
3221:
3217:
3212:
3192:
3185:
3181:
3176:
3172:
3163:
3160:
3155:
3150:
3146:
3142:
3118:
3098:
3089:. In general,
3078:
3058:
3034:
3029:
3025:
3021:
3016:
3012:
3008:
2988:
2983:
2979:
2975:
2970:
2966:
2962:
2946:in kJ/mol and
2935:
2911:
2888:
2883:
2879:
2873:
2870:
2864:
2858:
2854:
2848:
2844:
2817:
2811:
2808:
2804:
2800:
2795:
2789:
2785:
2779:
2775:
2751:
2727:
2707:
2685:
2680:
2676:
2673:
2667:
2658:
2619:
2615:
2611:
2599:
2589:
2583:
2580:
2576:
2566:
2562:
2559:
2556:
2530:
2503:
2466:
2463:
2444:
2438:
2435:
2430:
2427:
2421:
2413:
2409:
2405:
2400:
2396:
2388:
2384:
2380:
2375:
2371:
2364:
2358:
2354:
2348:
2344:
2318:
2315:
2312:
2308:
2303:
2298:
2294:
2290:
2285:
2281:
2277:
2274:
2271:
2267:
2262:
2257:
2253:
2249:
2244:
2240:
2236:
2233:
2209:
2185:
2161:
2141:
2138:
2134:
2129:
2126:
2123:
2119:
2114:
2111:
2108:
2105:
2102:
2099:
2095:
2068:
2063:
2059:
2054:
2050:
2045:
2041:
2036:
2009:
2004:
2000:
1996:
1991:
1987:
1966:
1946:
1934:
1931:
1918:
1915:
1895:
1875:
1855:
1852:
1849:
1845:
1842:
1838:
1834:
1814:
1811:
1808:
1805:
1782:
1778:
1773:
1769:
1765:
1738:
1732:
1729:
1725:
1721:
1716:
1710:
1706:
1700:
1696:
1672:
1669:
1665:
1661:
1658:
1655:
1652:
1648:
1643:
1639:
1635:
1612:
1607:
1604:
1599:
1596:
1593:
1573:
1570:
1567:
1564:
1539:
1534:
1530:
1527:
1521:
1518:
1515:
1493:
1488:
1484:
1480:
1473:
1470:
1466:
1446:
1443:
1439:
1434:
1431:
1428:
1424:
1404:
1401:
1397:
1392:
1389:
1386:
1382:
1377:
1374:
1371:
1367:
1346:
1319:
1299:
1296:
1292:
1287:
1284:
1281:
1277:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1244:
1213:
1209:
1205:
1200:
1196:
1192:
1189:
1186:
1164:
1160:
1156:
1151:
1147:
1143:
1140:
1137:
1117:
1111:
1108:
1103:
1100:
1094:
1088:
1084:
1078:
1074:
1050:
1030:
1002:
999:
995:
987:
983:
977:
973:
966:
963:
959:
931:
911:
908:
904:
897:
892:
886:
883:
878:
875:
869:
864:
861:
857:
830:
827:
823:
816:
811:
805:
802:
797:
794:
788:
783:
780:
776:
741:
738:
734:
727:
722:
716:
713:
708:
705:
699:
694:
691:
687:
680:
675:
669:
666:
661:
658:
652:
647:
644:
640:
619:
596:
569:
550:
547:
525:
522:
507:
502:
498:
495:
489:
486:
465:
459:
454:
450:
444:
441:
435:
429:
425:
419:
415:
385:
366:
363:
346:
343:
319:
316:
292:
268:
259:at any point,
244:
240:
235:
231:
227:
206:
200:
197:
192:
189:
183:
177:
174:
170:
166:
161:
155:
151:
145:
141:
97:
94:
92:
89:
47:vapor pressure
26:
9:
6:
4:
3:
2:
6774:
6763:
6760:
6758:
6755:
6753:
6750:
6748:
6745:
6743:
6740:
6739:
6737:
6724:
6708:
6704:
6700:
6690:
6686:
6673:
6667:
6663:
6658:
6654:
6648:
6644:
6640:
6635:
6631:
6625:
6621:
6616:
6615:
6601:
6597:
6593:
6589:
6585:
6581:
6574:
6565:
6560:
6557:(1): 011025.
