Critical point (thermodynamics)

Liquid–vapor critical point

Overview

The liquid–vapor critical point was the first critical point to be discovered, and it remains the best known and most studied one.

The figure shows the schematic P-T diagram of a pure substance (as opposed to mixtures, which have additional state variables and richer phase diagrams, discussed below). The commonly known phases solid, liquid and vapor are separated by phase boundaries, i.e. pressure–temperature combinations where two phases can coexist. At the triple point, all three phases can coexist. However, the liquid–vapor boundary terminates in an endpoint at some critical temperature T_c and critical pressure p_c. This is the critical point.

The critical point of water occurs at 647.096 K (373.946 °C; 705.103 °F) and 22.064 megapascals (3,200.1 psi; 217.75 atm; 220.64 bar).

In the vicinity of the critical point, the physical properties of the liquid and the vapor change dramatically, with both phases becoming even more similar. For instance, liquid water under normal conditions is nearly incompressible, has a low thermal expansion coefficient, has a high dielectric constant, and is an excellent solvent for electrolytes. Near the critical point, all these properties change into the exact opposite: water becomes compressible, expandable, a poor dielectric, a bad solvent for electrolytes, and mixes more readily with nonpolar gases and organic molecules.

At the critical point, only one phase exists. The heat of vaporization is zero. There is a stationary inflection point in the constant-temperature line (critical isotherm) on a PV diagram. This means that at the critical point:

$${\displaystyle \begin{array}{cc}{\left(\frac{\mathrm{\partial }p}{\mathrm{\partial }V}\right)}_T& =0\\ {\left(\frac{{\mathrm{\partial }}^{2}p}{\mathrm{\partial }{V}^2}\right)}_T& =0\end{array}}$$

Above the critical point there exists a state of matter that is continuously connected with (can be transformed without phase transition into) both the liquid and the gaseous state. It is called supercritical fluid. The common textbook knowledge that all distinction between liquid and vapor disappears beyond the critical point has been challenged by Fisher and Widom, who identified a p–T line that separates states with different asymptotic statistical properties (Fisher–Widom line). Note that even at temperatures above the critical point temperature, sufficient pressure can still potentially compress the matter into a solid.

Sometimes the critical point does not manifest in most thermodynamic or mechanical properties, but is "hidden" and reveals itself in the onset of inhomogeneities in elastic moduli, marked changes in the appearance and local properties of non-affine droplets, and a sudden enhancement in defect pair concentration.

History

The existence of a critical point was first discovered by Charles Cagniard de la Tour in 1822 and named by Dmitri Mendeleev in 1860 and Thomas Andrews in 1869. Cagniard showed that CO₂ could be liquefied at 31 °C at a pressure of 73 atm, but not at a slightly higher temperature, even under pressures as high as 3000 atm.

Theory

Solving the above condition ${\displaystyle (\mathrm{\partial }p/\mathrm{\partial }V{)}_T=0}$ for the van der Waals equation, one can compute the critical point as

${\displaystyle T_{\text{c}}=\frac{8a}{27Rb},V_{\text{c}}=3nb,p_{\text{c}}=\frac{a}{27b^2}.}$

However, the van der Waals equation, based on a mean-field theory, does not hold near the critical point. In particular, it predicts wrong scaling laws.

To analyse properties of fluids near the critical point, reduced state variables are sometimes defined relative to the critical properties

${\displaystyle T_{\text{r}}=\frac{T}{T_{\text{c}}},p_{\text{r}}=\frac{p}{p_{\text{c}}},V_{\text{r}}=\frac{V}{R{T}_{\text{c}}/p_{\text{c}}}.}$

The principle of corresponding states indicates that substances at equal reduced pressures and temperatures have equal reduced volumes. This relationship is approximately true for many substances, but becomes increasingly inaccurate for large values of p_r.

For some gases, there is an additional correction factor, called Newton's correction, added to the critical temperature and critical pressure calculated in this manner. These are empirically derived values and vary with the pressure range of interest.

Table of liquid–vapor critical temperature and pressure for selected substances