81:. At vapor–liquid equilibrium, a liquid with individual components in certain concentrations will have an equilibrium vapor in which the concentrations or partial pressures of the vapor components have certain values depending on all of the liquid component concentrations and the temperature. The converse is also true: if a vapor with components at certain concentrations or partial pressures is in vapor–liquid equilibrium with its liquid, then the component concentrations in the liquid will be determined dependent on the vapor concentrations and on the temperature. The equilibrium concentration of each component in the liquid phase is often different from its concentration (or vapor pressure) in the vapor phase, but there is a relationship. The VLE concentration data can be determined experimentally or approximated with the help of theories such as
1453:
1303:
1435:
complexity, such boiling-point diagrams are rarely seen. Alternatively, the three-dimensional curved surfaces can be represented on a two-dimensional graph by the use of curved isotherm lines at graduated intervals, similar to iso-altitude lines on a map. Two sets of such isotherm lines are needed on such a two-dimensional graph: one set for the bubble point surface and another set for the dew point surface.
3308:
1779:
1771:
1426:
of the triangle represent a mixture of the two components at each end of the edge. Any point inside the triangle represents the composition of a mixture of all three components. The mole fraction of each component would correspond to where a point lies along a line starting at that component's corner and perpendicular to the opposite edge. The
1323:. At any given temperature (or pressure) where both phases are present, vapor with a certain mole fraction is in equilibrium with liquid with a certain mole fraction. The two mole fractions often differ. These vapor and liquid mole fractions are represented by two points on the same horizontal isotherm (constant
1425:
can be used. Two of the dimensions would be used to represent the composition mole fractions, and the third dimension would be the temperature. Using two dimensions, the composition can be represented as an equilateral triangle in which each corner represents one of the pure components. The edges
2661:
For many kinds of mixtures, particularly where there is interaction between components beyond simply the effects of dilution, Raoult's law does not work well for determining the shapes of the curves in the boiling point or VLE diagrams. Even in such mixtures, there are usually still differences in
142:
When a temperature is reached such that the sum of the equilibrium vapor pressures of the liquid components becomes equal to the total pressure of the system (it is otherwise smaller), then vapor bubbles generated from the liquid begin to displace the gas that was maintaining the overall pressure,
1434:
data would become curved surfaces inside a triangular prism, which connect the three boiling points on the vertical temperature "axes". Each face of this triangular prism would represent a two-dimensional boiling-point diagram for the corresponding binary mixture. Due to their three-dimensional
1958:
For each component in a binary mixture, one could make a vapor–liquid equilibrium diagram. Such a diagram would graph liquid mole fraction on a horizontal axis and vapor mole fraction on a vertical axis. In such VLE diagrams, liquid mole fractions for components 1 and 2 can be represented as
2566:
s as a function of liquid composition in terms of mole fractions have been determined, these values can be inserted into the above equations to obtain corresponding vapor compositions in terms of mole fractions. When this is finished over a complete range of liquid mole fractions and their
1475:
constant. There can be VLE data for mixtures of four or more components, but such a boiling-point diagram is hard to show in either tabular or graphical form. For such multi-component mixtures, as well as binary mixtures, the vapor–liquid equilibrium data are represented in terms of
149:
of the liquid mixture at the given pressure. (It is assumed that the total pressure is held steady by adjusting the total volume of the system to accommodate the specific volume changes that accompany boiling.) The boiling point at an overall pressure of 1 atm is called the
134:
Binary mixtures are those having two components. Three-component mixtures are called ternary mixtures. There can be VLE data for mixtures with even more components, but such data is often hard to show graphically. VLE data is a function of the total pressure, such as
2080:) needed to distill a given composition binary feed mixture into one distillate fraction and one bottoms fraction. Corrections can also be made to take into account the incomplete efficiency of each tray in a distillation column when compared to a theoretical plate.
2345:
may be used to approximate how the vapor pressure varies as a function of temperature. This makes each of the partial pressures dependent on temperature also regardless of whether Raoult's law applies or not. When Raoult's law is valid these expressions become:
2662:
the vapor and liquid equilibrium concentrations at most points, and distillation is often still useful for separating components at least partially. For such mixtures, empirical data is typically used in determining such boiling point and VLE diagrams.
1176:
1327:) line. When an entire range of temperatures vs. vapor and liquid mole fractions is graphed, two (usually curved) lines result. The lower one, representing the mole fraction of the boiling liquid at various temperatures, is called the
2494:
At boiling temperatures if Raoult's law applies, a number of the preceding equations in this section can be combined to give the following expressions for vapor mole fractions as a function of liquid mole fractions and temperature:
1918:
1923:
which is a measure of the relative ease or difficulty of separating the two components. Large-scale industrial distillation is rarely undertaken if the relative volatility is less than 1.05 with the volatile component being
975:
633:
326:
1310:
The preceding equilibrium equations are typically applied for each phase (liquid or vapor) individually, but the result can be plotted in a single diagram. In a binary boiling-point diagram, temperature (
2245:
is approximately valid for mixtures of components between which there is very little interaction other than the effect of dilution by the other components. Examples of such mixtures includes mixtures of
1702:
1212:
Binary mixture VLE data at a certain overall pressure, such as 1 atm, showing mole fraction vapor and liquid concentrations when boiling at various temperatures can be shown as a two-dimensional
182:
If the liquid and vapor are pure, in that they consist of only one molecular component and no impurities, then the equilibrium state between the two phases is described by the following equations:
1404:. When they meet, they meet tangently; the dew-point temperature always lies above the boiling-point temperature for a given composition when they are not equal. The meeting point is called an
1629:
224:
168:
describes when vapor–liquid equilibrium is possible, and its properties. Much of the analysis depends on whether the vapor and liquid consist of a single component, or if they are mixtures.
905:
863:
266:
1539:
1068:
1027:
748:
692:
517:
481:
2666:
have done a significant amount of research trying to develop equations for correlating and/or predicting VLE data for various kinds of mixtures which do not obey Raoult's law well.
527:) within the liquid and vapor, respectively. In other words, the temperature, pressure and molar Gibbs free energy are the same between the two phases when they are at equilibrium.
1767:
are correlated empirically or theoretically in terms of temperature, pressure and phase compositions in the form of equations, tables or graph such as the DePriester charts.
