Critical points: Difference between revisions

From SklogWiki
Jump to navigation Jump to search
No edit summary
No edit summary
Line 21: Line 21:
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.
===Gamma divergence===
===Gamma divergence===
When approaching the critical point along the critical isochore (<math>T > T_c</math>) the divergence is of the form
When approaching the critical point along the critical isochore (<math>T > T_c</math>) the divergence is of the form
Line 40: Line 62:
# M. I. Bagatskii and A. V. Voronel and B. G. Gusak "", Journal of Experimental and Theoretical Physics '''16''' pp. 517- (1963)
# 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/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)]
[[category: statistical mechanics]]
[[category: statistical mechanics]]
[[category:classical thermodynamics]]
[[category:classical thermodynamics]]

Revision as of 13:49, 19 September 2007

Critical points are singularities in the partition function. In the critical point vicinity (Ref. 1 Eq. 17a)

and

For a review of the critical region see the work of Michael E. Fisher (Ref. 2).

... Turning now to the question of specific heats, it has long been known
that real gases exhibit a large ``anomalous" specific-heat maximum
above  which lies near the critical isochore and which is not expected on classical theory..." (Ref. 3)

also

... measurements (Ref 4) of  for argon along the critical isochore suggest strongly that
. Such a result is again inconsistent with classical theory."

Thus in the vicinity of the liquid-vapour critical point, both the isothermal compressibility and the heat capacity at constant pressure diverge to infinity.

Critical exponents

Specific heat, C

Magnetic order parameter, m,

Susceptibility

Correlation length

where is the reduced distance from the critical temperature, i.e.

Note that this implies a certain symmetry when the critical point is approached from either 'above' or 'below', which is not necessarily the case.

Gamma divergence

When approaching the critical point along the critical isochore () the divergence is of the form

where 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

where is 2/3 for the Van der Waals equation of state, and is usually 0.75 to 0.8.

References

  1. G. A. Martynov; G. N. Sarkisov "Exact equations and the theory of liquids. V", Molecular Physics 49 pp. 1495-1504 (1983)
  2. Michael E. Fisher "Correlation Functions and the Critical Region of Simple Fluids", Journal of Mathematical Physics 5 pp. 944-962 (1964)
  3. 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)
  4. M. I. Bagatskii and A. V. Voronel and B. G. Gusak "", Journal of Experimental and Theoretical Physics 16 pp. 517- (1963)
  5. Robert B. Griffiths and John C. Wheeler "Critical Points in Multicomponent Systems", Physical Review A 2 1047 - 1064 (1970)
  6. Michael E. Fisher "The renormalization group in the theory of critical behavior", Reviews of Modern Physics 46 pp. 597 - 616 (1974)