Editing Critical points

[[Image:press_temp.png|thumb|right]]
The '''critical point''', discovered in 1822 by Charles Cagniard de la Tour <ref>Charles Cagniard de la Tour "", Annales de chimie et de physique '''21''' pp. 127- (1822)</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) &sect; 83 p. 258</ref>. However, recent work using the [[Z1 and Z2 potentials |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]]
==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)]
*[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)]
;Books
* H. Eugene Stanley "Introduction to Phase Transitions and Critical Phenomena", Oxford University Press (1971) ISBN 9780195053166
* Cyril Domb "The Critical Point: A Historical Introduction To The Modern Theory Of Critical Phenomena", Taylor and Francis (1996) ISBN 9780748404353

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