Taylor series

2014-01-24T12:39:03Z

81.152.62.83: /* Definition */

{{For|other uses|Tripoint (disambiguation)|tripoint}}
In [[thermodynamics]], the '''triple point''' of a substance is the [[temperature]] and [[pressure]] at which the three [[Phase (matter)|phases]] ([[gas]], [[liquid]], and [[solid]]) of that substance coexist in [[thermodynamic equilibrium]].<ref name=gold>{{GoldBookRef | title=Triple point |file=T06502 |year=1994}}</ref> For example, the triple point of [[Mercury (element)|mercury]] occurs at a temperature of −38.8344 °C and a pressure of 0.2 m[[Pascal (unit)|Pa]].

In addition to the triple point between solid, liquid, and gas, there can be triple points involving more than one solid phase, for substances with multiple [[polymorphism (materials science)|polymorphs]]. [[Helium-4]] is a special case that presents a triple point involving two different fluid phases (see [[lambda point]]). In general, for a system with ''p'' possible phases, there are <math>{p\choose 3} = \tfrac16p(p-1)(p-2)</math> triple points.<ref name=gold/>

The triple point of [[water]] is used to define the [[kelvin]], the [[SI]] [[SI base unit|base unit]] of thermodynamic temperature.<ref>[http://www1.bipm.org/en/si/base_units/ Definition of the kelvin] at BIPM</ref> The number given for the temperature of the triple point of water is an exact definition rather than a measured quantity. The triple points of several substances are used to define points in the [[ITS-90]] international temperature scale, ranging from the triple point of hydrogen (13.8033 K) to the triple point of water (273.16 K, or 0.01 °C).

==Triple points of water==
===Gas–liquid–solid triple point===
{{main|Water (molecule)#Triple point}}
[[Image:Phase-diag2.svg|thumb|350px|A typical phase diagram. The solid green line applies to most substances; the dotted green line gives the anomalous behaviour of water]]
The single combination of pressure and temperature at which liquid [[water]], solid [[ice]], and [[water vapour]] can coexist in a stable equilibrium occurs at exactly 273.16 K (0.01 [[°C]]) and a partial vapour pressure of 611.73 [[pascal (unit)|pascal]]s (ca. 6.1173 [[bar (unit)|millibars]], 0.0060373 [[atmosphere (unit)|atm]]). At that point, it is possible to change all of the substance to ice, water, or vapour by making arbitrarily small changes in pressure and temperature. Even if the total pressure of a system is well above triple point of water, provided the [[partial pressure]] of the water vapour is 611.73 pascals then the system can still be brought to the triple point of water. Strictly speaking, the surfaces separating the different phases should also be perfectly flat, to negate the effects of surface tensions.

The gas–liquid–solid triple point of water corresponds to the minimum pressure at which liquid water can exist. At pressures below the triple point (as in [[outer space]]), solid ice when heated at constant pressure is converted directly into water vapour in a process known as [[Sublimation (phase transition)|sublimation]]. Above the triple point, solid ice when heated at constant pressure first melts to form liquid water, and then evaporates or boils to form vapor at a higher temperature.

For most substances the gas–liquid–solid triple point is also the minimum temperature at which the liquid can exist. For water, however, this is not true because the melting point of ordinary ice decreases as a function of pressure, as shown by the dotted green line in the phase diagram. At temperatures just below the triple point, compression at constant temperature transforms water vapor first to solid and then to liquid.

The triple point pressure of water was used during the [[Mariner 9]] mission to [[Mars]] as a reference point to define "sea level". More recent missions use [[laser altimetry]] and gravity measurements instead of pressure to define elevation on Mars.<ref>{{cite book |first=Michael H. |last=Carr |title=The Surface of Mars |location= |publisher=Cambridge University Press |year=2007 |page=5 |isbn=0-521-87201-4 }}</ref>

===Other triple points of water at high pressure===
At high pressures water has a complex phase diagram with 15 known [[Ice#Phases|phases of ice]] and a number of triple points including ten whose coordinates are shown in the diagram. For example, the triple point at 251 K (−22 °C) and 210 MPa (2070 atm) corresponds to the conditions for the coexistence of [[ice Ih]] (ordinary ice), [[ice III]] and liquid water, all at equilibrium. There are also triple points for the coexistence of three solid phases, for example [[ice II]], ice V and ice VI at 218 K (−55 °C) and 620 MPa (6120 atm).

For those high-pressure forms of ice which can exist in equilibrium with liquid, the diagram shows that melting points increase with pressure. At temperatures above 273 K (0 °C), increasing the pressure on water vapor results first in liquid water and then a high-pressure form of ice. In the range 251–273 K, ice I is formed first, followed by liquid water and then ice III or ice V, followed by other still denser high-pressure forms.

[[File:Phase_diagram_of_water.svg|thumb|725px|centre|Phase diagram of water including high-pressure forms ice II, ice III, etc. The pressure axis is logarithmic. For detailed descriptions of these phases, see [[Ice#Phases]].]]