2119:
441:
414:
383:
356:
2790:
818:
In a multicomponent system, where the vapor and liquid consist of more than one type of compounds, describing the equilibrium state is more complicated. For all components
2237:
1088:
1732:
1410:
for that particular pair of substances. It is characterized by an azeotrope temperature and an azeotropic composition, often expressed as a mole fraction. There can be
800:
77:(a part of the total gas pressure) if any other gas(es) are present with the vapor. The equilibrium vapor pressure of a liquid is in general strongly dependent on
2953:
3750:
1803:
3245:
1951:
2851:
2094:
At boiling and higher temperatures the sum of the individual component partial pressures becomes equal to the overall pressure, which can symbolized as
108:, which is a particular specialty of chemical engineers. Distillation is a process used to separate or partially separate components in a mixture by
2258:
in many ways, so there is little attraction or repulsion between the molecules. Raoult's law states that for components 1, 2, etc. in a mixture:
3056:
2886:
2861:
2827:
912:
540:
1422:
2753:
273:
1452:
1381:, pure component 1). The temperatures at those two points correspond to the boiling points of each of the two pure components.
123:. The mole fraction of a given component of a mixture in a particular phase (either the vapor or the liquid phase) is the number of
2946:
2742:
1643:
3147:
2466:
may not be mathematically analytical (i.e., may require a numerical solution or approximation). For a binary mixture at a given
2342:
3512:
2748:
2738:
1071:
3755:
3582:
3507:
3238:
530:
An equivalent, more common way to express the vapor–liquid equilibrium condition in a pure system is by using the concept of
3694:
3522:
3322:
3109:
172:
3704:
2939:
1582:
3018:
188:
3125:
1471:
The tendency of a given chemical species to partition itself preferentially between liquid and vapor phases is the
870:
828:
231:
1490:
3765:
3231:
3033:
3001:
1032:
991:
697:
641:
66:
2073:
1082:) within the liquid and vapor, respectively, for each phase. The partial molar Gibbs free energy is defined by:
3729:
3628:
3258:
3174:
3169:
3130:
486:
450:
116:. Distillation takes advantage of differences in concentrations of components in the liquid and vapor phases.
3623:
2711:
2675:
1939:
105:
3383:
2333:
Recall from the first section that vapor pressures of liquids are very dependent on temperature. Thus the
119:
In mixtures containing two or more components, the concentrations of each component are often expressed as
3648:
3638:
3388:
3193:
3083:
2778:
3760:
3186:
3181:
3159:
3104:
3011:
2991:
2462:
can be solved for to give the liquid mixture's boiling point or bubble point, although the solution for
3567:
3327:
3068:
3051:
2962:
1735:
1334:. The upper one, representing the mole fraction of the vapor at various temperatures, is called the
1171:{\displaystyle {\bar {G}}_{i}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\partial G}{\partial n_{i}}}}
822:
in the system, the equilibrium state between the two phases is described by the following equations:
152:
97:
35:
2097:
1950:
1302:
419:
392:
361:
334:
3542:
3434:
3424:
3337:
3292:
3164:
3142:
3028:
2986:
2817:
2696:
1977:
respectively, and vapor mole fractions of the corresponding components are commonly represented as
1791:
1481:
1447:
1421:
If one wants to represent a VLE data for a three-component mixture as a boiling point "diagram", a
1197:
1079:
524:
2784:
2215:
3689:
3618:
3452:
3152:
3043:
2976:
2773:
2680:
2491:) and this function can be shown on a two-dimensional graph like a binary boiling point diagram.
1710:
1186:
1075:
2571:
function of vapor composition mole fractions. This function effectively acts as the dew point
1341:
These two curves necessarily meet where the mixture becomes purely one component, namely where
3719:
3714:
3684:
3643:
3532:
3484:
3469:
3362:
3332:
3198:
3078:
3006:
2996:
2882:
2857:
2823:
2726:
2706:
2691:
2663:
2255:
2251:
2077:
520:
1384:
For certain pairs of substances, the two curves also coincide at some point strictly between
3674:
3297:
3137:
3023:
1746:
74:
47:
771:
3664:
3517:
3254:
3099:
2686:
2242:
2125:
2089:
1635:
1574:
86:
82:
2789:
George
Schlowsky, Alan Erickson, and Thomas A. Schafer, Modular Process Systems, Inc.,
1220:. The mole fraction of component 1 in the mixture can be represented by the symbol
3462:
3457:
3414:
3347:
3342:
3210:
1472:
165:
128:
90:
70:
31:
3744:
3699:
3679:
3602:
3562:
3497:
3429:
3352:
3063:
2716:
1913:{\displaystyle \alpha ={\frac {K_{i}}{K_{j}}}={\frac {(y_{i}/x_{i})}{(y_{j}/x_{j})}}}
145:
120:
62:
3724:
3597:
3592:
3587:
3552:
3502:
3419:
2721:
1427:
1329:
113:
101:
55:
1778:
1770:
3633:
3527:
3439:
3307:
2701:
2316:, etc. are the vapor pressures of components 1, 2, etc. when they are pure, and
444:
124:
78:
51:
2769:
17:
3572:
3547:
3474:
3444:
3378:
3357:
3073:
1461:
136:
2405:
At boiling temperatures if Raoult's law applies, the total pressure becomes:
2981:
2756:(Chemical Engineering Dept., Prof. Richard Rowley, Brigham Young University)
1431:
1414:, where the azeotrope temperature is at a maximum in the boiling curves, or
1406:
807:
3223:
3709:
3537:
1754:
1465:
1418:, where the azeotrope temperature is at a minimum in the boiling curves.
1270:
In multi-component mixtures in general with n components, this becomes:
1213:
751:
531:
386:
2759:
2567:
corresponding temperatures, one effectively obtains a temperature
2330:, etc. are mole fractions of the corresponding component in the liquid.
3669:
3557:
3492:
3409:
3404:
2337:
pure vapor pressures for each component are a function of temperature (
1457:
970:{\displaystyle {\bar {G}}_{i}^{\text{liq}}={\bar {G}}_{i}^{\text{vap}}}
628:{\displaystyle f^{\text{liq}}(T_{s},P_{s})=f^{\text{vap}}(T_{s},P_{s})}
534:. Under this view, equilibrium is described by the following equation:
109:
2931:
2785:
Can. J. Chem. Eng. ternary and multicomponent systems from binary ones
131:
divided by the total number of moles of all components in that phase.
3278:
2247:
2770:
Some VLE data sets and diagrams for mixtures of 30 common components
2035:
Such VLE diagrams are square with a diagonal line running from the (
321:{\displaystyle {\tilde {G}}^{\text{liq}}={\tilde {G}}^{\text{vap}}}
3287:
3273:
1777:
1769:
1451:
1782:
K-Values for systems of light hydrocarbons High
Temperature Range
754:
of the liquid and vapor, respectively, at the system temperature
96:
Such vapor–liquid equilibrium information is useful in designing
2763:
1774:
K-Values for systems of light hydrocarbons Low
Temperature Range
143:
and the mixture is said to boil. This temperature is called the
3227:
2935:
3283:
1697:{\displaystyle K_{i}={\frac {\gamma _{i}P_{i}^{\star }}{P}}}
2791:
Operations & Maintenance - Generating your own VLE Data
2341:): For example, commonly for a pure liquid component, the
1995:. Similarly for binary mixtures in these VLE diagrams:
1229:. The mole fraction of component 2, represented by
2779:
Where can I get the vapor-liquid phase equilibrium data?
