# 临界点（热力学）

[[Image:CriticalPointMeasurementEthane.jpg|thumb|right|upright=1.5|

[[Image:CriticalPointMeasurementEthane.jpg|thumb|right|upright=1.5|

[图片: 临界点测量乙烷 jpg | thumb | right | upright = 1.5 |

{{ordered list

{{ordered list

{有序列表

|Subcritical ethane, liquid and gas phase coexist.

|Subcritical ethane, liquid and gas phase coexist.

|Critical point (32.17 °C, 48.72 bar), opalescence.

|Critical point (32.17 °C, 48.72 bar), opalescence.

|Supercritical ethane, fluid.[1]

|Supercritical ethane, fluid.

}}]]

}}]]

}}]]

In thermodynamics, a critical point (or critical state) is the end point of a phase equilibrium curve. The most prominent example is the liquid–vapor critical point, the end point of the pressure–temperature curve that designates conditions under which a liquid and its vapor can coexist. At higher temperatures, the gas cannot be liquefied by pressure alone. At the critical point, defined by a critical temperature Tc and a critical pressure pc, phase boundaries vanish. Other examples include the liquid–liquid critical points in mixtures.

In thermodynamics, a critical point (or critical state) is the end point of a phase equilibrium curve. The most prominent example is the liquid–vapor critical point, the end point of the pressure–temperature curve that designates conditions under which a liquid and its vapor can coexist. At higher temperatures, the gas cannot be liquefied by pressure alone. At the critical point, defined by a critical temperature Tc and a critical pressure pc, phase boundaries vanish. Other examples include the liquid–liquid critical points in mixtures.

## Liquid–vapor critical point液-汽临界点

### Overview 总览

The liquid–vapor critical point in a pressure–temperature phase diagram is at the high-temperature extreme of the liquid–gas phase boundary. The dotted green line shows the anomalous behavior of water.

The liquid–vapor critical point in a pressure–temperature phase diagram is at the high-temperature extreme of the liquid–gas phase boundary. The dotted green line shows the anomalous behavior of water.

For simplicity and clarity, the generic notion of critical point is best introduced by discussing a specific example, the liquid–vapor critical point. This was the first critical point to be discovered, and it is still the best known and most studied one.

For simplicity and clarity, the generic notion of critical point is best introduced by discussing a specific example, the liquid–vapor critical point. This was the first critical point to be discovered, and it is still the best known and most studied one.

The figure to the right shows the schematic PT diagram of a pure substance (as opposed to mixtures, which have additional state variables and richer phase diagrams, discussed below). The commonly known phases solid, liquid and vapor are separated by phase boundaries, i.e. pressure–temperature combinations where two phases can coexist. At the triple point, all three phases can coexist. However, the liquid–vapor boundary terminates in an endpoint at some critical temperature Tc and critical pressure pc. This is the critical point.

The figure to the right shows the schematic PT diagram of a pure substance (as opposed to mixtures, which have additional state variables and richer phase diagrams, discussed below). The commonly known phases solid, liquid and vapor are separated by phase boundaries, i.e. pressure–temperature combinations where two phases can coexist. At the triple point, all three phases can coexist. However, the liquid–vapor boundary terminates in an endpoint at some critical temperature Tc and critical pressure pc. This is the critical point.

In water, the critical point occurs at 模板:Convert and 模板:Convert.[3]

In water, the critical point occurs at and .

In the vicinity of the critical point, the physical properties of the liquid and the vapor change dramatically, with both phases becoming ever more similar. For instance, liquid water under normal conditions is nearly incompressible, has a low thermal expansion coefficient, has a high dielectric constant, and is an excellent solvent for electrolytes. Near the critical point, all these properties change into the exact opposite: water becomes compressible, expandable, a poor dielectric, a bad solvent for electrolytes, and prefers to mix with nonpolar gases and organic molecules.[4]

In the vicinity of the critical point, the physical properties of the liquid and the vapor change dramatically, with both phases becoming ever more similar. For instance, liquid water under normal conditions is nearly incompressible, has a low thermal expansion coefficient, has a high dielectric constant, and is an excellent solvent for electrolytes. Near the critical point, all these properties change into the exact opposite: water becomes compressible, expandable, a poor dielectric, a bad solvent for electrolytes, and prefers to mix with nonpolar gases and organic molecules.