==Triple point cells==
Triple point cells are used in the [[calibration]] of [[thermometer]]s. For exacting work, triple point cells are typically filled with a highly pure chemical substance such as hydrogen, argon, mercury, or water (depending on the desired temperature). The purity of these substances can be such that only one part in a million is a contaminant; what is called "six-nines" because it is 99.9999% pure. When it is a water-based cell, a special [[isotope|isotopic]] composition called [[Vienna Standard Mean Ocean Water|VSMOW]] is used because it is very pure and produces temperatures that are more comparable from lab to lab. Triple point cells are so effective at achieving highly precise, reproducible temperatures, an international calibration standard for thermometers called [[International Temperature Scale of 1990|ITS–90]] relies upon triple point cells of [[hydrogen]], [[neon]], [[oxygen]], [[argon]], [[mercury (element)|mercury]], and [[water (molecule)|water]] for delineating six of its defined temperature points.

==Table of triple points==
This table lists the gas–liquid–solid triple points of common substances. Unless otherwise noted, the data comes from the U.S. [[National Bureau of Standards]] (now [[NIST]], National Institute of Standards and Technology).<ref>{{cite book |first=Yunus A. |last=Cengel |first2=Robert H. |last2=Turner |title=Fundamentals of thermal-fluid sciences |location=Boston |publisher=McGraw-Hill |year=2004 |page=78 |isbn=0-07-297675-6 }}</ref>