2781:
Reference to the various phase equilibrium data sources
774:
2793:, Chemical Engineering, March 1995, McGraw-Hill, Inc.
2762:(Describes the extensive VLE database available from
2749:
2218:
2100:
1938:
values are widely used in the design calculations of
1806:
1713:
1646:
1585:
1493:
1091:
1035:
994:
988:
are the temperature and pressure for each phase, and
915:
873:
831:
700:
644:
543:
489:
453:
422:
395:
364:
337:
276:
234:
191:
160:
Thermodynamic description of vapor–liquid equilibrium
3657:
3611:
3483:
3397:
3371:
3315:
3266:
3118:
3092:
3042:
2969:
2458:such as 1 atm and a given liquid composition,
27:
Concentration of a vapor in contact with its liquid
2231:
2113:
2076:to determine the number of equilibrium stages (or
1912:
1726:
1696:
1623:
1533:
1170:
1062:
1021:
969:
899:
857:
794:
742:
686:
627:
511:
475:
435:
408:
377:
350:
320:
260:
218:
1624:{\displaystyle K_{i}={\frac {P_{i}^{\star }}{P}}}
1942:columns for distilling multicomponent mixtures.
139:or at the pressure the process is conducted at.
2592:and the above equations can be expressed as:
219:{\displaystyle P^{\text{liq}}=P^{\text{vap}}\,}
2877:Seader, J. D. & Henley, Ernest J. (1998).
2156:Then for each component in the vapor phase:
3239:
2947:
900:{\displaystyle T^{\text{liq}}=T^{\text{vap}}}
858:{\displaystyle P^{\text{liq}}=P^{\text{vap}}}
768:. It is often convenient to use the quantity
261:{\displaystyle T^{\text{liq}}=T^{\text{vap}}}
8:
2072:These types of VLE diagrams are used in the
1790:values for the two components is called the
1534:{\displaystyle K_{i}={\frac {y_{i}}{x_{i}}}}
1063:{\displaystyle {\bar {G}}_{i}^{\text{vap}}}
1022:{\displaystyle {\bar {G}}_{i}^{\text{liq}}}
743:{\displaystyle f^{\text{vap}}(T_{s},P_{s})}
687:{\displaystyle f^{\text{liq}}(T_{s},P_{s})}
3246:
3232:
3224:
2954:
2940:
2932:
65:in contact with its liquid, especially at
2223:
2217:
2105:
2099:
1898:
1889:
1883:
1865:
1856:
1850:
1840:
1829:
1819:
1813:
1805:
1718:
1712:
1682:
1677:
1667:
1660:
1651:
1645:
1610:
1605:
1599:
1590:
1584:
1558:are the mole fractions of component
1523:
1513:
1507:
1498:
1492:
1314:) (or sometimes pressure) is graphed vs.
1159:
1141:
1123:
1122:
1117:
1115:
1114:
1105:
1094:
1093:
1090:
1054:
1049:
1038:
1037:
1034:
1013:
1008:
997:
996:
993:
961:
956:
945:
944:
934:
929:
918:
917:
914:
891:
878:
872:
849:
836:
830:
784:
773:
731:
718:
705:
699:
675:
662:
649:
643:
616:
603:
590:
574:
561:
548:
542:
512:{\displaystyle {\tilde {G}}^{\text{vap}}}
503:
492:
491:
488:
476:{\displaystyle {\tilde {G}}^{\text{liq}}}
467:
456:
455:
452:
427:
421:
400:
394:
369:
363:
342:
336:
312:
301:
300:
290:
279:
278:
275:
252:
239:
233:
215:
209:
196:
190:
2811:
2809:
2807:
2239:= partial pressure of component 2, etc.
1949:
1301:
2850:Perry, R.H.; Green, D.W., eds. (1997).
2845:
2843:
2841:
2839:
2803:
1928:and the less volatile component being
1786:For binary mixtures, the ratio of the
2760:NIST Standard Reference Database 103b
1442:values and relative volatility values
7:
2853:Perry's Chemical Engineers' Handbook
2745:(scroll down to Relative Volatility)
3751:Chemical engineering thermodynamics
2906:Chemical Engineering Thermodynamics
2900:
2898:
2683:(includes a collection of VLE data)
2212:= partial pressure of component 1,
1152:
1144:
1130:
1127:
1124:
46:) describes the distribution of a
25:
2919:Chem. Eng. Prog. Symposium Series
2743:University of Newcastle upon Tyne
2578:In the case of a binary mixture,
2128:would be in effect as follows:
1946:Vapor–liquid equilibrium diagrams
447:within the liquid and vapor, and
69:, is often expressed in terms of
3306:
1954:Vapor-Liquid Equilibrium Diagram
1482:vapor–liquid distribution ratios
1247:in a binary mixture as follows:
2575:function of vapor composition.
1072:partial molar Gibbs free energy
178:Pure (single-component) systems
2114:{\displaystyle P_{\text{tot}}}
1904:
1876:
1871:
1843:
1099:
1043:
1002:
950:
923:
737:
711:
681:
655:
622:
596:
580:
554:
497:
461:
436:{\displaystyle T^{\text{vap}}}
409:{\displaystyle T^{\text{liq}}}
378:{\displaystyle P^{\text{vap}}}
351:{\displaystyle P^{\text{liq}}}
306:
284:
1:
3695:Macroscopic quantum phenomena
2879:Separation Process Principles
2856:(7th ed.). McGraw-hill.
2822:(1st ed.). McGraw-hill.
389:within the liquid and vapor,
3705:Order and disorder (physics)
2232:{\displaystyle P_{\text{2}}}
1460:Best-Fit Curve), Mixture of
1361:, pure component 2) or
2343:Clausius–Clapeyron relation
1727:{\displaystyle \gamma _{i}}
112:(vaporization) followed by
3782:
2087:
1445:
1416:minimum-boiling azeotropes
1412:maximum-boiling azeotropes
170:
127:of that component in that
3304:
3034:Thermodynamic equilibrium
2904:Balzhiser et al. (1972),
2816:Kister, Henry Z. (1992).
2477:can become a function of
1189:) Gibbs free energy, and
3730:Thermo-dielectric effect
3629:Enthalpy of vaporization
3323:Bose–Einstein condensate
3187:Distribution coefficient
3131:Hammett acidity function
3110:Liquid–liquid extraction
3019:Le Chatelier's principle
2772:, a small subset of the
2473:, the bubble point
2069:) corner for reference.
