Critical points

Critical points are singularities in the partition function. In the critical point vicinity (Ref. 1 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 (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 $T_{c}$ which lies near the critical isochore and which is not expected on classical theory..." (Ref. 3)

also

... measurements (Ref 4) 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.

Critical exponents

Specific heat, C

$\left.C\right.=C_{0}\epsilon ^{-\alpha }$

Magnetic order parameter, m,

$\left.m\right.=m_{0}\epsilon ^{\beta }$

Susceptibility

$\left.\chi \right.=\chi _{0}\epsilon ^{-\gamma }$

Correlation length

$\left.\xi \right.=\xi _{0}\epsilon ^{-\nu }$

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

$\epsilon =\left|1-{\frac {T}{T_{c}}}\right|$

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

$\alpha +2\beta +\gamma =2$

Gamma divergence

When approaching the critical point along the critical isochore ($T>T_{c}$) the divergence is of the form

$\left.\right.C_{p}\sim \kappa _{T}\sim (T-T_{c})^{-\gamma }\sim (p-p_{c})^{-\gamma }$

where $\gamma$ 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

$\left.\right.\kappa _{T}\sim (p-p_{c})^{-\epsilon }$

where $\epsilon$ 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)