6556:
6552:
6548:
6541:
6533:
6529:
6522:
6515:
6511:
6507:
6501:
6493:
6489:
6485:
6481:
6477:
6473:
6466:
6462:
6456:
6446:
6441:
6437:
6433:
6429:
6425:
6421:
6414:
6405:
6400:
6396:
6392:
6391:
6383:
6375:
6371:
6364:
6349:
6347:9780495126713
6343:
6339:
6338:
6330:
6316:on 2007-06-07
6315:
6311:
6307:
6300:
6292:
6286:
6282:
6278:
6271:
6269:
6253:
6249:
6242:
6226:
6222:
6215:
6207:
6201:
6197:
6190:
6188:
6186:
6184:
6182:
6180:
6178:
6176:
6174:
6157:
6153:
6146:
6138:
6132:
6128:
6121:
6113:
6109:
6105:
6101:
6097:
6093:
6087:
6080:
6068:
6064:
6060:
6054:
6046:
6043:(in French).
6042:
6039:
6034:
6029:
6022:
6013:
6008:
6004:
6000:
5996:
5992:
5988:
5985:(in German).
5984:
5980:
5973:
5969:
5959:
5956:
5954:
5951:
5949:
5946:
5945:
5939:
5937:
5919:
5911:
5902:
5898:
5884:
5880:
5876:
5870:
5867:
5862:
5858:
5849:
5831:
5823:
5814:
5810:
5796:
5792:
5788:
5782:
5779:
5771:
5753:
5749:
5724:
5720:
5714:
5709:
5703:
5693:
5682:
5672:
5669:
5665:
5659:
5655:
5651:
5646:
5643:
5639:
5633:
5629:
5621:
5612:
5608:
5604:
5599:
5595:
5590:
5585:
5583:
5571:
5564:
5554:
5538:
5534:
5528:
5524:
5520:
5515:
5511:
5505:
5501:
5494:
5491:
5486:
5480:
5477:
5473:
5469:
5464:
5461:
5457:
5449:
5440:
5436:
5432:
5427:
5423:
5418:
5413:
5411:
5401:
5397:
5386:
5381:
5359:
5349:
5347:
5343:
5338:
5325:
5316:
5304:
5301:
5298:
5292:
5284:
5245:
5239:
5225:
5222:
5218:
5214:
5211:
5208:
5205:
5202:
5192:
5189:
5167:
5164:
5161:
5154:
5131:
5118:
5114:
5110:
5107:
5104:
5101:
5094:
5093:
5091:
5077:
5074:
5064:
5057:
5053:
5048:
5045:
5021:
5003:
5001:
5000:exponentially
4985:
4977:
4973:
4968:
4966:
4962:
4961:Magnus–Tetens
4958:
4953:
4939:
4931:
4915:
4907:
4889:
4885:
4864:
4860:
4854:
4851:
4848:
4843:
4840:
4834:
4830:
4827:
4824:
4821:
4815:
4807:
4803:
4794:
4792:
4788:
4784:
4764:
4756:
4752:
4738:
4720:
4716:
4708:
4705:
4701:
4683:
4679:
4671:
4668:
4652:
4645:
4642:
4624:
4620:
4612:
4611:
4609:
4606:
4593:
4585:
4581:
4575:
4571:
4563:
4559:
4552:
4544:
4540:
4533:
4527:
4515:
4511:
4494:
4492:
4488:
4484:
4483:precipitation
4480:
4476:
4473:
4464:
4462:
4446:
4442:
4438:
4418:
4415:
4412:
4404:
4400:
4384:
4376:
4372:
4368:
4352:
4332:
4324:
4308:
4300:
4293:
4289:
4273:
4253:
4233:
4230:
4227:
4222:
4219:
4212:
4209:
4204:
4201:
4198:
4195:
4192:
4173:
4171:
4154:
4151:
4147:
4143:
4140:
4119:
4109:
4106:
4102:
4098:
4095:
4088:
4084:
4079:
4073:
4063:
4035:
4031:
4027:
4002:
3998:
3994:
3974:
3948:
3945:
3941:
3937:
3934:
3913:
3906:
3903:
3899:
3895:
3892:
3885:
3873:
3863:
3852:
3849:
3829:
3826:
3817:
3814:
3811:
3791:
3766:
3746:
3733:
3729:
3724:
3709:
3705:
3700:
3695:
3692:
3686:
3680:
3668:
3664:
3658:
3654:
3650:
3647:
3644:
3633:
3630:
3624:
3612:
3599:
3595:
3591:
3587:
3582:
3569:
3565:
3557:
3553:
3549:
3544:
3537:
3533:
3529:
3523:
3517:
3514:
3509:
3506:
3499:
3495:
3489:
3485:
3479:
3476:
3456:
3449:
3445:
3437:
3433:
3429:
3426:
3421:
3416:
3413:
3404:
3399:
3396:
3391:
3388:
3381:
3377:
3369:
3365:
3361:
3358:
3346:
3343:
3340:
3321:
3314:
3310:
3305:
3290:
3286:
3278:
3274:
3269:
3263:
3260:
3255:
3250:
3246:
3231:
3227:
3219:
3215:
3210:
3190:
3183:
3179:
3174:
3161:
3158:
3153:
3148:
3144:
3116:
3096:
3076:
3056:
3048:
3027:
3023:
3019:
3014:
3010:
2981:
2977:
2973:
2968:
2964:
2951:
2933:
2925:
2909:
2886:
2881:
2877:
2871:
2868:
2862:
2856:
2846:
2831:
2815:
2809:
2802:
2798:
2793:
2787:
2777:
2749:
2741:
2725:
2705:
2696:
2683:
2678:
2674:
2671:
2665:
2656:
2647:
2646:ideal gas law
2643:
2639:
2617:
2613:
2609:
2597:
2587:
2581:
2578:
2574:
2564:
2560:
2557:
2528:
2501:
2492:
2488:
2484:
2480:
2476:
2472:
2462:
2460:
2455:
2442:
2436:
2428:
2419:
2411:
2407:
2403:
2398:
2394:
2386:
2382:
2378:
2373:
2369:
2362:
2356:
2346:
2329:
2316:
2313:
2310:
2296:
2292:
2288:
2283:
2279:
2272:
2269:
2255:
2251:
2247:
2242:
2238:
2231:
2223:
2207:
2199:
2183:
2175:
2159:
2136:
2127:
2124:
2121:
2112:
2109:
2103:
2100:
2097:
2084:
2079:
2066:
2061:
2057:
2048:
2043:
2039:
2025:
2020:
2007:
2002:
1998:
1994:
1989:
1985:
1964:
1944:
1930:
1916:
1893:
1873:
1850:
1843:
1836:
1832:
1809:
1803:
1796:
1780:
1771:
1767:
1754:
1749:
1736:
1730:
1723:
1719:
1714:
1708:
1698:
1683:), we obtain
1670:
1663:
1659:
1653:
1650:
1641:
1637:
1623:
1610:
1605:
1602:
1597:
1594:
1571:
1565:
1562:
1555:
1550:
1537:
1532:
1528:
1519:
1516:
1504:
1491:
1486:
1482:
1471:
1468:
1444:
1441:
1432:
1429:
1426:
1402:
1399:
1390:
1387:
1384:
1375:
1372:
1369:
1344:
1337:
1333:
1317:
1297:
1294:
1285:
1282:
1279:
1270:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1233:
1229:
1211:
1207:
1203:
1198:
1194:
1190:
1187:
1162:
1158:
1154:
1149:
1145:
1141:
1138:
1115:
1109:
1101:
1092:
1086:
1076:
1048:
1028:
1020:
1016:
1000:
997:
985:
975:
964:
961:
949:
945:
929:
909:
906:
895:
890:
884:
876:
867:
862:
859:
846:
841:
828:
825:
814:
809:
803:
795:
786:
781:
778:
766:. Therefore,
765:
761:
760:closed system
757:
752:
739:
736:
725:
720:
714:
706:
697:
692:
689:
678:
673:
667:
659:
650:
645:
642:
617:
610:
594:
587:
583:
567:
560:
556:
543:
539:
535:
534:phase diagram
530:
521:
518:
505:
500:
496:
493:
487:
484:
476:
463:
457:
452:
448:
442:
439:
433:
427:
417:
401:
399:
383:
376:
372:
371:ideal gas law
362:
360:
344:
333:
317:
306:
290:
282:
266:
258:
242:
233:
229:
204:
198:
190:
181:
175:
168:
164:
159:
153:
143:
127:
123:
119:
115:
111:
107:
103:
88:
85:
84:James Thomson
80:
78:
74:
73:
68:
64:
60:
56:
52:
48:
44:
40:
33:
32:ideal gas law
19:
6722:
6689:
6661:
6642:
6619:
6611:Bibliography
6583:
6579:
6573:
6554:
6550:
6540:
6531:
6521:
6500:
6475:
6471:
6455:
6427:
6423:
6413:
6389:
6382:
6373:
6369:
6363:
6351:. Retrieved
6336:
6329:
6318:. Retrieved
6314:the original
6309:
6299:
6276:
6256:. Retrieved
6252:HyperPhysics
6251:
6241:
6229:. Retrieved
6224:
6214:
6195:
6160:. Retrieved
6155:
6145:
6126:
6120:
6103:
6099:
6086:
6078:
6071:. Retrieved
6066:
6053:
6044:
6031:
6021:
5986:
5982:
5972:
5355:
5339:
5270:
5009:
4969:
4960:
4956:
4954:
4781:
4743:
4737:gas constant
4607:
4495:
4479:meteorologic
4470:
4375:vaporization
4292:gas constant
4184:
4176:Applications
4169:
3752:
3593:
3583:
2952:
2829:
2697:
2638:temperatures
2468:
2456:
2330:
2224:) to obtain
2080:
2021:
1936:
1752:
1750:
1624:
1551:
1505:
1357:, we obtain
1061:, to obtain
1015:factored out
842:
753:
552:
519:
477:
402:
397:
368:
113:
109:
99:
81:
70:
53:between two
38:
36:
6073:13 December
6036: [
5342:ice skating
4972:denominator
4704:evaporation
4667:temperature
4475:water vapor
4472:Atmospheric
4399:sublimation
4290:(i.e., the
3586:equilibrium
2083:Gibbs–Duhem
1019:integrating
609:temperature
582:homogeneous
557:, take the
524:Derivations
305:temperature
281:latent heat
106:temperature
77:Lord Kelvin
67:Sadi Carnot
6736:Categories
6376:: 153–190.
6320:2007-10-11
6258:2007-10-16
6231:1 February
6162:1 February
5964:References
5271:we obtain
4299:molar mass
2648:, so that
2222:molar mass
553:Using the
532:A typical
373:using the
91:Definition
6664:. Wiley.