1760:The values of the ratio
173:Wong–Sandler mixing rule
63:concentration of a vapor
40:vapor–liquid equilibrium
3624:Enthalpy of sublimation
2739:Distillation Principals
2712:Margules activity model
2676:Continuous distillation
2124:Under such conditions,
1940:continuous distillation
1456:K Values Diagram (with
1423:three-dimensional graph
156:of the liquid mixture.
106:fractional distillation
3639:Latent internal energy
3389:Color-glass condensate
3148:Coordination complexes
3084:Thermodynamic activity
2562:Once the bubble point
2233:
2115:
1955:
1914:
1783:
1775:
1728:
1698:
1625:
1535:
1468:
1307:
1208:Boiling-point diagrams
1172:
1064:
1023:
971:
901:
859:
814:Multicomponent systems
796:
795:{\textstyle \phi =f/P}
744:
688:
629:
513:
477:
437:
410:
379:
352:
322:
262:
220:
3756:Equilibrium chemistry
3449:Magnetically ordered
3160:Dissociation constant
3105:Equilibrium unfolding
2992:Equilibrium chemistry
2234:
2116:
1953:
1915:
1781:
1773:
1729:
1699:
1626:
1536:
1455:
1306:Boiling-point diagram
1305:
1218:boiling-point diagram
1173:
1078:(units of energy per
1065:
1024:
972:
902:
860:
797:
745:
689:
630:
523:(units of energy per
514:
478:
438:
411:
380:
353:
323:
263:
221:
171:Further information:
3328:Fermionic condensate
3069:Predominance diagram
3052:Equilibrium constant
2216:
2098:
2074:McCabe–Thiele method
1804:
1736:activity coefficient
1711:
1644:
1583:
1491:
1089:
1033:
992:
913:
871:
829:
806:, which is 1 for an
804:fugacity coefficient
802:, the dimensionless
772:
698:
642:
541:
487:
451:
420:
393:
362:
335:
274:
232:
189:
153:normal boiling point
36:chemical engineering
3543:Chemical ionization
3435:Programmable matter
3425:Quantum spin liquid
3293:Supercritical fluid
3143:Binding selectivity
3119:Specific equilibria
3029:Reversible reaction
2987:Dynamic equilibrium
2963:Chemical equilibria
2881:. New York: Wiley.
2819:Distillation Design
2254:, relatively inert
2083:
1792:relative volatility
1687:
1615:
1448:Relative volatility
1198:amount of substance
1080:amount of substance
1059:
1018:
966:
939:
525:amount of substance
521:Gibbs free energies
3690:Leidenfrost effect
3619:Enthalpy of fusion
3384:Quark–gluon plasma
3153:Macrocyclic effect
2977:Chemical stability
2917:DePriester, C.L.,
2774:Dortmund Data Bank
2754:VLE Thermodynamics
2681:Dortmund Data Bank
2664:Chemical engineers
2229:
2111:
2078:theoretical plates
1956:
1910:
1784:
1776:
1724:
1694:
1673:
1621:
1601:
1531:
1469:
1308:
1200:of component
1168:
1076:chemical potential
1060:
1036:
1019:
995:
967:
943:
916:
897:
855:
792:
740:
684:
625:
509:
473:
433:
406:
375:
348:
318:
258:
216:
73:, which will be a
3738:
3737:
3720:Superheated vapor
3715:Superconductivity
3685:Equation of state
3533:Flash evaporation
3485:Phase transitions
3470:String-net liquid
3363:Photonic molecule
3333:Degenerate matter
3221:
3220:
3199:Common-ion effect
3126:Acid dissociation
3079:Reaction quotient
2997:Equilibrium stage
2888:978-0-471-58626-5
2863:978-0-07-049841-9
2829:978-0-07-034909-4
2741:by Ming T. Tham,
2727:Superheated steam
2707:Van Laar equation
2692:Flash evaporation
2226:
2108:
2052:) corner to the (
1908:
1835:
1692:
1619:
1529:
1166:
1140:
1135:
1113:
1102:
1057:
1046:
1016:
1005:
964:
953:
937:
926:
894:
881:
852:
839:
708:
652:
593:
551:
506:
500:
470:
464:
430:
403:
372:
345:
315:
309:
293:
287:
255:
242:
212:
199:
16:(Redirected from
3773:
3766:Phases of matter
3675:Compressed fluid
3310:
3255:States of matter
3248:
3241:
3234:
3225:
3138:Binding constant
3024:Phase separation
2956:
2949:
2942:
2933:
2926:
2915:
2909:
2902:
2893:
2892:
2874:
2868:
2867:
2847:
2834:
2833:
2813:
2250:, which are non-
2238:
2236:
2235:
2230:
2228:
2227:
2224:
2120:
2118:
2117:
2112:
2110:
2109:
2106:
2068:
2051:
2031:
2014:
1994:
1985:
1976:
1967:
1937:
1931:
1927:
1919:
1917:
1916:
1911:
1909:
1907:
1903:
1902:
1893:
1888:
1887:
1874:
1870:
1869:
1860:
1855:
1854:
1841:
1836:
1834:
1833:
1824:
1823:
1814:
1797:
1789:
1766:
1752:
1747:partial pressure
1744:
1733:
1731:
1730:
1725:
1723:
1722:
1703:
1701:
1700:
1695:
1693:
1688:
1686:
1681:
1672:
1671:
1661:
1656:
1655:
1630:
1628:
1627:
1622:
1620:
1614:
1609:
1600:
1595:
1594:
1569:
1565:
1561:
1557:
1550:
1540:
1538:
1537:
1532:
1530:
1528:
1527:
1518:
1517:
1508:
1503:
1502:
1479:
1403:
1393:
1380:
1370:
1360:
1350:
1326:
1322:
1313:
1298:
1266:
1246:
1238:, is related to
1237:
1228:
1203:
1195:
1184:
1177:
1175:
1174:
1169:
1167:
1165:
1164:
1163:
1150:
1142:
1138:
1137:
1136:
1134:
1133:
1121:
1116:
1111:
1110:
1109:
1104:
1103:
1095:
1069:
1067:
1066:
1061:
1058:
1055:
1053:
1048:
1047:
1039:
1028:
1026:
1025:
1020:
1017:
1014:
1012:
1007:
1006:
998:
987:
983:
976:
974:
973:
968:
965:
962:
960:
955:
954:
946:
938:
935:
933:
928:
927:
919:
906:
904:
903:
898:
896:
895:
892:
883:
882:
879:
864:
862:
861:
856:
854:
853:
850:
841:
840:
837:
821:
801:
799:
798:
793:
788:
767:
760:
749:
747:
746:
741:
736:
735:
723:
722:
710:
709:
706:
693:
691:
690:
685:
680:
679:
667:
666:
654:
653:
650:
634:
632:
631:
626:
621:
620:
608:
607:
595:
594:
591:
579:
578:
566:
565:
553:
552:
549:
518:
516:
515:
510:
508:
507:
504:
502:
501:
493:
482:
480:
479:
474:
472:
471:
468:
466:
465:
457:
442:
440:
439:
434:
432:
431:
428:
415:
413:
412:
407:
405:
404:
401:
384:
382:
381:
376:
374:
373:
370:
357:
355:
354:
349:
347:
346:
343:
327:
325:
324:
319:
317:
316:
313:
311:
310:
302:
295:
294:
291:
289:
288:
280:
267:
265:
264:
259:
257:
256:
253:
244:
243:
240:
225:
223:
222:
217:
214:
213:
210:
201:
200:
197:
75:partial pressure
48:chemical species
21:
3781:
3780:
3776:
3775:
3774:
3772:
3771:
3770:
3741:
3740:
3739:
3734:
3665:Baryonic matter
3653:
3607:
3578:Saturated fluid
3518:Crystallization
3479:
3453:Antiferromagnet
3393:
3367:
3311:
3302:
3262:
3252:
3222:
3217:
3170:Self-ionization
3114:
3100:Buffer solution
3088:
3038:
2965:
2960:
2930:
2929:
2916:
2912:
2903:
2896:
2889:
2876:
2875:
2871:
2864:
2849:
2848:
2837:
2830:
2815:
2814:
2805:
2800:
2735:
2687:Fenske equation
2672:
2657:
2651: /
2647:
2640:
2633:
2624:
2618: /
2614:
2608:
2601:
2591:
2584:
2557:
2551: /
2547:
2541:
2534:
2527:
2521: /
2517:
2511:
2504:
2490:
2483:
2472:
2457:
2443:
2437:
2427:
2421:
2414:
2397:
2391:
2381:
2371:
2365:
2355:
2329:
2322:
2315:
2311:
2300:
2294:
2287:
2280:
2274:
2267:
2219:
2214:
2213:
2211:
2200:
2194: /
2193:
2186:
2179:
2173: /
2172:
2165:
2151:
2144:
2137:
2101:
2096:
2095:
2092:
2086:
2066:
2059:
2053:
2049:
2042:
2036:
2029:
2022:
2016:
2012:
2005:
1999:
1993:
1987:
1984:
1978:
1975:
1969:
1966:
1960:
1948:
1935:
1929:
1925:
1894:
1879:
1875:
1861:
1846:
1842:
1825:
1815:
1802:
1801:
1795:
1787:
1765:
1761:
1750:
1743:
1739:
1714:
1709:
1708:
1663:
1662:
1647:
1642:
1641:
1586:
1581:
1580:
1567:
1563:
1559:
1556:
1552:
1549:
1545:
1519:
1509:
1494:
1489:
1488:
1477:
1450:
1444:
1401:
1395:
1391:
1385:
1378:
1372:
1368:
1362:
1358:
1352:
1348:
1342:
1336:dew point curve
1324:
1321:
1315:
1311:
1296:
1287:
1280:
1274:
1264:
1257:
1251:
1245:
1239:
1236:
1230:
1227:
1221:
1210:
1201:
1194:
1190:
1182:
1155:
1151:
1143:
1092:
1087:
1086:
1031:
1030:
990:
989:
985:
981:
911:
910:
887:
874:
869:
868:
845:
832:
827:
826:
819:
816:
770:
769:
766:
762:
759:
755:
727:
714:
701:
696:
695:
671:
658:
645:
640:
639:
612:
599:
586:
570:
557:
544:
539:
538:
490:
485:
484:
454:
449:
448:
423:
418:
417:
396:
391:
390:
365:
360:
359:
338:
333:
332:
299:
277:
272:
271:
248:
235:
230:
229:
205:
192:
187:
186:
180:
175:
162:
28:
23:
22:
18:Saturated fluid
15:
12:
11:
5:
3779:
3777:
3769:
3768:
3763:
3758:
3753:
3743:
3742:
3736:
3735:
3733:
3732:
3727:
3722:
3717:
3712:
3707:
3702:
3697:
3692:
3687:
3682:
3677:
3672:
3667:
3661:
3659:
3655:
3654:
3652:
3651:
3646:
3644:Trouton's rule
3641:
3636:
3631:
3626:
3621:
3615:
3613:
3609:
3608:
3606:
3605:
3600:
3595:
3590:
3585:
3580:
3575:
3570:
3565:
3560:
3555:
3550:
3545:
3540:
3535:
3530:
3525:
3520:
3515:
3513:Critical point
3510:
3505:
3500:
3495:
3489:
3487:
3481:
3480:
3478:
3477:
3472:
3467:
3466:
3465:
3460:
3455:
3447:
3442:
3437:
3432:
3427:
3422:
3417:
3415:Liquid crystal
3412:
3407:
3401:
3399:
3395:
3394:
3392:
3391:
3386:
3381:
3375:
3373:
3369:
3368:
3366:
3365:
3360:
3355:
3350:
3348:Strange matter
3345:
3343:Rydberg matter
3340:
3335:
3330:
3325:
3319:
3317:
3313:
3312:
3305:
3303:
3301:
3300:
3295:
3290:
3281:
3276:
3270:
3268:
3264:
3263:
3253:
3251:
3250:
3243:
3236:
3228:
3219:
3218:
3216:
3215:
3214:
3213:
3203:
3202:
3201:
3191:
3190:
3189:
3179:
3178:
3177:
3167:
3162:
3157:
3156:
3155:
3145:
3140:
3135:
3134:
3133:
3122:
3120:
3116:
3115:
3113:
3112:
3107:
3102:
3096:
3094:
3090:
3089:
3087:
3086:
3081:
3076:
3071:
3066:
3061:
3060:
3059:
3048:
3046:
3040:
3039:
3037:
3036:
3031:
3026:
3021:
3016:
3015:
3014:
3009:
2999:
2994:
2989:
2984:
2979:
2973:
2971:
2967:
2966:
2961:
2959:
2958:
2951:
2944:
2936:
2928:
2927:
2910:
2894:
2887:
2869:
2862:
2835:
2828:
2802:
2801:
2799:
2796:
2795:
2794:
2787:
2782:
2776:
2767:
2757:
2751:
2746:
2734:
2733:External links
2731:
2730:
2729:
2724:
2719:
2714:
2709:
2704:
2699:
2694:
2689:
2684:
2678:
2671:
2668:
2659:
2658:
2655:
2645:
2638:
2631:
2626:
2622:
2612:
2606:
2599:
2589:
2582:
2560:
2559:
2555:
2545:
2539:
2532:
2528:,
2525:
2515:
2509:
2502:
2488:
2481:
2470:
2455:
2449:
2448:
2441:
2435:
2425:
2419:
2412:
2403:
2402:
2395:
2389:
2379:
2375:,
2369:
2363:
2353:
2327:
2320:
2313:
2309:
2303:
2302:
2298:
2292:
2285:
2281:,
2278:
2272:
2265:
2222:
2209:
2203:
2202:
2198:
2191:
2184:
2180:,
2177:
2170:
2163:
2154:
2153:
2149:
2142:
2135:
2104:
2088:Main article:
2085:
2082:
2064:
2057:
2047:
2040:
2033:
2032:
2027:
2020:
2010:
2003:
1991:
1982:
1973:
1964:
1947:
1944:
1921:
1920:
1906:
1901:
1897:
1892:
1886:
1882:
1878:
1873:
1868:
1864:
1859:
1853:
1849:
1845:
1839:
1832:
1828:
1822:
1818:
1812:
1809:
1763:
1741:
1721:
1717:
1705:
1704:
1691:
1685:
1680:
1676:
1670:
1666:
1659:
1654:
1650:
1632:
1631:
1618:
1613:
1608:
1604:
1598:
1593:
1589:
1570:respectively.