At the critical point, only one phase exists. The heat of vaporization is zero. There is a stationary inflection point in the constant-temperature line (critical isotherm) on a PV diagram. This means that at the critical point:[5][6][7]

At the critical point, only one phase exists. The heat of vaporization is zero. There is a stationary inflection point in the constant-temperature line (critical isotherm) on a PV diagram. This means that at the critical point:

The critical isotherm with the critical point K

$\displaystyle{ \left(\frac{\partial p}{\partial V}\right)_T = 0, }$

Above the critical point there exists a state of matter that is continuously connected with (can be transformed without phase transition into) both the liquid and the gaseous state. It is called supercritical fluid. The common textbook knowledge that all distinction between liquid and vapor disappears beyond the critical point has been challenged by Fisher and Widom,[10] who identified a pT line that separates states with different asymptotic statistical properties (Fisher–Widom line).

Above the critical point there exists a state of matter that is continuously connected with (can be transformed without phase transition into) both the liquid and the gaseous state. It is called supercritical fluid. The common textbook knowledge that all distinction between liquid and vapor disappears beyond the critical point has been challenged by Fisher and Widom, who identified a p–T line that separates states with different asymptotic statistical properties (Fisher–Widom line).

$\displaystyle{ \left(\frac{\partial^2p}{\partial V^2}\right)_T = 0. }$

Some times the critical point does not manifest in most thermodynamic or mechanical properties, but is hidden and reveals itself in the onset of inhomogeneities in elastic moduli, marked changes in the appearance and local properties of non-affine droplets and a sudden enhancement in defect pair concentration. In those cases we have a hidden critical point, otherwise we have an exposed critical point.

The critical isotherm with the critical point K

Some times the critical point does not manifest in most thermodynamic or mechanical properties, but is hidden and reveals itself in the onset of inhomogeneities in elastic moduli, marked changes in the appearance and local properties of non-affine droplets and a sudden enhancement in defect pair concentration. In those cases we have a hidden critical point, otherwise we have an exposed critical point.[11]

### History历史

Critical carbon dioxide exuding fog while cooling from supercritical to critical temperature.

Solving the above condition $\displaystyle{ (\partial p / \partial V)_T = 0 }$ for the van der Waals equation, one can compute the critical point as

The existence of a critical point was first discovered by Charles Cagniard de la Tour in 1822[12][13] and named by Dmitri Mendeleev in 1860[14][15] and Thomas Andrews in 1869.[16] Cagniard showed that CO2 could be liquefied at 31 °C at a pressure of 73 atm, but not at a slightly higher temperature, even under pressures as high as 3000 atm.

The existence of a critical point was first discovered by Charles Cagniard de la Tour in 1822 and named by Dmitri Mendeleev in 1860 and Thomas Andrews in 1869. Cagniard showed that CO2 could be liquefied at 31 °C at a pressure of 73 atm, but not at a slightly higher temperature, even under pressures as high as 3000 atm.

However, the van der Waals equation, based on a mean-field theory, does not hold near the critical point. In particular, it predicts wrong scaling laws.

### Theory理论

To analyse properties of fluids near the critical point, reduced state variables are sometimes defined relative to the critical properties

Solving the above condition $\displaystyle{ (\partial p / \partial V)_T = 0 }$ for the van der Waals equation, one can compute the critical point as

$\displaystyle{ T_\text{r} = \frac{T}{T_\text{c}}, 如果你想知道更多的信息，请访问我的网站, : \lt math\gt T_\text{c} = \frac{8a}{27Rb}, \quad p_\text{r} = \frac{p}{p_\text{c}}, 4.1.1.1.2.2.2.2.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3 \quad V_\text{c} = 3nb, \quad V_\text{r} = \frac{V}{RT_\text{c} / p_\text{c}}. }$


4 v _ text { r } = frac { v }{ RT _ text { c }/p _ text { c } . </math >

 \quad p_\text{c} = \frac{a}{27b^2}.[/itex]

 --Miyasaki（讨论）这里内容有待整理


However, the van der Waals equation, based on a mean-field theory, does not hold near the critical point. In particular, it predicts wrong scaling laws.

