# Triple point Phase diagram of an "ordinary" substance and water, each with a representation of the triple point and the critical point .

In the thermodynamics is a triple point (also triple point ) a state of a group consisting of a single material component system in which temperature and pressure of three phases in the thermodynamic equilibrium are.

The phases involved, the three states of aggregation represent the substance, as well as various modifications of the solid or the liquid phase; an example of this are the various possible crystal structures of water ice. At triple points, depending on the substance, a solid, a liquid and a vapor phase can coexist, but also two solid phases and a liquid, or two solid and one vapor, or three solid, in rare cases two liquid and one vapor.

In a pressure-temperature diagram ( p - T diagram for short ) a triple point state is represented as a point. In particular, if it is a p - T phase diagram and the three coexisting phases are the three different states of aggregation, then the triple point is the intersection of the two phase boundary lines, saturation vapor pressure and melting curve: the liquid and vapor phases are in equilibrium on the saturation vapor pressure curve the melting curve the liquid and the solid phase, at the intersection of the two curves all three.

## Number of triple points

If there are possible phases in the system, then it has triple points. For example, sulfur can exist in four phases: one vapor, one liquid and two solid (one with a rhombic and one with a monoclinic crystal lattice). There are therefore four triple points in the phase diagram of sulfur. If water only had the three phases solid, liquid and vapor, then it would have exactly one triple point. Due to the existence of various ice modifications, there are further triple points. ${\ displaystyle p}$ ${\ displaystyle \ textstyle {\ frac {p!} {(p-3)! 3!}}}$ ## Triple point and Gibbs phase rule

A triple point represents a special case of Gibbs' phase rule:

${\ displaystyle f = N-P + 2}$ The degree of freedom of the system (here a one-component system:, with three phases:) is always in triple points according to the phase rule : If you change an intensive state variable, the phase equilibrium is left immediately. This is also the reason why there cannot be 4 phase boundary lines in a one-component system that meet at one point (would be here ). ${\ textstyle f}$ ${\ textstyle N = 1}$ ${\ textstyle P = 3}$ ${\ textstyle f = 0}$ ${\ textstyle f = -1}$ ## Example: triple point of water

The triple point of water is the point in the p - T diagram of water in which the three phases of pure liquid water, pure water ice and water vapor are in equilibrium. The pressure common to all three phases is equal to the saturation vapor pressure of the pure water at the temperature common to all three phases.

According to the internationally accepted best value from Guildner, Johnson and Jones, this pressure is 611.657 (± 0.010)  Pa (approx. 6  mbar ).

The triple point temperature was until May 19, 2019 - as the defining fixed point on the temperature scale - exactly 273.16  K (or 0.01  ° C ). Since the redefinition of the SI units in 2019 , the temperature scale has been defined independently of the water, so that the triple point temperature again has to be determined experimentally with a certain measurement uncertainty . When the new definition was introduced, this uncertainty was 100 µK; within this uncertainty the numerical value is still 273.16 K.

If the three phases are brought into equilibrium in a container (e.g. in a triple point cell), the triple point temperature and triple point pressure are maintained for a longer period of time, even if a small heat flow flows through the not perfectly insulating wall of the container. As with all phase transitions in water, the inflow or outflow of heat is compensated for by the corresponding latent heat conversion, in that the proportion of the phases shifts accordingly due to melting, freezing, evaporation, condensation or sublimation processes. Pressure and temperature remain constant until one of the phases is used up. Because of this property and the precisely defined triple point, such a triple point cell is suitable for calibration purposes.

In addition to the coexistence point of liquid, frozen and vaporous water described here, there are other but less significant triple points in the phase diagram of water, in which two or three different ice modifications are involved (see the table below).

## Triple area In the three-dimensional phase diagram of water, the system moves along the triple line when its volume is changed while it has triple point pressure and temperature.

