Editing Critical points
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==Introduction== | |||
For an interesting discourse on the "discovery" of the gas-liquid critical point, the Bakerian Lecture of [[Thomas Andrews]] | |||
For an interesting discourse on the "discovery" of the liquid | makes interesting reading (Ref. 1). | ||
makes | Critical points are singularities in the [[partition function]]. | ||
In the critical point vicinity (Ref. | In the critical point vicinity (Ref. 2 Eq. 17a) | ||
:<math> \left.\frac{\partial P}{\partial n}\right\vert_{T} \simeq 0</math> | :<math> \left.\frac{\partial P}{\partial n}\right\vert_{T} \simeq 0</math> | ||
Line 12: | Line 12: | ||
:<math>n \int_0^{\infty} c(r) ~4 \pi r^2 ~{\rm d}r \simeq 1</math> | :<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 | For a review of the critical region see the work of Michael E. Fisher (Ref. 3). | ||
"... Turning now to the question of specific heats, it has long been known | "... Turning now to the question of specific heats, it has long been known | ||
that real gases exhibit a large ``anomalous" specific-heat maximum | 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..." | above <math>T_c</math> which lies near the critical isochore and which is not expected on classical theory..." (Ref. 3) | ||
also | also | ||
"... measurements (Ref | "... measurements (Ref 4) 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." | <math>C_V(T) \rightarrow \infty ~{\rm as} ~ T \rightarrow T_c \pm</math>. Such a result is again inconsistent with classical theory." | ||
Thus in the vicinity of the liquid-vapour critical point, both the [[Compressibility | isothermal compressibility]] | 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. | and the [[heat capacity]] at constant pressure diverge to infinity. | ||
== | ==Critical exponents== | ||
[[Heat capacity |Specific heat]], ''C'' | |||
< | :<math>\left. C\right.=C_0 \epsilon^{-\alpha}</math> | ||
< | |||
Magnetic order parameter, ''m'', | |||
:<math>\left. m\right. = m_0 \epsilon^\beta</math> | |||
[[Susceptibility]] | |||
:<math>\left. \chi \right. = \chi_0 \epsilon^{-\gamma}</math> | |||
Correlation length | |||
:<math>\left. \xi \right.= \xi_0 \epsilon^{-\nu}</math> | |||
where <math>\epsilon</math> is the reduced distance from the critical [[temperature]], i.e. | |||
:<math>\epsilon = \left| 1 -\frac{T}{T_c}\right|</math> | |||
Note that this implies a certain symmetry when the critical point is approached from either 'above' or 'below', which is not necessarily the case. | |||
Rushbrooke equality | |||
:<math>\alpha + 2\beta + \gamma =2</math> | |||
====Gamma divergence==== | |||
When approaching the critical point along the critical isochore (<math>T > T_c</math>) the divergence is of the form | |||
:<math>\left. \right. C_p \sim \kappa_T \sim (T-T_c)^{-\gamma} \sim (p-p_c)^{-\gamma}</math> | |||
where <math>\gamma</math> is 1.0 for the [[Van der Waals equation of state]], and is usually 1.2 to 1.3. | |||
====Epsilon divergence==== | |||
When approaching the critical point along the critical isotherm the divergence is of the form | |||
:<math>\left. \right. \kappa_T \sim (p-p_c)^{-\epsilon}</math> | |||
where <math>\epsilon</math> is 2/3 for the [[Van der Waals equation of state]], and is usually 0.75 to 0.8. | |||
==Tricritical points== | ==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/10.1103/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)] | *[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)] | ||
==See also== | ==See also== | ||
*[[Binder cumulant]] | *[[Binder cumulant]] | ||
==References== | ==References== | ||
#[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)] | |||
''' | #[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)] | ||
#[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)] | |||
#[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)] | |||
# 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)] | |||
[[category: statistical mechanics]] | [[category: statistical mechanics]] | ||
[[category:classical thermodynamics]] | [[category:classical thermodynamics]] |