The principle of corresponding states indicates that substances at equal reduced pressures and temperatures have equal reduced volumes. This relationship is approximately true for many substances, but becomes increasingly inaccurate for large values of pr.

To analyse properties of fluids near the critical point, reduced state variables are sometimes defined relative to the critical properties[22]

For some gases, there is an additional correction factor, called Newton's correction, added to the critical temperature and critical pressure calculated in this manner. These are empirically derived values and vary with the pressure range of interest.[24]

For some gases, there is an additional correction factor, called Newton's correction, added to the critical temperature and critical pressure calculated in this manner. These are empirically derived values and vary with the pressure range of interest.

$\displaystyle{ T_\text{r} = \frac{T}{T_\text{c}}, \quad p_\text{r} = \frac{p}{p_\text{c}}, \quad V_\text{r} = \frac{V}{RT_\text{c} / p_\text{c}}. }$

< 中心 >

{ | class = “ wikitable sortable” style = “ text-align: center; ” The principle of corresponding states indicates that substances at equal reduced pressures and temperatures have equal reduced volumes. This relationship is approximately true for many substances, but becomes increasingly inaccurate for large values of pr.
Substance 物质

Critical temperature 临界温度

Critical pressure (absolute) 临界压力(绝对值)

### Table of liquid–vapor critical temperature and pressure for selected substances

Argon 氩气
}} }}
 Substance[26][27] Critical temperature Critical pressure (absolute) }} }} Ammonia (NH3) 氨(NH < sub > 3 ) }} }} }} }} Argon 模板:Sort R-134a R-134a 模板:Sort }} }} }} }} Ammonia (NH3)[28] 模板:Sort 模板:Sort R-410A R-410A }} }} }} }} R-134a 模板:Sort 模板:Sort Bromine 溴 }} }} }} }} R-410A 模板:Sort Caesium 铯 模板:Sort }} }} }} }} Bromine Chlorine 氯气 模板:Sort }} }} 模板:Sort }} }} Caesium Ethanol (C2H5OH) 乙醇(c < sub > 2 h < sub > 5 OH) 模板:Sort }} }} 模板:Sort }} }} Chlorine Fluorine 氟 模板:Sort }} }} 模板:Sort }} }} Ethanol (C2H5OH) Helium 氦气 模板:Sort }} }} 模板:Sort }} }} Fluorine Hydrogen 氢气 模板:Sort }} }} 模板:Sort }} }} Helium Krypton 氪星 模板:Sort }} }} 模板:Sort }} }} Hydrogen Methane (CH4) 甲烷(CH < sub > 4 ) 模板:Sort }} }} 模板:Sort }} }} Krypton Neon 霓虹灯 模板:Sort }} }} 模板:Sort }} }} Methane (CH4) Nitrogen 氮气 模板:Sort }} }} 模板:Sort }} }} Neon Oxygen (O2) 氧气(o < sub > 2 ) 模板:Sort }} }} 模板:Sort }} }} Nitrogen Carbon dioxide (CO2) 二氧化碳(CO < sub > 2 ) 模板:Sort }} }} 模板:Sort }} }} Oxygen (O2) Nitrous oxide (N2O) 氧化亚氮(n < sub > 2 o) 模板:Sort }} }} 模板:Sort }} }} Carbon dioxide (CO2) Sulfuric acid (H2SO4) 硫酸(h < sub > 2 SO < sub > 4 ) 模板:Sort }} }} 模板:Sort }} }} Nitrous oxide (N2O) Xenon 氙气 模板:Sort }} }} 模板:Sort }} }} Sulfuric acid (H2SO4) Lithium Lithium 模板:Sort }} }} 模板:Sort }} }} Xenon Mercury 水星 模板:Sort }} }} 模板:Sort }} }} Lithium Sulfur 硫磺 模板:Sort }} }} 模板:Sort }} }} Mercury Iron 铁 模板:Sort }} }} 模板:Sort Sulfur Gold 黄金 模板:Sort }} }} 模板:Sort }} }} Iron Aluminium 铝 模板:Sort }} }} Gold Water (H2O) 水(h < sub > 2 o) 模板:Sort }} }} 模板:Sort }} }} Aluminium

|

|-

| Water (H2O)[3][29]

