# Critical points

The critical point, discovered in 1822 by Charles Cagniard de la Tour [1][2] , 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 ${\displaystyle (T_{c})}$ and the pressure is known as the critical pressure ${\displaystyle (P_{c})}$. For an interesting discourse on the "discovery" of the liquid-vapour critical point, the Bakerian Lecture of Thomas Andrews makes good reading [3]. Critical points are singularities in the partition function. In the critical point vicinity (Ref. [4] Eq. 17a)

${\displaystyle \left.{\frac {\partial P}{\partial n}}\right\vert _{T}\simeq 0}$

and

${\displaystyle n\int _{0}^{\infty }c(r)~4\pi r^{2}~{\rm {d}}r\simeq 1}$

For a review of the critical region see the work of Michael E. Fisher [5]

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

also

"... measurements (Ref. [6] ) of ${\displaystyle C_{V}(T)}$ for argon along the critical isochore suggest strongly that ${\displaystyle C_{V}(T)\rightarrow \infty ~{\rm {as}}~T\rightarrow T_{c}\pm }$. 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 [7]. However, recent work using the Z2 potential suggests that this may not be the last word on the subject. [8].

## Critical exponents

Main article: Critical exponents

## Yang-Yang anomaly

Main article: Yang-Yang anomaly

## References

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