1562:in the phases
1554:
1547:
1542:
1541:
1526:
1522:
1516:
1512:
1506:
1501:
1497:
1446:Main article:
1443:
1437:
1399:
1389:
1376:
1366:
1356:
1346:
1319:
1300:
1299:
1292:
1285:
1278:
1268:
1267:
1262:
1255:
1243:
1234:
1225:
1209:
1206:
1192:
1179:
1178:
1162:
1158:
1154:
1149:
1146:
1132:
1129:
1126:
1120:
1108:
1101:
1098:
1052:
1045:
1042:
1011:
1004:
1001:
978:
977:
959:
952:
949:
942:
932:
925:
922:
908:
890:
886:
877:
866:
848:
844:
835:
815:
812:
791:
787:
783:
780:
777:
764:
757:
739:
734:
730:
726:
721:
717:
713:
704:
683:
678:
674:
670:
665:
661:
657:
648:
636:
635:
624:
619:
615:
611:
606:
602:
598:
589:
585:
582:
577:
573:
569:
564:
560:
556:
547:
519:are the molar
499:
496:
463:
460:
426:
399:
368:
341:
329:
328:
308:
305:
298:
286:
283:
269:
251:
247:
238:
227:
208:
204:
195:
179:
176:
166:thermodynamics
161:
158:
121:mole fractions
71:vapor pressure
32:thermodynamics
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3778:
3767:
3764:
3762:
3759:
3757:
3754:
3752:
3749:
3748:
3746:
3731:
3728:
3726:
3723:
3721:
3718:
3716:
3713:
3711:
3708:
3706:
3703:
3701:
3700:Mpemba effect
3698:
3696:
3693:
3691:
3688:
3686:
3683:
3681:
3680:Cooling curve
3678:
3676:
3673:
3671:
3668:
3666:
3663:
3662:
3660:
3656:
3650:
3647:
3645:
3642:
3640:
3637:
3635:
3632:
3630:
3627:
3625:
3622:
3620:
3617:
3616:
3614:
3610:
3604:
3603:Vitrification
3601:
3599:
3596:
3594:
3591:
3589:
3586:
3584:
3581:
3579:
3576:
3574:
3571:
3569:
3568:Recombination
3566:
3564:
3563:Melting point
3561:
3559:
3556:
3554:
3551:
3549:
3546:
3544:
3541:
3539:
3536:
3534:
3531:
3529:
3526:
3524:
3521:
3519:
3516:
3514:
3511:
3509:
3508:Critical line
3506:
3504:
3501:
3499:
3498:Boiling point
3496:
3494:
3491:
3490:
3488:
3486:
3482:
3476:
3473:
3471:
3468:
3464:
3461:
3459:
3456:
3454:
3451:
3450:
3448:
3446:
3443:
3441:
3438:
3436:
3433:
3431:
3430:Exotic matter
3428:
3426:
3423:
3421:
3418:
3416:
3413:
3411:
3408:
3406:
3403:
3402:
3400:
3396:
3390:
3387:
3385:
3382:
3380:
3377:
3376:
3374:
3370:
3364:
3361:
3359:
3356:
3354:
3351:
3349:
3346:
3344:
3341:
3339:
3336:
3334:
3331:
3329:
3326:
3324:
3321:
3320:
3318:
3314:
3309:
3299:
3296:
3294:
3291:
3289:
3285:
3282:
3280:
3277:
3275:
3272:
3271:
3269:
3265:
3260:
3256:
3249:
3244:
3242:
3237:
3235:
3230:
3229:
3226:
3212:
3209:
3208:
3207:
3204:
3200:
3197:
3196:
3195:
3192:
3188:
3185:
3184:
3183:
3180:
3176:
3173:
3172:
3171:
3168:
3166:
3163:
3161:
3158:
3154:
3151:
3150:
3149:
3146:
3144:
3141:
3139:
3136:
3132:
3129:
3128:
3127:
3124:
3123:
3121:
3117:
3111:
3108:
3106:
3103:
3101:
3098:
3097:
3095:
3091:
3085:
3082:
3080:
3077:
3075:
3072:
3070:
3067:
3065:
3064:Phase diagram
3062:
3058:
3057:determination
3055:
3054:
3053:
3050:
3049:
3047:
3045:
3041:
3035:
3032:
3030:
3027:
3025:
3022:
3020:
3017:
3013:
3010:
3008:
3005:
3004:
3003:
3000:
2998:
2995:
2993:
2990:
2988:
2985:
2983:
2980:
2978:
2975:
2974:
2972:
2968:
2964:
2957:
2952:
2950:
2945:
2943:
2938:
2937:
2934:
2924:
2920:
2914:
2911:
2907:
2901:
2899:
2895:
2890:
2884:
2880:
2873:
2870:
2865:
2859:
2855:
2854:
2846:
2844:
2842:
2840:
2836:
2831:
2825:
2821:
2820:
2812:
2810:
2808:
2804:
2797:
2792:
2788:
2786:
2783:
2780:
2777:
2775:
2771:
2768:
2765:
2761:
2758:
2755:
2752:
2750:
2747:
2744:
2740:
2737:
2736:
2732:
2728:
2725:
2723:
2720:
2718:
2717:Pervaporation
2715:
2713:
2710:
2708:
2705:
2703:
2700:
2698:
2697:DECHEMA model
2695:
2693:
2690:
2688:
2685:
2682:
2679:
2677:
2674:
2673:
2669:
2667:
2665:
2654:
2650:
2644:
2637:
2630:
2627:
2621:
2617:
2611:
2605:
2598:
2595:
2594:
2593:
2588:
2581:
2576:
2574:
2570:
2565:
2554:
2550:
2544:
2538:
2531:
2524:
2520:
2514:
2508:
2501:
2498:
2497:
2496:
2492:
2487:
2480:
2476:
2469:
2465:
2461:
2454:
2446:
2440:
2434:
2430:
2424:
2418:
2411:
2408:
2407:
2406:
2400:
2394:
2388:
2384:
2378:
2374:
2368:
2362:
2358:
2352:
2349:
2348:
2347:
2344:
2340:
2336:
2331:
2326:
2319:
2308:
2297:
2291:
2284:
2277:
2271:
2264:
2261:
2260:
2259:
2257:
2253:
2249:
2244:
2240:
2220:
2208:
2197:
2190:
2183:
2176:
2169:
2162:
2159:
2158:
2157:
2148:
2141:
2134:
2131:
2130:
2129:
2127:
2122:
2102:
2091:
2081:
2079:
2075:
2070:
2063:
2056:
2046:
2039:
2026:
2019:
2009:
2002:
1998:
1997:
1996:
1990:
1981:
1972:
1963:
1952:
1945:
1943:
1941:
1933:
1899:
1895:
1890:
1884:
1880:
1866:
1862:
1857:
1851:
1847:
1837:
1830:
1826:
1820:
1816:
1810:
1807:
1800:
1799:
1798:
1793:
1780:
1772:
1768:
1758:
1756:
1748:
1737:
1719:
1715:
1689:
1683:
1678:
1674:
1668:
1664:
1657:
1652:
1648:
1640:
1639:
1638:
1637:
1634:For modified
1616:
1611:
1606:
1602:
1596:
1591:
1587:
1579:
1578:
1577:
1576:
1571:
1524:
1520:
1514:
1510:
1504:
1499:
1495:
1487:
1486:
1485:
1484:) defined by
1483:
1474:
1467:
1463:
1459:
1454:
1449:
1441:
1438:
1436:
1433:
1429:
1424:
1419:
1417:
1413:
1409:
1408:
1398:
1388:
1382:
1375:
1365:
1355:
1345:
1339:
1337:
1333:
1331:
1318:
1304:
1295:
1291:
1284:
1277:
1273:
1272:
1271:
1261:
1254:
1250:
1249:
1248:
1242:
1233:
1224:
1219:
1215:
1207:
1205:
1199:
1188:
1160:
1156:
1147:
1118:
1106:
1096:
1085:
1084:
1083:
1081:
1077:
1073:
1050:
1040:
1009:
999:
957:
947:
940:
930:
920:
909:
888:
884:
875:
867:
846:
842:
833:
825:
824:
823:
813:
811:
809:
805:
789:
785:
781:
778:
775:
761:and pressure
753:
732:
728:
724:
719:
715:
702:
676:
672:
668:
663:
659:
646:
617:
613:
609:
604:
600:
587:
583:
575:
571:
567:
562:
558:
545:
537:
536:
535:
533:
528:
526:
522:
494:
458:
446:
424:
397:
388:
366:
339:
303:
296:
281:
270:
249:
245:
236:
228:
206:
202:
193:
185:
184:
183:
177:
174:
169:
167:
164:The field of
159:
157:
155:
154:
148:
147:
146:boiling point
140:
138:
132:
130:
126:
122:
117:
115:
111:
107:
104:, especially
103:
99:
94:
92:
88:
84:
80:
76:
72:
68:
64:
59:
57:
53:
49:
45:
41:
37:
33:
19:
3761:Distillation
3725:Superheating
3598:Vaporization
3593:Triple point
3588:Supercooling
3577:
3553:Lambda point
3503:Condensation
3420:Time crystal
3398:Other states
3338:Quantum Hall
3206:Vapor–liquid
3205:
3093:Applications
2925:, pages 1–43
2922:
2918:
2913:
2905:
2878:
2872:
2852:
2818:
2722:Supercooling
2660:
2652:
2648:
2642:
2635:
2628:
2619:
2615:
2609:
2603:
2596:
2586:
2579:
2577:
2572:
2568:
2563:
2561:
2552:
2548:
2542:
2536:
2529:
2522:
2518:
2512:
2506:
2499:
2493:
2485:
2478:
2474:
2467:
2463:
2459:
2452:
2450:
2444:
2438:
2432:
2428:
2422:
2416:
2409:
2404:
2398:
2392:
2386:
2382:
2376:
2372:
2366:
2360:
2356:
2350:
2338:
2334:
2332:
2324:
2317:
2306:
2304:
2295:
2289:
2282:
2275:
2269:
2262:
2243:Raoult's law
2241:
2206:
2204:
2195:
2188:
2181:
2174:
2167:
2160:
2155:
2146:
2139:
2132:
2126:Dalton's law
2123:
2093:
2090:Raoult's law
2084:Raoult's law
2071:
2061:
2054:
2044:
2037:
2034:
2024:
2017:
2007:
2000:
1988:
1979:
1970:
1961:
1957:
1934:
1922:
1785:
1759:
1706:
1636:Raoult's law
1633:
1575:Raoult's law
1572:
1543:
1470:
1439:
1428:bubble point
1420:
1415:
1411:
1405:
1396:
1386:
1383:
1373:
1363:
1353:
1343:
1340:
1335:
1330:bubble point
1328:
1316:
1309:
1293:
1289:
1282:
1275:
1269:
1259:
1252:
1240:
1231:
1222:
1217:
1211:
1180:
1074:also called
979:
817:
803:
637:
529:
445:temperatures
330:
181:
163:
151:
144:
141:
133:
118:
114:condensation
102:distillation
95:
87:Dalton's law
83:Raoult's law
60:
56:liquid phase
50:between the
43:
39:
29:
3634:Latent heat
3583:Sublimation
3528:Evaporation
3463:Ferromagnet
3458:Ferrimagnet
3440:Dark matter
3372:High energy
3211:Henry's law
3002:Free energy
2702:Hand boiler
2451:At a given
2401:, ... etc.
2301:, ... etc.
1794:denoted by
1473:Henry's law
91:Henry's law
79:temperature
67:equilibrium
52:vapor phase
3745:Categories
3649:Volatility
3612:Quantities
3573:Regelation
3548:Ionization
3523:Deposition
3475:Superglass
3445:Antimatter
3379:QCD matter
3358:Supersolid
3353:Superfluid
3316:Low energy
3194:Solubility
3165:Hydrolysis
3074:Phase rule
2798:References
2558:, ... etc.
2447:+ ... etc.
2201:, ... etc.