A plot of typical polymer solution phase behavior including two critical points: a LCST and an UCST

The liquid–liquid critical point of a solution, which occurs at the critical solution temperature, occurs at the limit of the two-phase region of the phase diagram. In other words, it is the point at which an infinitesimal change in some thermodynamic variable (such as temperature or pressure) leads to separation of the mixture into two distinct liquid phases, as shown in the polymer–solvent phase diagram to the right. Two types of liquid–liquid critical points are the upper critical solution temperature (UCST), which is the hottest point at which cooling induces phase separation, and the lower critical solution temperature (LCST), which is the coldest point at which heating induces phase separation.

|-

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From a theoretical standpoint, the liquid–liquid critical point represents the temperature–concentration extremum of the spinodal curve (as can be seen in the figure to the right). Thus, the liquid–liquid critical point in a two-component system must satisfy two conditions: the condition of the spinodal curve (the second derivative of the free energy with respect to concentration must equal zero), and the extremum condition (the third derivative of the free energy with respect to concentration must also equal zero or the derivative of the spinodal temperature with respect to concentration must equal zero).

## Mixtures: liquid–liquid critical point混合物：液体-液体临界点

A plot of typical polymer solution phase behavior including two critical points: a LCST and an UCST

The liquid–liquid critical point of a solution, which occurs at the critical solution temperature, occurs at the limit of the two-phase region of the phase diagram. In other words, it is the point at which an infinitesimal change in some thermodynamic variable (such as temperature or pressure) leads to separation of the mixture into two distinct liquid phases, as shown in the polymer–solvent phase diagram to the right. Two types of liquid–liquid critical points are the upper critical solution temperature (UCST), which is the hottest point at which cooling induces phase separation, and the lower critical solution temperature (LCST), which is the coldest point at which heating induces phase separation. 在“临界溶液温度”下，溶液的液-液临界点出现在相图两相区的极限处。换言之，它是某个热力学变量（如温度或压力）的微小变化导致混合物分离为两个不同的液相的点，如右侧的聚合物-溶剂相图所示。两种类型的液-液临界点是上临界溶液温度（UCST），这是冷却导致相分离的最热点，和下临界溶液温度（LCST），这是加热导致相分离的最冷点。

### Mathematical definition数学定义

From a theoretical standpoint, the liquid–liquid critical point represents the temperature–concentration extremum of the spinodal curve (as can be seen in the figure to the right). Thus, the liquid–liquid critical point in a two-component system must satisfy two conditions: the condition of the spinodal curve (the second derivative of the free energy with respect to concentration must equal zero), and the extremum condition (the third derivative of the free energy with respect to concentration must also equal zero or the derivative of the spinodal temperature with respect to concentration must equal zero). 从理论上看，从液体的临界点（从理论上看，是指液体的临界温度）。因此，双组分体系中的液-液临界点必须满足两个条件：旋节曲线的条件（自由能相对于浓度的“二阶”导数必须等于零）和极值条件（自由能相对于浓度的“第三”导数）也必须等于零，或者旋节温度对浓度的导数必须等于零）。