5871:−
5859:κ
5780:α
5666:κ
5652:−
5640:κ
5605:−
5535:α
5521:−
5512:α
5492:−
5470:−
5433:−
5302:−
5290:Δ
5282:Δ
5223:−
5215:×
5209:−
5200:Δ
5111:×
5072:Δ
5062:Δ
5043:Δ
5019:Δ
4831:
4706:of water,
4416:
4205:−
4196:
4148:−
4103:−
3942:−
3900:−
3701:−
3659:−
3651:
3645:⋅
3631:≅
3545:−
3510:−
3507:≅
3480:
3405:⋅
3392:−
3389:≅
3344:
3270:∫
3256:≅
3211:∫
3154:≅
3077:β
3057:α
2807:Δ
2614:≈
2582:−
2555:Δ
2475:gas phase
2469:When the
2434:Δ
2426:Δ
2412:α
2404:−
2399:β
2387:α
2379:−
2374:β
2297:α
2289:−
2284:β
2256:α
2248:−
2243:β
2232:−
2110:−
2098:μ
2085:relation
2062:β
2058:μ
2044:α
2040:μ
2003:β
1999:μ
1990:α
1986:μ
1965:β
1945:α
1914:Δ
1848:Δ
1728:Δ
1668:Δ
1657:Δ
1592:Δ
1569:Δ
1526:Δ
1514:Δ
1283:−
1262:δ
1253:δ
1212:α
1204:−
1199:β
1191:≡
1185:Δ
1163:α
1155:−
1150:β
1142:≡
1136:Δ
1107:Δ
1099:Δ
1049:β
1029:α
882:∂
874:∂
801:∂
793:∂
712:∂
704:∂
665:∂
657:∂
342:Δ
315:Δ
196:Δ
188:Δ
173:Δ
6510:Archived
6463:(2005).
6094:(1849).
6061:(1963).
5942:See also
4976:exponent
4487:dynamics
4155:′
4144:″
4110:′
4099:″
3949:′
3938:″
3907:′
3896:″
2642:pressure
764:pressure
126:tangents
102:pressure
6588:Bibcode
6480:Bibcode
6432:Bibcode
6353:3 April
5991:Bibcode
5934:is the
5846:is the
5768:is the
5346:thimble
5188:kelvins
5006:Example
4974:of the
4793:formula
4735:is the
4698:is the
4365:is the
4321:is the
4286:is the
3594:without
2922:is the
2738:is the
2636:at low
2220:is the
2196:is the
2174:entropy
2152:(where
1793:of the
1330:is the
1234:holds:
357:is the
330:is the
303:is the
124:of the
69:in his
6668:
6649:
6626:
6344:
6287:
6202:
6133:
5850:, and
5308:
5231:
5124:
4957:Magnus
4928:is in
4908:, and
4904:is in
4877:where
4855:243.04
4841:17.625
4825:6.1094
4791:Magnus
4783:August
4608:where
4325:, and
4246:where
3926:where
3713:
3672:
3637:
2742:, and
2698:where
2479:liquid
2200:, and
1584:gives
1310:where
1128:where
922:where
847:gives
580:for a
217:where
6681:Notes
6468:(PDF)
6040:]
4787:Roche
4493:) is
4373:) of
2640:. If
2483:solid
1017:when
758:in a
122:slope
100:On a
41:, in
6666:ISBN
6647:ISBN
6624:ISBN
6395:NOAA
6355:2020
6342:ISBN
6285:ISBN
6233:2023
6200:ISBN
6164:2023
6131:ISBN
6075:2023
5305:13.5
5212:9.05
5108:3.34
4431:and
4369:(or
3681:8.31
3069:and
2999:and
2953:Let
1957:and
1177:and
607:and
61:and
37:The
6596:doi
6559:doi
6488:doi
6440:doi
6399:doi
6374:XIV
6108:doi
6007:hdl
5999:doi
5987:155
5312:MPa
5168:273
4959:or
4906:hPa
4828:exp
4702:of
4665:is
4639:is
4301:),
3710:373
3669:700
3648:exp
3641:bar
3617:vap
3588:or
3469:or
2481:or
6738::
6641:.
6594:.
6584:82
6582:.
6555:11
6553:.
6549:.
6530:.
6508:.
6486:.
6476:86
6474:.
6470:.
6438:.
6428:35
6426:.
6422:.
6397:,
6393:,
6372:.
6308:.
6267:^
6250:.
6223:.
6172:^
6154:.
6104:16
6102:.
6098:.
6077:.
6065:.
6045:23
6038:fr
6030:.
6005:.
5997:.
5938:.
5219:10
5190:),
5115:10
4967:.
4413:ln
4193:ln
3665:40
3477:ln
3341:ln
2461:.
2317:0.
2176:,
2026:,
630:.
307:,
6709:T
6705:/
6701:1
6674:.
6655:.
6632:.
6602:.
6598::
6590::
6567:.
