Critical points

The critical point, discovered in 1822 by Charles Cagniard de la Tour , 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 \((T_c)\) and the pressure is known as the critical pressure \((P_c)\). For an interesting discourse on the "discovery" of the liquid-vapour critical point, the Bakerian Lecture of Thomas Andrews makes good reading . Critical points are singularities in the partition function. In the critical point vicinity (Ref. Eq. 17a)

\[ \left.\frac{\partial P}{\partial n}\right\vert_{T} \simeq 0\]

and

\[n \int_0^{\infty} c(r) ~4 \pi r^2 ~{\rm d}r \simeq 1\]

For a review of the critical region see the work of Michael E. Fisher

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

also

"... measurements (Ref. ) of \(C_V(T)\) for argon along the critical isochore suggest strongly that \(C_V(T) \rightarrow \infty ~{\rm as} ~ T \rightarrow T_c \pm\). 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.

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 . However, recent work using the Z2 potential suggests that this may not be the last word on the subject. .

Tricritical points

• Robert B. Griffiths "Thermodynamics Near the Two-Fluid Critical Mixing Point in He³ - He⁴", Physical Review Letters 24 715-717 (1970)
• 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

Related reading

• M. I. Bagatskii and A. V. Voronel and B. G. Gusak "", Journal of Experimental and Theoretical Physics 16 pp. 517- (1963)
• Robert B. Griffiths and John C. Wheeler "Critical Points in Multicomponent Systems", Physical Review A 2 1047 - 1064 (1970)
• Michael E. Fisher "The renormalization group in the theory of critical behavior", Reviews of Modern Physics 46 pp. 597 - 616 (1974)
• 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)
• Kamakshi Jagannathan and Arun Yethiraj "Molecular Dynamics Simulations of a Fluid near Its Critical Point", Physical Review Letters 93 015701 (2004)
• 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