Vapor—liquid coexistence

Figure 7.1 Schematic phase diagram of water (not to scale), showing phase boundaries (heavy solid lines), triple point (triangle), critical point (circle-x), and a representative point (circle, dotted lines) at 25°C on the liquid-vapor coexistence curve.

What does it mean that (25°C, 23.8 Torr) is a point on the liquid-vapor coexistence line Consider a beaker of liquid water at 25°C, covered with a lid and allowed to come into equilibrium with its own vapor ... 

Figure 7.1 also shows the critical point (circle-x, dashed lines) of water, the terminus of the liquid-vapor coexistence line. Beyond this point (which occurs at Tc = 374°C, Pc = 217.7 atm), there is no longer a sensible distinction between liquid and vapor, so one should only speak of a supercritical fluid (or simply fluid ) beyond the dashed lines. A sample of water above Pc can never exhibit a boiling point, no matter how far the temperature is increased, nor can a sample above Tc exhibit condensation, no matter how far the pressure is increased. 

The critical state is evidently an invariant point (terminus of a line) in this case, because it lies at a dimensional boundary between states of / =2 (p = 1) and /= 1 (p = 2). The critical point is therefore a uniquely specified state for a pure substance, and it plays an important role (Section 2.5) as a type of origin or reference state for description of all thermodynamic properties. Note that a limiting critical terminus appears to be a universal feature of liquid-vapor coexistence lines, whereas (as shown in Fig. 7.1) solid-liquid and solid-vapor lines extend indefinitely or form closed networks with other coexistence lines. 

Vaporization Transition Clausius-Clapeyron Equation For the liquid-vapor coexistence line ( vapor-pressure curve ), the Clapeyron equation (7.29) becomes... 

Figure 7.3 provides a more complete picture of the H20 phase diagram in the range of pressures up to lOkbar. (For comparison, the pressure at the bottom of Earth s deepest oceans is about 0.5 kbar and that in planetary interiors ranges above 1 Mbar.) The region covered in Fig. 7.1 is the narrow sliver under the dotted line in the lower right comer, with the entire liquid-vapor coexistence curve visually indistinguishable from... 

The principal features of elemental sulfur in the displayed T, P range are the usual liquid and vapor phases and two solid forms, a-sulfur ( red sulfur, of orthorhombic crystalline form) and /3-sulfur ( yellow sulfur, monoclinic needle-like crystals), both of which are available as common stockroom species. The stable phase ranges for each elemental form are shown by the solid lines in Fig. 7.5. The liquid-vapor coexistence line terminates in a critical point at 1041°C, and will not be discussed further. 

Starting with a study on the liquid-vapor coexistence of ammonium chloride (NH4C1) [34], there have been repeated reports on classical ionic criticality [4], but none of these studies allows unambiguous conclusions [14]. In 1990 more decisive results were reported by Singh and Pitzer [35], who observed a parabolic liquid-liquid coexistence curve of an electrolyte solution. This apparent classical behavior was the stimulus for most theoretical and experimental work reported here. 

Studies of liquid-vapor coexistence are, generally, best addressed in the framework of an open ensemble thus the state variables here comprise both the particle coordinates r and the particle number N. A path with the appropriate credentials can be constructed by identifying pairs of values of the chemical potential p and the temperature T which trace out some rough approximation to the coexistence curve in the p—7 plane, but extend into the one-phase region beyond the critical point. Once again there is some circularity here to which we shall return. Making the relevant variables explicit, the sampling distribution [Eq. (26)] takes the form... 

In the context of liquid-vapor coexistence the particle number N (or equivalently the number density p N/ V) plays the role of an order parameter. Estimates of... 

This strategy has been applied to the study of a range of coexistence problems, initially focused on lattice models in magnetism [41] and particle physics [42]. Figure 5 [43—4-5] shows the results of an application to liquid-vapor coexistence in a Lennard-Jones system with the particle number density chosen as an order parameter. 

J. Harris, K. Yung, Carbon dioxide s liquid-vapor coexistence curve and critical properties as predicted by a simple molecular model. J. Phys. Chem. 99, 12021 (1995)... 

Liquid-Vapor Coexistence Curve and the Critical Point... 

We conclude this section by giving a topical example of the utility of conditional averages in considering molecularly complex systems (Ashbaugh et al, 2004). We considered the RPLC system discussed above (p. 5), but without methanol n-Ci8 alkyl chains, tethered to a planar support, with water as the mobile phase. The backside of the liquid water phase contacts a dilute water vapor truncated by a repulsive wall see Fig. 1.2, p. 7. Thus, it is appropriate to characterize the system as consistent with aqueous liquid-vapor coexistence at low pressure. A standard CHARMM force-field model (MacKerell Jr. et al, 1998) is used, as are standard molecular dynamics procedures - including periodic boimdary conditions - to acquire the data considered here. Our interest is in the interface between the stationary alkyl and the mobile liquid water phases at 300 K. 

The second characteristic in our answer is the variation of the liquid density along the liquid-vapor coexistence curve in the temperature regimes of interest here. The coefficient of thermal expansion along the coexistence curve, plotted in Fig. 8.12 for several solvents, is typically more than five times smaller for water than for common organic solvents. It is a secondary curiosity that liquid water has... 

J.J. De Pablo, T.M. Prausnitz, H.J. Strauch and P.T. Cummings, Molecular simulation of water along the liquid-vapor coexistence curve from 25 C to the critical point, J. Chem. Phys., 93, (1990) 7355-7359. 

Figure 6, Profiles of the density as a fimction of z, the distance from the center of a parallel-walled slh. The vertical lines show the planes of solid that make up the pore. The density is shown for a conqjletely wet (part a) and a con letely dry (part b) surface. Both the fluid adsorbate and the solid adsorbent are made up of Lennard-Jones atoms with well-depth ratios % /% = 0.85 (part a) and 0.30 (part b). The simulations were performed under conditions such that each system was at bulk liquid-vapor coexistence for 0.7. From Ref [31], J. Stat. Phys.

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