Joback 方法 Klingewicz方法 Lydersen 方法（从分子结构估算临界温度、压力和体积）

Néel点

Rushbrooke不等式

Widom缩放 模板:Colend

## Footnotes脚注

1. Horstmann, Sven (2000). Theoretische und experimentelle Untersuchungen zum Hochdruckphasengleichgewichtsverhalten fluider Stoffgemische für die Erweiterung der PSRK-Gruppenbeitragszustandsgleichung [Theoretical and experimental investigations of the high-pressure phase equilibrium behavior of fluid mixtures for the expansion of the PSRK group contribution equation of state] (Ph.D.) (in Deutsch). Oldenburg, Germany: Carl-von-Ossietzky Universität Oldenburg. ISBN 3-8265-7829-5. OCLC 76176158.
2. Horstmann, Sven (2000). Theoretische und experimentelle Untersuchungen zum Hochdruckphasengleichgewichtsverhalten fluider Stoffgemische für die Erweiterung der PSRK-Gruppenbeitragszustandsgleichung [Theoretical and experimental investigations of the high-pressure phase equilibrium behavior of fluid mixtures for the expansion of the PSRK group contribution equation of state] (Ph.D.) (in Deutsch). Oldenburg, Germany: Carl-von-Ossietzky Universität Oldenburg. ISBN 3-8265-7829-5. OCLC 76176158.
3. Wagner, W.; Pruß, A. (June 2002). "The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use". Journal of Physical and Chemical Reference Data. 31 (2): 398. doi:10.1063/1.1461829.
4. Anisimov, Sengers, Levelt Sengers (2004):Near-critical behavior of aqueous systems. 水体系的近临界行为 Chapter 2 inAqueous System at Elevated Temperatures and Pressures 高温高压下的水体系 Palmer et al., eds. $\displaystyle{ \left(\frac{\partial p}{\partial V}\right)_T = 0, }$ 左(frac { partial p }{ partial v } right) _ t = 0, Elsevier.
5. P. Atkins and J. de Paula, Physical Chemistry, 8th ed. (W. H. Freeman 2006), p. 21.
6. K. J. Laidler and J. H. Meiser, Physical Chemistry (Benjamin/Cummings 1982), p. 27.
7. P. A. Rock, Chemical Thermodynamics (MacMillan 1969), p. 123.
8. K. J. Laidler and J. H. Meiser, Physical Chemistry (Benjamin/Cummings 1982), p. 27.
9. P. A. Rock, Chemical Thermodynamics (MacMillan 1969), p. 123.
10. Fisher, Widom: Decay of Correlations in Linear Systems, J. Chem. Phys. 50, 3756 (1969).
11. Das, Tamoghna; Ganguly, Saswati; Sengupta, Surajit; Rao, Madan (3 June 2015). "Pre-Yield Non-Affine Fluctuations and A Hidden Critical Point in Strained Crystals". Scientific Reports. 5 (1): 10644. Bibcode:2015NatSR...510644D. doi:10.1038/srep10644. PMC 4454149. PMID 26039380.
12. Charles Cagniard de la Tour (1822). "Exposé de quelques résultats obtenu par l'action combinée de la chaleur et de la compression sur certains liquides, tels que l'eau, l'alcool, l'éther sulfurique et l'essence de pétrole rectifiée" [Presentation of some results obtained by the combined action of heat and compression on certain liquids, such as water, alcohol, sulfuric ether (i.e., diethyl ether), and distilled petroleum spirit]. Annales de Chimie et de Physique (in français). 21: 127–132.
13. Berche, B., Henkel, M., Kenna, R (2009) Critical phenomena: 150 years since Cagniard de la Tour. Journal of Physical Studies 13 (3), pp. 3001-1–3001-4.