2015:and
1462:Chloroform
752:fugacities
3182:Partition
3012:Helmholtz
2982:Chelation
2634:= (1 −
2256:compounds
1808:α
1716:γ
1684:⋆
1665:γ
1612:⋆
1432:dew point
1407:azeotrope
1216:called a
1187:extensive
1153:∂
1145:∂
1100:¯
1044:¯
1003:¯
951:¯
924:¯
808:ideal gas
776:ϕ
498:~
462:~
387:pressures
307:~
285:~
3710:Spinodal
3658:Concepts
3538:Freezing
3175:of water
2970:Concepts
2670:See also
1755:pressure
1480:values (
1466:Methanol
1185:is the (
1070:are the
750:are the
532:fugacity
443:are the
385:are the
3670:Binodal
3558:Melting
3493:Boiling
3410:Crystal
3405:Colloid
2248:alkanes
1753:is the
1745:is the
1734:is the
1458:UNIQUAC
1288:+ ⋯ +
1196:is the
135:1
110:boiling
98:columns
3298:Plasma
3279:Liquid
3044:Models
2885:
2860:
2826:
2585:= 1 −
2305:where
2205:where
1707:where
1544:where
1181:where
1139:
1112:
980:where
638:where
331:where
89:, and
54:and a
38:, the
3288:Vapor
3274:Solid
3267:State
3007:Gibbs
2921:, 7,
2625:, and
2252:polar
2152:+ ...
2060:= 1,
2043:= 0,
1371:(and
1351:(and
1332:curve
1214:graph
907:; and
268:; and
129:phase
125:moles
3259:list
2883:ISBN
2858:ISBN
2824:ISBN
2764:NIST
2484:(or
1986:and
1968:and
1749:and
1573:For
1566:and
1551:and
1430:and
1394:and
1029:and
984:and
694:and
483:and
416:and
358:and
100:for
61:The
34:and
3284:Gas
2656:tot
2623:tot
2556:tot
2526:tot
2471:tot
2456:tot
2413:tot
2312:, P
2199:tot
2178:tot
2136:tot
2107:tot
2067:= 1
2050:= 0
2030:= 1
2023:+
2013:= 1
2006:+
1402:= 1
1392:= 0
1379:= 0
1369:= 1
1359:= 1
1349:= 0
1297:= 1
1281:+
1265:= 1
1258:+
1056:vap
1015:liq
963:vap
936:liq
893:vap
880:liq
851:vap
838:liq
707:vap
651:liq
592:vap
550:liq
505:vap
469:liq
429:vap
402:liq
371:vap
344:liq
314:vap
292:liq
254:vap
241:liq
211:vap
198:liq
137:atm
44:VLE
30:In
3747::
3286:/
2923:49
2897:^
2838:^
2806:^
2641:)
2602:=
2564:T'
2535:=
2505:=
2431:+
2415:=
2385:=
2359:=
2323:,
2288:=
2268:=
2187:=
2166:=
2145:+
2138:=
2121:.
1932:.
1757:.
1738:,
1338:.
1325:T
1312:T
1204:.
810:.
93:.
85:,
58:.
3261:)
3257:(
3247:e
3240:t
3233:v
2955:e
2948:t
2941:v
2908:.
2891:.
2866:.
2832:.
2766:)
2653:P
2649:T
2646:2
2643:P
2639:1
2636:x
2632:2
2629:y
2620:P
2616:T
2613:1
2610:P
2607:1
2604:x
2600:1
2597:y
2590:1
2587:x
2583:2
2580:x
2573:T
2569:T
2553:P
2549:T
2546:2
2543:P
2540:2
2537:x
2533:2
2530:y
2523:P
2519:T
2516:1
2513:P
2510:1
2507:x
2503:1
2500:y
2489:2
2486:x
2482:1
2479:x
2475:T
2468:P
2464:T
2460:T
2453:P
2445:T
2442:2
2439:P
2436:2
2433:x
2429:T
2426:1
2423:P
2420:1
2417:x
2410:P
2399:T
2396:2
2393:P
2390:2
2387:x
2383:T
2380:2
2377:P
2373:T
2370:1
2367:P
2364:1
2361:x
2357:T
2354:1
2351:P
2339:T
2335:P
2328:2
2325:x
2321:1
2318:x
2314:2
2310:1
2307:P
2299:2
2296:P
2293:2
2290:x
2286:2
2283:P
2279:1
2276:P
2273:1
2270:x
2266:1
2263:P
2225:2
2221:P
2210:1
2207:P
2196:P
2192:2
2189:P
2185:2
2182:y
2175:P
2171:1
2168:P
2164:1
2161:y
2150:2
2147:P
2143:1
2140:P
2133:P
2103:P
2065:1
2062:y
2058:1
2055:x
2048:1
2045:y
2041:1
2038:x
2028:2
2025:y
2021:1
2018:y
2011:2
2008:x
2004:1
2001:x
1992:2
1989:y
1983:1
1980:y
1974:2
1971:x
1965:1
1962:x
1936:K
1930:j
1926:i
1905:)
1900:j
1896:x
1891:/
1885:j
1881:y
1877:(
1872:)
1867:i
1863:x
1858:/
1852:i
1848:y
1844:(
1838:=
1831:j
1827:K
1821:i
1817:K
1811:=
1796:α
1788:K
1764:i
1762:K
1751:P
1742:i
1740:P
1720:i
1690:P
1679:i
1675:P
1669:i
1658:=
1653:i
1649:K
1617:P
1607:i
1603:P
1597:=
1592:i
1588:K
1568:x
1564:y
1560:i
1555:i
1553:x
1548:i
1546:y
1525:i
1521:x
1515:i
1511:y
1505:=
1500:i
1496:K
1478:K
1464:/
1440:K
1400:1
1397:x
1390:1
1387:x
1377:2
1374:x
1367:1
1364:x
1357:2
1354:x
1347:1
1344:x
1320:1
1317:x
1294:n
1290:x
1286:2
1283:x
1279:1
1276:x
1263:2
1260:x
1256:1
1253:x
1244:1
1241:x
1235:2
1232:x
1226:1
1223:x
1202:i
1193:i
1191:n
1183:G
1161:i
1157:n
1148:G
1131:f
1128:e
1125:d
1119:=
1107:i
1097:G
1051:i
1041:G
1010:i
1000:G
986:T
982:P
958:i
948:G
941:=
931:i
921:G
889:T
885:=
876:T
865:;
847:P
843:=
834:P
820:i
790:P
786:/
782:f
779:=
765:s
763:P
758:s
756:T
738:)
733:s
729:P
725:,
720:s
716:T
712:(
703:f
682:)
677:s
673:P
669:,
664:s
660:T
656:(
647:f
623:)
618:s
614:P
610:,
605:s
601:T
597:(
588:f
584:=
581:)
576:s
572:P
568:,
563:s
559:T
555:(
546:f
495:G
459:G
425:T
398:T
367:P
340:P
304:G
297:=
282:G
250:T
246:=
237:T
226:;
207:P
203:=
194:P
42:(
20:)
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