Critical points

2010-08-30T08:37:36Z

Myrrhman: /* See also */

[[Image:press_temp.png|thumb|right]]
The '''critical point''', discovered in 1822 by Charles Cagniard de la Tour <ref>Charles Cagniard de la Tour "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", Annales de chimie et de physique '''21''' pp. 127-132 (1822)</ref><ref>[http://dx.doi.org/10.1590/S1806-11172009000200015 Bertrand Berche, Malte Henkel, and Ralph Kenna "Critical phenomena: 150 years since Cagniard de la Tour", Revista Brasileira de Ensino de Física '''31''' pp.2602.1-2602.4 (2009)] (in English [http://arxiv.org/abs/0905.1886v1 arXiv:0905.1886v1])</ref> , is a point found at the end of the liquid-vapour coexistence curve (the red point shown on the [[pressure-temperature]] plot on the right). At this point the [[temperature]] is known as the ''critical temperature'' <math>(T_c)</math>
and the [[pressure]] is known as the ''critical pressure'' <math>(P_c)</math>.
For an interesting discourse on the "discovery" of the liquid-vapour critical point, the Bakerian Lecture of [[Thomas Andrews]]
makes good reading <ref>[http://links.jstor.org/sici?sici=0261-0523%281869%29159%3C575%3ATBLOTC%3E2.0.CO%3B2-0 Thomas Andrews "The Bakerian Lecture: On the Continuity of the Gaseous and Liquid States of Matter", Philosophical Transactions of the Royal Society of London '''159''' pp. 575-590 (1869)]</ref>. Critical points are singularities in the [[partition function]].
In the critical point vicinity (Ref. <ref>[http://dx.doi.org/10.1080/00268978300102111 G. A. Martynov; G. N. Sarkisov "Exact equations and the theory of liquids. V", Molecular Physics '''49''' pp. 1495-1504 (1983)]</ref> Eq. 17a)

:<math> \left.\frac{\partial P}{\partial n}\right\vert_{T} \simeq 0</math>

and

:<math>n \int_0^{\infty} c(r) ~4 \pi r^2 ~{\rm d}r \simeq 1</math>

For a review of the critical region see the work of Michael E. Fisher <ref>[http://dx.doi.org/10.1063/1.1704197 Michael E. Fisher "Correlation Functions and the Critical Region of Simple Fluids", Journal of Mathematical Physics '''5''' pp. 944-962 (1964)]</ref>
<blockquote>
"... Turning now to the question of specific heats, it has long been known
that real gases exhibit a large ``anomalous" specific-heat maximum
above <math>T_c</math> which lies near the critical isochore and which is not expected on classical theory..."
</blockquote>
also
<blockquote>
"... measurements (Ref. <ref>[http://dx.doi.org/10.1016/S0031-8914(58)80093-2 A. Michels, J.M. Levelt and G.J. Wolkers "Thermodynamic properties of argon at temperatures between 0°C and −140°C and at densities up to 640 amagat (pressures up to 1050 atm.)", Physica '''24''' pp. 769-794 (1958)]</ref> ) of <math>C_V(T)</math> for argon along the critical isochore suggest strongly that
<math>C_V(T) \rightarrow \infty ~{\rm as} ~ T \rightarrow T_c \pm</math>. Such a result is again inconsistent with classical theory."
</blockquote>
Thus in the vicinity of the liquid-vapour critical point, both the [[Compressibility | isothermal compressibility]]
and the [[heat capacity]] at constant pressure diverge to infinity.
==Liquid-liquid critical point==
==Solid-liquid critical point==
It is widely held that there is no solid-liquid critical point. The reasoning behind this was given on the grounds of symmetry by Landau and Lifshitz
<ref>L. D. Landau and E. M. Lifshitz, "Statistical Physics" (Course of Theoretical Physics, Volume 5) 3rd Edition Part 1, Chapter XIV, Pergamon Press (1980) § 83 p. 258</ref>. However, recent work using the [[Z2 potential]] suggests that this may not be the last word on the subject.
<ref>[http://dx.doi.org/10.1063/1.3213616 Måns Elenius and Mikhail Dzugutov "Evidence for a liquid-solid critical point in a simple monatomic system", Journal of Chemical Physics 131, 104502 (2009)]</ref>.
==Tricritical points==
*[http://dx.doi.org/10.1103/PhysRevLett.24.715 Robert B. Griffiths "Thermodynamics Near the Two-Fluid Critical Mixing Point in He<sup>3</sup> - He<sup>4</sup>", Physical Review Letters '''24''' 715-717 (1970)]
*[http://dx.doi.org/10.1063/1.451007 Lech Longa "On the tricritical point of the nematic–smectic A phase transition in liquid crystals", Journal of Chemical Physics '''85''' pp. 2974-2985 (1986)]
==Critical exponents==
:''Main article: [[Critical exponents]]''
==Yang-Yang anomaly==
:''Main article: [[Yang-Yang anomaly]]''
==See also==
*[[Binder cumulant]]
*[[Law of corresponding states]]
*[http://www.mac-how.net Mac How]

==References==
<references/>
'''Related reading'''
* M. I. Bagatskii and A. V. Voronel and B. G. Gusak "", Journal of Experimental and Theoretical Physics '''16''' pp. 517- (1963)
* [http://dx.doi.org/10.1103/PhysRevA.2.1047 Robert B. Griffiths and John C. Wheeler "Critical Points in Multicomponent Systems", Physical Review A '''2''' 1047 - 1064 (1970)]
* [http://dx.doi.org/10.1103/RevModPhys.46.597 Michael E. Fisher "The renormalization group in the theory of critical behavior", Reviews of Modern Physics '''46''' pp. 597 - 616 (1974)]
* [http://dx.doi.org/10.1146/annurev.pc.37.100186.001201 J. V. Sengers and J. M. H. Levelt Sengers "Thermodynamic Behavior of Fluids Near the Critical Point", Annual Review of Physical Chemistry '''37''' pp. 189-222 (1986)]
* [http://dx.doi.org/10.1103/PhysRevLett.93.015701 Kamakshi Jagannathan and Arun Yethiraj "Molecular Dynamics Simulations of a Fluid near Its Critical Point", Physical Review Letters '''93''' 015701 (2004)]
* Cyril Domb "The Critical Point: A Historical Introduction To The Modern Theory Of Critical Phenomena", Taylor and Francis (1996) ISBN 9780748404353
*[http://dx.doi.org/10.1080/00268976.2010.495734 Kurt Binder "Computer simulations of critical phenomena and phase behaviour of fluids", Molecular Physics '''108''' pp. 1797-1815 (2010)]

[[category: statistical mechanics]]
[[category:classical thermodynamics]]

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Critical points