As described, the triple point is a point in the p - T diagram, so that the pressure and temperature of a system located in the triple point are clearly determined. Nevertheless, when the system is at this point, it can assume various states of equilibrium - as long as none of the phases completely disappear. The system can exchange heat with the environment and its volume can change; the relative proportions of the three phases change. The extensive coordinates of internal energy and total volume can be used as state coordinates. In the UV diagram, the states of simultaneous coexistence of the three phases fill a two-dimensional area, while pressure and temperature have degenerated into a constant value here. In addition to the recognizability, it is precisely this insensitivity to small fluctuations in the volume and the heat supply and removal that distinguishes the triple point for use as a temperature reference.

If the states of equilibrium are represented as an area in a pressure - volume - temperature space, see graphic, the triple area shrinks to a one-dimensional line, the triple line , which is characterized by the constant triple point values ​​for pressure and temperature. Along the triple line, different volume values ​​correspond to changes in the proportions of the phases.

The variability of the extensive quantities volume and internal energy does not contradict Gibbs' phase rule, since this rule only makes statements about intensive variables.

## Scale fixed points and other values

The uniqueness of the triple point provides particularly good temperature fixed points for calibrating the scales of thermometers .

Common triple point temperature specifications, e.g. B. According to the International Temperature Scale of 1990 , are:

• Water : 273.16 K (0.01 ° C) at 611.657 ± 0.010 Pa
• Mercury : ITS-90 fixed point 234.3156 K (−38.8344 ° C) at 1.65 · 10 −4  Pa

The ITS-90 supplies further temperatures for triple points .

Triple points of some substances
material temperature pressure
Surname Molecular formula K ° C kPa
water H 2 O 273.16 −000.01 000.611657
oxygen O 2 054,361 −218.789 000.14633
Carbon dioxide CO 2 216,592 0−56.558 517.95
nitrogen N 2 063,151 −209.999 012,523
ammonia NH 3 195.5 0−77.65 006.1
Triple points of water
involved phases temperature Pressure (MPa)
liquid water, ice I h , water vapor −00.01 ° C 000.000 611 657
liquid water, ice I h , ice III −22 ° C 209.9
liquid water, ice III, ice V −17 ° C 350.1
liquid water, ice V, ice VI −00.16 ° C 632.4
Ice I h , ice II, ice III −35 ° C 213
Ice II, Ice III, Ice V −24 ° C 344
Ice II, Ice V, Ice VI −70 ° C 626

## Higher states with no degree of freedom

As the phase rule shows, a state of a one-component system in which three phases exist in equilibrium has no remaining degrees of freedom (it is an "invariant" or "non-variant" state), and no more than three phases can coexist in such a system . From the phase rule, however, it also follows that in systems that consist of several independent components, more than three phases can be in equilibrium at invariant points.

If a system consisting of two independent components can break down into four different phases, then a state in which all four phases are in equilibrium is an invariant state. An example of such a quadruple point is the eutectic point of a binary system with those in equilibrium

• two solid phases,
• a liquid phase and
• a gaseous phase.

### Quintuple point

If a system consisting of three independent components can break down into five different phases, then a state in which all five phases are in equilibrium is an invariant state. An example is a system made up of three components

• Water, H 2 O
• Sodium sulfate, Na 2 SO 4
• Magnesium sulfate, MgSO 4

consists. At a temperature of 22 ° C, the following five phases can coexist:

• Steam
• a liquid mixture of water, sodium sulfate, and magnesium sulfate
• Crystals of sodium sulfate decahydrate, Na 2 SO 4 · 10 H 2 O
• Crystals of Magnesium Sulphate Decahydrate, MgSO 4 · 10 H 2 O
• Crystals of sodium magnesium sulfate tetrahydrate, Na 2 Mg [SO 4 ] 2 · 4 H 2 O

## Trivia

The only known substance that does not have a solid / liquid / gaseous triple point is helium . It has a triple point where liquid helium I , liquid helium II and gaseous helium coexist, and a triple point where liquid helium I, liquid helium II and solid helium are in equilibrium.