{| class="wikitable sortable" style="text-align:center"
!width="150"|Substance !!width="125"| T [[[Kelvin|K]]] ([[Celsius|°C]]) !!width="200"| p [[[Pascal (unit)|kPa]]]*
|-
|align="left"| [[Acetylene]]
| {{sort|0192.4|{{convert|192.4|K|C}}}} || {{ntsh|120}} {{convert|120|kPa|atm|abbr=on}}
|-
|align="left"| [[Ammonia]]
| {{sort|0195.40|{{convert|195.40|K|C}}}} || {{ntsh|6.076}} {{convert|6.076|kPa|atm|abbr=on}}
|-
|align="left"| [[Argon]]
| {{sort|0083.81|{{convert|83.81|K|C}}}} || {{ntsh|68.9}} {{convert|68.9|kPa|atm|abbr=on}}
|-
|align="left"| [[Arsenic]]
| {{sort|1090|{{convert|1090|K|C}}}} || {{ntsh|3628}} {{convert|3628|kPa|atm|abbr=on}}
|-
|align="left"| [[Butane]]<ref>See [[Butane (data page)]]</ref>
| {{sort|0134.6|{{convert|134.6|K|C}}}} || {{ntsh|7e-4}} 7× 10−4 kPa
|-
|align="left"| Carbon ([[graphite]])
| {{sort|4765|{{convert|4765|K|C}}}} || {{ntsh|10132}} {{convert|10132|kPa|atm|abbr=on}}
|-
|align="left"| [[Carbon dioxide]]
| {{sort|0216.55|{{convert|216.55|K|C}}}} || {{ntsh|517}} {{convert|517|kPa|atm|abbr=on}}
|-
|align="left"| [[Carbon monoxide]]
| {{sort|0068.10|{{convert|68.10|K|C}}}} || {{ntsh|15.37}} {{convert|15.37|kPa|atm|abbr=on}}
|-
|align="left"| [[Chloroform]]<ref>See [[Chloroform (data page)]]</ref>
| {{sort|0175.43|{{convert|175.43|K|C}}}} || {{ntsh|0.870}} {{convert|0.870|kPa|atm|abbr=on}}
|-
|align="left"| [[Deuterium]]
| {{sort|0018.63|{{convert|18.63|K|C}}}} || {{ntsh|17.1}} {{convert|17.1|kPa|atm|abbr=on}}
|-
|align="left"| [[Ethane]]
| {{sort|0089.89|{{convert|89.89|K|C}}}} || {{ntsh|8e-4}} 8 × 10−4 kPa
|-
|align="left"| [[Ethanol]]<ref>See [[Ethanol (data page)]]</ref>
| {{sort|0150|{{convert|150|K|C}}}} || {{ntsh|4.3e-7}} 4.3 × 10−7 kPa
|-
|align="left"| [[Ethylene]]
| {{sort|0104.0|{{convert|104.0|K|C}}}} || {{ntsh|0.12}} {{convert|0.12|kPa|atm|abbr=on}}
|-
|align="left"| [[Formic acid]]<ref>See [[Formic acid (data page)]]</ref>
| {{sort|0281.40|{{convert|281.40|K|C}}}} || {{ntsh|2.2}} {{convert|2.2|kPa|atm|abbr=on}}
|-
|align="left"| [[Helium-4]] ([[lambda point]])
| {{sort|0002.19|{{convert|2.19|K|C}}}} || {{ntsh|5.1}} {{convert|5.1|kPa|atm|abbr=on}}
|-
|align="left"| [[Hexafluoroethane]]<ref>See [[Hexafluoroethane (data page)]]</ref>
| {{sort|0173.08|{{convert|173.08|K|C}}}} || {{ntsh|26.60}} {{convert|26.60|kPa|atm|abbr=on}}
|-
|align="left"| [[Hydrogen]]
| {{sort|0013.84|{{convert|13.84|K|C}}}} || {{ntsh|7.04}} {{convert|7.04|kPa|atm|abbr=on}}
|-
|align="left"| [[Hydrogen chloride]]
| {{sort|0158.96|{{convert|158.96|K|C}}}} || {{ntsh|13.9}} {{convert|13.9|kPa|atm|abbr=on}}
|-
|align="left"| [[Iodine]]<ref>{{cite book |last=Walas |first=S. M. |title=Chemical Process Equipment – Selection and Design |location=Amsterdam |publisher=Elsevier |year=1990 |page=639 |isbn=0-7506-7510-1 }}</ref>
| {{sort|0386.65|{{convert|386.65|K|C}}}} || {{ntsh|12.07}} {{convert|12.07|kPa|atm|abbr=on}}
|-
|align="left"| [[Isobutane]]<ref>See [[Isobutane (data page)]]</ref>
| {{sort|0113.55|{{convert|113.55|K|C}}}} || {{ntsh|1.9481e-5 }} 1.9481 × 10−5 kPa
|-
|align="left"| [[Mercury (element)|Mercury]]
| {{sort|0234.2|{{convert|234.2|K|C}}}} || {{ntsh|1.65e-7}} 1.65 × 10−7 kPa
|-
|align="left"| [[Methane]]
| {{sort|0090.68|{{convert|90.68|K|C}}}} || {{ntsh|11.7}} {{convert|11.7|kPa|atm|abbr=on}}
|-
|align="left"| [[Neon]]
| {{sort|0024.57|{{convert|24.57|K|C}}}} || {{ntsh|43.2}} {{convert|43.2|kPa|atm|abbr=on}}
|-
|align="left"| [[Nitric oxide]]
| {{sort|0109.50|{{convert|109.50|K|C}}}} || {{ntsh|21.92}} {{convert|21.92|kPa|atm|abbr=on}}
|-
|align="left"| [[Nitrogen]]
|{{sort|0063.18|{{convert|63.18|K|C}}}} || {{ntsh|12.6}} {{convert|12.6|kPa|atm|abbr=on}}
|-
|align="left"| [[Nitrous oxide]]
| {{sort|0182.34|{{convert|182.34|K|C}}}} || {{ntsh|87.85}} {{convert|87.85|kPa|atm|abbr=on}}
|-
|align="left"| [[Oxygen]]
| {{sort|0054.36|{{convert|54.36|K|C}}}} || {{ntsh|0.152}} {{convert|0.152|kPa|atm|abbr=on}}
|-
|align="left"| [[Palladium]]
| {{sort|1825|{{convert|1825|K|C}}}} || {{ntsh|3.5e-3}} 3.5 × 10−3 kPa
|-
|align="left"| [[Platinum]]
| {{sort|2045|{{convert|2045|K|C}}}} || {{ntsh|2e-4}} 2.0 × 10−4 kPa
|-
|align="left"| [[Sulfur dioxide]]
| {{sort|0197.69|{{convert|197.69|K|C}}}} || {{ntsh|1.67}} {{convert|1.67|kPa|atm|abbr=on}}
|-
|align="left"| [[Titanium]]
| {{sort|1941|{{convert|1941|K|C}}}} || {{ntsh|5.3e-3}} 5.3 × 10−3 kPa
|-
|align="left"| [[Uranium hexafluoride]]
| {{sort|0337.17|{{convert|337.17|K|C}}}} || {{ntsh|151.7}} {{convert|151.7|kPa|atm|abbr=on}}
|-
|align="left"| [[Water]]
| {{sort|0273.16|{{convert|273.16|K|C}}}} || {{ntsh|0.6117}} {{convert|0.6117|kPa|atm|abbr=on}}
|-
|align="left"| [[Xenon]]
| {{sort|0161.3|{{convert|161.3|K|C}}}} || {{ntsh|81.5}} {{convert|81.5|kPa|atm|abbr=on}}
|-
|align="left"| [[Zinc]]
| {{sort|0692.65|{{convert|692.65|K|C}}}} || {{ntsh|0.065}} {{convert|0.065|kPa|atm|abbr=on}}
|} * Note: for comparison, typical atmospheric pressure is 101.325 kPa (1 atm).

==See also==
*[[Critical point (thermodynamics)|Critical point]]
*[[Gibbs' phase rule]]

==References==
{{Reflist}}

{{Phase of matter}}

{{DEFAULTSORT:Triple Point}}
[[Category:Chemical properties]]
[[Category:Phase transitions]]
[[Category:Thermodynamics]]
[[Category:Threshold temperatures]]

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Taylor series