14. Mendeleev called the critical point the "absolute temperature of boiling" (模板:Lang-ru; 模板:Lang-de). $\displaystyle{ T_\text{c} = \frac{8a}{27Rb}, 8 a }{27Rb } , * {{cite journal |last1=Менделеев |first1=Д. |title=О расширении жидкостей от нагревания выше температуры кипения |journal=Горный Журнал [Mining Journal] |date=1861 |volume=4 |pages=141–152 |trans-title=On the expansion of liquids from heating above the temperature of boiling |language=ru}} The "absolute temperature of boiling" is defined on p. 151. Available at [https://upload.wikimedia.org/wikipedia/commons/e/e6/%D0%93%D0%BE%D1%80%D0%BD%D1%8B%D0%B9_%D0%B6%D1%83%D1%80%D0%BD%D0%B0%D0%BB%2C_1861%2C_%E2%84%9604_%28%D0%B0%D0%BF%D1%80%D0%B5%D0%BB%D1%8C%29.pdf Wikimedia] \quad V_\text{c} = 3nb, 3nb, * German translation: {{cite journal |last1=Mendelejeff |first1=D. |title=Ueber die Ausdehnung der Flüssigkeiten beim Erwärmen über ihren Siedepunkt |journal=Annalen der Chemie und Pharmacie |date=1861 |volume=119 |pages=1–11 |url=https://babel.hathitrust.org/cgi/pt?id=uc1.c036497486;view=1up;seq=13 |trans-title=On the expansion of fluids during heating above their boiling point |language=de |doi=10.1002/jlac.18611190102 }} The "absolute temperature of boiling" is defined on p. 11: "{{lang|de|2=Als absolute Siedetemperatur müssen wir den Punkt betrachten, bei welchem 1) die Cohäsion der Flüssigkeit = 0° ist und a\lt sup\gt 2\lt /sup\gt = 0, bei welcher 2) die latente Verdamfungswärme auch = 0 ist und bei welcher sich 3) die Flüssigkeit in Dampf verwandelt, unabhängig von Druck und Volum."}} (As the "absolute temperature of boiling" we must regard the point at which (1) the cohesion of the liquid equals 0° and ''a''\lt sup\gt 2\lt /sup\gt = 0 [where ''a''\lt sup\gt 2\lt /sup\gt is the coefficient of capillarity, p. 6], at which (2) the latent heat of vaporization also equals zero, and at which (3) the liquid is transformed into vapor, independently of the pressure and the volume.) \quad p_\text{c} = \frac{a}{27b^2}. }$ 27b ^ 2} . </math >
15. Landau, Lifshitz, Theoretical Physics, Vol. V: Statistical Physics, Ch. 83 [German edition 1984].
16. Andrews, Thomas (1869). "The Bakerian lecture: On the continuity of the gaseous and liquid states of matter". Philosophical Transactions of the Royal Society. London. 159: 575–590. doi:10.1098/rstl.1869.0021. The term "critical point" appears on page 588.
17. Charles Cagniard de la Tour (1822). "Exposé de quelques résultats obtenu par l'action combinée de la chaleur et de la compression sur certains liquides, tels que l'eau, l'alcool, l'éther sulfurique et l'essence de pétrole rectifiée" [Presentation of some results obtained by the combined action of heat and compression on certain liquids, such as water, alcohol, sulfuric ether (i.e., diethyl ether), and distilled petroleum spirit]. Annales de Chimie et de Physique (in français). 21: 127–132.
18. Berche, B., Henkel, M., Kenna, R (2009) Critical phenomena: 150 years since Cagniard de la Tour. Journal of Physical Studies 13 (3), pp. 3001-1–3001-4.
19. Mendeleev called the critical point the "absolute temperature of boiling" (模板:Lang-ru; 模板:Lang-de). $\displaystyle{ T_\text{c} = \frac{8a}{27Rb}, 8 a }{27Rb } , * {{cite journal |last1=Менделеев |first1=Д. |title=О расширении жидкостей от нагревания выше температуры кипения |journal=Горный Журнал [Mining Journal] |date=1861 |volume=4 |pages=141–152 |trans-title=On the expansion of liquids from heating above the temperature of boiling |language=ru}} The "absolute temperature of boiling" is defined on p. 151. Available at [https://upload.wikimedia.org/wikipedia/commons/e/e6/%D0%93%D0%BE%D1%80%D0%BD%D1%8B%D0%B9_%D0%B6%D1%83%D1%80%D0%BD%D0%B0%D0%BB%2C_1861%2C_%E2%84%9604_%28%D0%B0%D0%BF%D1%80%D0%B5%D0%BB%D1%8C%29.pdf