6561::
6534:.
6516:.
6494:.
6490::
6482::
6448:.
6442::
6434::
6401::
6357:.
6323:.
6293:.
6261:.
6235:.
6208:.
6166:.
6139:.
6114:.
6110::
6015:.
6009::
6001::
5993::
5920:T
5916:)
5912:P
5908:d
5903:/
5899:v
5895:d
5891:(
5888:)
5885:v
5881:/
5877:1
5874:(
5868:=
5863:T
5832:P
5828:)
5824:T
5820:d
5815:/
5811:v
5807:d
5803:(
5800:)
5797:v
5793:/
5789:1
5786:(
5783:=
5754:p
5750:c
5725:,
5721:]
5715:2
5710:)
5704:T
5700:d
5694:P
5690:d
5683:(
5678:)
5673:1
5670:T
5660:1
5656:v
5647:2
5644:T
5634:2
5630:v
5626:(
5622:[
5613:1
5609:v
5600:2
5596:v
5591:1
5586:+
5572:]
5565:T
5561:d
5555:P
5551:d
5544:)
5539:1
5529:1
5525:v
5516:2
5506:2
5502:v
5498:(
5495:2
5487:T
5481:1
5478:p
5474:c
5465:2
5462:p
5458:c
5450:[
5441:1
5437:v
5428:2
5424:v
5419:1
5414:=
5402:2
5398:T
5393:d
5387:P
5382:2
5377:d
5326:.
5322:K
5317:/
5299:=
5293:T
5285:P
5253:g
5250:k
5246:/
5240:3
5236:m
5226:5
5206:=
5203:v
5173:K
5165:=
5162:T
5139:g
5136:k
5132:/
5128:J
5119:5
5105:=
5102:L
5078:,
5075:T
5065:v
5058:T
5054:L
5049:=
5046:P
5022:T
4986:T
4940:T
4916:T
4890:s
4886:e
4865:,
4861:)
4852:+
4849:T
4844:T
4835:(
4822:=
4819:)
4816:T
4813:(
4808:s
4804:e
4789:–
4785:–
4768:)
4765:T
4762:(
4757:v
4753:L
4721:v
4717:R
4684:v
4680:L
4669:,
4653:T
4643:,
4625:s
4621:e
4594:,
4586:2
4582:T
4576:v
4572:R
4564:s
4560:e
4556:)
4553:T
4550:(
4545:v
4541:L
4534:=
4528:T
4524:d
4516:s
4512:e
4507:d
4447:T
4443:/
4439:1
4419:P
4385:L
4353:L
4333:c
4309:T
4295:R
4274:R
4254:P
4234:,
4231:c
4228:+
4223:T
4220:1
4213:R
4210:L
4202:=
4199:P
4152:v
4141:v
4120:,
4114:)
4107:v
4096:v
4092:(
4089:T
4085:L
4080:=
4074:T
4070:d
4064:p
4060:d
4036:T
4032:/
4028:T
4024:d
4003:Q
3999:/
3995:W
3975:x
3971:d
3946:V
3935:V
3914:,
3911:)
3904:V
3893:V
3889:(
3886:T
3882:d
3874:T
3870:d
3864:p
3860:d
3853:=
3850:W
3830:,
3827:x
3823:d
3818:L
3815:=
3812:Q
3792:T
3788:d
3767:x
3763:d
3734:.
3730:]
3725:)
3717:K
3706:1
3696:T
3693:1
3687:(
3676:K
3655:[
3634:1
3628:)
3625:T
3622:(
3613:P
3602:R
3570:.