The triple point temperature of water (273.16 K) is the temperature at which pure water and pure ice are in equilibrium with their vapor under the condition that the pressure in all three phases is equal to the saturation vapor pressure of the water at this temperature. If one assumes a different pressure than the water's own saturation vapor pressure, then the three phases are in equilibrium at a different temperature. At a pressure of 1013.25 hPa, air-saturated liquid water, air-saturated water ice and water-vapor-saturated air are in equilibrium at a temperature that is around 0.01 degrees lower than the triple point temperature, namely at the ice point temperature 273.15 K.

## Individual evidence

1. a b c Entry on triple point . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.T06502 Version: 2.3.3. : "The point in a one-component system at which the temperature and pressure of three phases are in equilibrium."
2. ^ EA Guggenheim: Modern Thermodynamics by the Methods of Willard Gibbs. Methuen & Co., London 1933, p. 59, Text Archive - Internet Archive
3. CF Bohren, BA Albrecht: Atmospheric Thermodynamics. Oxford University Press, New York / Oxford 1998, ISBN 978-0-19-509904-1 , pp. 221f.
4. LA Guildner, DP Johnson, FE Jones: Vapor Pressure of Water at Its Triple Point . In: Journal of Research of the National Bureau of Standards - A. Physics and Chemistry , Vol. 80A, No. 3, May-June 1976, pp. 505-521; nist.gov (PDF; 18.1 MB)
5. International Association for the Properties of Water and Steam (IAPWS): Guideline on the Use of Fundamental Physical Constants and Basic Constants of Water. IAPWS G5-01 (2016) ( PDF, 40 KB ): "For the triple-point pressure, we recommend the value measured by Guildner et al. [...], which is (611.657 ± 0.010) Pa. "
6. Definition of the temperature unit Kelvin (K): SI Brochure 8th ed.Bureau International des Poids et Mesures, accessed on June 2, 2013 .
7. 26th CGPM (2018) - Resolutions adopted / Résolutions adoptées. (PDF; 1.2 MB) Versailles 13–16 November 2018. In: bipm.org. Bureau International des Poids et Mesures, November 19, 2018, pp. 2–5 , accessed on May 6, 2019 (English, French).
8. M. Stock, R. Davis, E. de Mirandés, MJT Milton: The revision of the SI - the result of three decades of progress in metrology . In: Metrologia , Volume 56, Number 2 (2019), pp. 8 and 9, doi: 10.1088 / 1681-7575 / ab0013
9. ^ Bureau International des Poids et Mesures: Le Système international d'unités, The International System of Units. 9 e édition, Sèvres 2019, ISBN 978-92-822-2272-0 , pp. 21, 133; bipm.org (PDF; 2 MB)
10. ^ Elliott H. Lieb , Jakob Yngvason : The Physics and Mathematics of the Second Law of Thermodynamics . In: Physics Reports . tape  310 , no. 1 , 1999, III Simple Systems, A Coordinates of simple systems, p. 37 u. 101 , see Fig. 8 , doi : 10.1016 / S0370-1573 (98) 00082-9 , arxiv : cond-mat / 9708200 (English).
11. VDI Society for Process Engineering and Chemical Engineering (ed.): VDI-Wärmeatlas . 11th edition. Springer-Verlag, Berlin / Heidelberg 2013, ISBN 978-3-642-19980-6 , Part D.3 Thermophysical material properties.
12. ^ P. Duhem: Thermodynamics and Chemistry. (GK Burgess transl.). John Wiley & Sons, New York 1903, p. 192 Textarchiv - Internet Archive
13. ^ Brockhaus ABC Chemie , VEB FA Brockhaus Verlag Leipzig 1965, p. 1151.
14. ^ A b P. Duhem: Thermodynamics and Chemistry. (GK Burgess transl.), John Wiley & Sons, New York 1903, p. 193 Textarchiv - Internet Archive
15. ^ AF Holleman , N. Wiberg : Inorganische Chemie . 103rd edition. Volume 1: Basics and main group elements. Walter de Gruyter, Berlin / Boston 2016, ISBN 978-3-11-049585-0 , p. 462 (Reading sample: Part A - Basics of the chemistry of hydrogen. Google book search ).