Wikimedia] \quad V_\text{c} = 3nb, 3nb, * German translation: {{cite journal |last1=Mendelejeff |first1=D. |title=Ueber die Ausdehnung der Flüssigkeiten beim Erwärmen über ihren Siedepunkt |journal=Annalen der Chemie und Pharmacie |date=1861 |volume=119 |pages=1–11 |url=https://babel.hathitrust.org/cgi/pt?id=uc1.c036497486;view=1up;seq=13 |trans-title=On the expansion of fluids during heating above their boiling point |language=de |doi=10.1002/jlac.18611190102 }} The "absolute temperature of boiling" is defined on p. 11: "{{lang|de|2=Als absolute Siedetemperatur müssen wir den Punkt betrachten, bei welchem 1) die Cohäsion der Flüssigkeit = 0° ist und a\lt sup\gt 2\lt /sup\gt = 0, bei welcher 2) die latente Verdamfungswärme auch = 0 ist und bei welcher sich 3) die Flüssigkeit in Dampf verwandelt, unabhängig von Druck und Volum."}} (As the "absolute temperature of boiling" we must regard the point at which (1) the cohesion of the liquid equals 0° and ''a''\lt sup\gt 2\lt /sup\gt = 0 [where ''a''\lt sup\gt 2\lt /sup\gt is the coefficient of capillarity, p. 6], at which (2) the latent heat of vaporization also equals zero, and at which (3) the liquid is transformed into vapor, independently of the pressure and the volume.) \quad p_\text{c} = \frac{a}{27b^2}. }$ 27b ^ 2} . </math >
20. Landau, Lifshitz, Theoretical Physics, Vol. V: Statistical Physics, Ch. 83 [German edition 1984].
21. Andrews, Thomas (1869). "The Bakerian lecture: On the continuity of the gaseous and liquid states of matter". Philosophical Transactions of the Royal Society. London. 159: 575–590. doi:10.1098/rstl.1869.0021. The term "critical point" appears on page 588.
22. Cengel, Yunus A.; Boles, Michael A. (2002). Thermodynamics: an engineering approach. Boston: McGraw-Hill. pp. 91–93. ISBN 978-0-07-121688-3.
23. Cengel, Yunus A.; Boles, Michael A. (2002). Thermodynamics: an engineering approach. Boston: McGraw-Hill. pp. 91–93. ISBN 978-0-07-121688-3.
24. Maslan, Frank D.; Littman, Theodore M. (1953). "Compressibility Chart for Hydrogen and Inert Gases". Ind. Eng. Chem. 45 (7): 1566–1568. doi:10.1021/ie50523a054.
25. Maslan, Frank D.; Littman, Theodore M. (1953). "Compressibility Chart for Hydrogen and Inert Gases". Ind. Eng. Chem. 45 (7): 1566–1568. doi:10.1021/ie50523a054.
26. Emsley, John (1991). The Elements (Second ed.). Oxford University Press. ISBN 978-0-19-855818-7.
27. Cengel, Yunus A.; Boles, Michael A. (2002). Thermodynamics: An Engineering Approach (Fourth ed.). McGraw-Hill. pp. 824. ISBN 978-0-07-238332-4.
28. "Ammonia - NH3 - Thermodynamic Properties". www.engineeringtoolbox.com. Retrieved 2017-04-07.
29. "Critical Temperature and Pressure". Purdue University. Retrieved 2006-12-19.

| publisher = Purdue University | url = http://www.chem.purdue.edu/gchelp/liquids/critical.html | accessdate = 2006-12-03 }}

| publisher = 普渡大学 | url = http://www.chem.purdue.edu/gchelp/liquids/critical.html | accessdate = 2006-12-03}

## References参考

Category:Condensed matter physics

Category:Conformal field theory

Category:Critical phenomena

• "Critical Temperature and Pressure". Department of Chemistry Category:Phase transitions 类别: 阶段转变. Purdue University. Retrieved 2006-12-03. line feed character in |work= at position 24 (help)

Category:Renormalization group

Category:Threshold temperatures

Category:Gases

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