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."

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)