Figure 9.3 shows the isotherms of the van der Waals equation of state plotted in the - plane. It can be seen that, for temperatures that exceed the critical temperature (i.e., ), reducing the molar volume of the fluid causes its pressure to rise monotonically, eventually becoming infinite when (i.e., when the molecules are closely packed). On the other hand, for temperatures less than the critical temperature (i.e., ), as is decreased the pressure rises, falls, and then rises again, which seems to imply that, for some molar volumes, compressing the fluid can cause its pressure to decrease. However, according to the analysis of Section 9.5, if a phase is such that compressing it causes its pressure to decrease (i.e., if its isothermal compressibility is negative) then the phase is unstable to density fluctuations. Thus, below the critical temperature, the stable states on a given isotherm are divided into two groups. The first group is characterized by relatively small molar volumes--these are liquid states. The second group is characterized by relatively large molar volumes--these are gas states. The liquid and gas states are separated by a set of unstable states. Above the critical temperature, the distinct liquid and gas states cease to exist--there is only a single stable fluid state. Consider the region of the - plane in which the states predicted by Equation (9.82) are unstable. It turns out that our analysis has broken down here because we assumed the existence of a single phase, whereas, in fact, in this region of the - plane, there exists a stable mixture of liquid and gas phases in equilibrium with one another. Let us investigate further.
At a given temperature and pressure, the true equilibrium state of a system is that which minimizes the Gibbs free energy. (See Section 9.4.) For the case of a van der Waals fluid,
Figure 9.4 shows plotted as a function of for . Also shown is the corresponding isotherm in the - plane. Although the van der Waals equation of state associates some pressures with more than one molar volume, the thermodynamically stable state is that with the lowest Gibbs free energy. Thus, the triangular loop in the graph of versus (points 2-3-4-5-6) corresponds to unstable states. As the pressure is gradually increased, the system will go straight from point 2 to point 6, with an abrupt reduction in molar volume. Of course, this corresponds to a gas-liquid phase transition. At intermediate molar volumes, the thermodynamically stable state is a mixture of gaseous and liquid phases at the transition pressure, as illustrated by the straight-line in the - graph. The curved portion of the isotherm that is cut off by this straight-line correctly indicates what the allowed states would be if the fluid were homogeneous. However, these homogeneous states are unstable, because there is always another mixed state at the same pressure that possesses a lower Gibbs free energy.
The pressure at the phase transition can easily be determined from the graph of versus . However, it is also possible to determine this pressure directly from the - isotherm. Note that the net change in in going around the triangular loop 2-3-4-5-6 in Figure 9.4 is zero. Hence,
Repeating the Maxwell construction for a variety of temperatures yields the phase diagram shown in Figure 9.5. [See J. Lekner, Am. J. Phys. 50, 161 (1982) for a practical method of implementing the Maxwell construction analytically. See, also, Exercise 7.] For each temperature, there is a well-defined pressure, known as the vapor pressure, at which the liquid-gas transformation takes place. Plotting this pressure versus the temperature yields the phase diagram shown in Figure 9.6. Note that the liquid-gas phase boundary disappears at the critical temperature ( ), because there is no phase transformation above this temperature. The point at which the phase boundary disappears is called the critical point. The corresponding temperature, pressure, and volume are known as the critical temperature, pressure, and volume, respectively. At the critical point, the properties of the liquid and gas phases become identical.
Consider a phase transformation at some temperature in which the gaseous phase has the molar volume , whereas the liquid phase has the molar volume . Now, the molar entropy of a van der Waals gas can be written