3566:)
3558:1
3554:T
3550:1
3538:2
3534:T
3530:1
3524:(
3518:R
3515:L
3500:1
3496:P
3490:2
3486:P
3457:,
3450:2
3446:T
3438:1
3434:T
3430:=
3427:T
3422:|
3417:T
3414:1
3400:R
3397:L
3382:2
3378:P
3370:1
3366:P
3362:=
3359:P
3353:|
3347:P
3322:,
3315:2
3311:T
3306:T
3302:d
3291:2
3287:T
3279:1
3275:T
3264:R
3261:L
3251:P
3247:P
3243:d
3232:2
3228:P
3220:1
3216:P
3191:,
3184:2
3180:T
3175:T
3171:d
3162:R
3159:L
3149:P
3145:P
3141:d
3117:L
3097:L
3033:)
3028:2
3024:T
3020:,
3015:2
3011:P
3007:(
2987:)
2982:1
2978:T
2974:,
2969:1
2965:P
2961:(
2948:R
2934:L
2910:L
2887:R
2882:2
2878:T
2872:L
2869:P
2863:=
2857:T
2853:d
2847:P
2843:d
2816:,
2810:v
2803:T
2799:L
2794:=
2788:T
2784:d
2778:P
2774:d
2750:T
2726:R
2706:P
2684:,
2679:P
2675:T
2672:R
2666:=
2661:g
2657:v
2622:g
2618:v
2610:)
2602:g
2598:v
2592:c
2588:v
2579:1
2575:(
2569:g
2565:v
2561:=
2558:v
2533:c
2529:v
2506:g
2502:v
2443:,
2437:v
2429:s
2420:=
2408:v
2395:v
2383:s
2370:s
2363:=
2357:T
2353:d
2347:P
2343:d
2314:=
2311:P
2307:d
2302:)
2293:v
2280:v
2276:(
2273:+
2270:T
2266:d
2261:)
2252:s
2239:s
2235:(
2208:M
2184:v
2160:s
2140:)
2137:P
2133:d
2128:v
2125:+
2122:T
2118:d
2113:s
2107:(
2104:M
2101:=
2094:d
2067:.
2053:d
2049:=
2035:d
2008:.
1995:=
1917:v
1894:T
1874:L
1854:)
1851:v
1844:T
1841:(
1837:/
1833:L
1813:)
1810:T
1807:(
1804:P
1781:T
1777:d
1772:/
1768:P
1764:d
1737:.
1731:v
1724:T
1720:L
1715:=
1709:T
1705:d
1699:P
1695:d
1671:v
1664:/
1660:s
1654:=
1651:T
1647:d
1642:/
1638:P
1634:d
1611:.
1606:T
1603:L
1598:=
1595:s
1572:h
1566:=
1563:L
1538:.
1533:T
1529:h
1520:=
1517:s
1492:.
1487:T
1483:h
1479:d
1472:=
1469:s
1465:d
1445:,
1442:s
1438:d
1433:T
1430:=
1427:h
1423:d
1403:,
1400:P
1396:d
1391:v
1388:+
1385:s
1381:d
1376:T
1373:=
1370:h
1366:d
1345:h
1318:u
1298:,
1295:v
1291:d
1286:P
1280:s
1276:d
1271:T
1268:=
1265:w
1259:+
1256:q
1250:=
1247:u
1243:d
1208:v
1195:v
1188:v
1159:s
1146:s
1139:s
1116:,
1110:v
1102:s
1093:=
1087:T
1083:d
1077:P
1073:d
1001:,
998:v
994:d
986:T
982:d
976:P
972:d
965:=
962:s
958:d
930:P
910:,
907:v
903:d
896:v
891:)
885:T
877:P
868:(
863:=
860:s
856:d
829:.
826:v
822:d
815:T
810:)
804:v
796:s
787:(
782:=
779:s
775:d
740:.
737:T
733:d
726:v
721:)
715:T
707:s
698:(
693:+
690:v
686:d
679:T
674:)
668:v
660:s
651:(
646:=
643:s
639:d
618:T
595:v
568:s
544:.
506:.
501:P
497:T
494:R
488:=
485:V
464:,
458:R
453:2
449:T
443:L
440:P
434:=
428:T
424:d
418:P
414:d
398:V
384:R
345:s
318:v
291:T
267:L
243:T
239:d
234:/
230:P
226:d
205:,
199:v
191:s
182:=
176:v
169:T
165:L
160:=
154:T
150:d
144:P
140:d
114:T
112:–
110:P
108:(
104:–
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
