Liquid—vapor phase coexistence

The grand-canonical ensemble is particularly well suited for studies of liquid-vapor phase coexistence (i) Fluctuations of the order parameter, i.e., the density, are efficiently relaxed. Since the density is not conserved, spatial fluctuations do not decay via slow diffusion of polymers but relax much faster through insertion/deletion moves. In the grand-canonical ensemble one controls the temperature, T, the volume, V, and the chemical potential, p,... 

In compressible polymer-solvent mixtures there are two coupled order parameters - the density and the composition - and in addition to liquid-vapor phase coexistence also liquid-liquid phase separation can occur [6, 7, 24, 21]. The interplay between composition and density gives rise to six distinct types of phase diagrams according to the classification of Van Konynenburg and Scott [24] (see also Fig. 3). 

The binodals of the liquid-vapor phase coexistence as a function of molar fraction and temperature resemble the binodals of a one-component system as a function of density and temperature At high temperature, there is a critical point. Upon decreasing temperature the polymer-rich phase becomes more concentrated in polymer, while the solvent concentration increases in the vapor phase. The spinodal of the polymer liquid, however, exhibits a non-monotonous temperature dependence of the composition. This dependence is parallel to the non-monotonous behavior of the nucleation barrier as we increase temperature. In fact, at the pressure considered, and even more so at lower pressures (cf. Fig. 20), there exists an extended temperature region, where the polymer-fraction at the spinodal of the liquid decreases upon increasing temperature. 

Liquid-vapor critical point 4, 5, 16, 55 Liquid-vapor interface 26, 27, 42 Liquid-vapor phase coexistence 21, 55, 93 Liquid-vapor phase equilibria 16 Liquid-vapor trausitiou 4, 14 Liquid-vapor unmixiug 19 Lorentz-Berthelot mixiug rule 24, 79... 

The shaded region is that part of the phase diagram where liquid and vapor phases coexist in equilibrium, somewhat in analogy to the boiling line for a pure fluid. The ordinary liquid state exists on the high-pressure, low-temperature side of the two-phase region, and the ordinary gas state exists on the other side at low pressure and high temperature. As with our earlier example, we can transform any Type I mixture... 

The lines separating the regions in a phase diagram are called phase boundaries. At any point on a boundary between two regions, the two neighboring phases coexist in dynamic equilibrium. If one of the phases is a vapor, the pressure corresponding to this equilibrium is just the vapor pressure of the substance. Therefore, the liquid-vapor phase boundary shows how the vapor pressure of the liquid varies with temperature. For example, the point at 80.°C and 0.47 atm in the phase diagram for water lies on the phase boundary between liquid and vapor (Fig. 8.10), and so we know that the vapor pressure of water at 80.°C is 0.47 atm. Similarly, the solid-vapor phase boundary shows how the vapor pressure of the solid varies with temperature (see Fig. 8.6). 

This approximation amounts to truncating the functional expansion of the excess free energy at second order in the density profile. This approach is accurate for Lennard-Jones fluids under some conditions, but has fallen out of favor because it is not capable of describing wetting transitions and coexisting liquid-vapor phases [105-107]. Incidentally, this approximation is identical to the hypemetted chain closure to the wall-OZ equation [103]. 

Within the PV region delimited by the two saturation boundary curves, liquid and vapor phases coexist stably at equilibrium. To the right of the vapor saturation curve, only vapor is present to the left of the liquid saturation curve, vapor is absent. Let us imagine inducing isothermal compression in a system composed of pure H2O at T = 350 °C, starting from an initial pressure of 140 bar. The H2O will initially be in the gaseous state up to P < 166 bar. At P = 166 bar, we reach the vapor saturation curve and the liquid phase begins to form. Any further... 

Fractional distillation can be represented on a liquid/vapor phase diagram by plotting temperature versus composition, as shown in Figure 11.18. The lower region of the diagram represents the liquid phase, and the upper region represents the vapor phase. Between the two is a thin equilibrium region where liquid and vapor coexist. 

When a system consists of saturated-liquid and saturated-vapor phases coexisting in equilibrium, the total value of any extensive property of the two-phase system is the sum of the total properties of the phases. Written for the volume, this relation is... 

The regions below ABC in Fig. 3.6 and above ABC in Fig. 3.7 represent subcooled liquid no vapor is present. The regions above ADC in Fig. 3.6 and below ADC in Fig. 3.7 represent superheated vapor no liquid is present. The area between the curves is the region where both liquid and vapor phases coexist. 

S. A system composed of ethane hydrate, water, and ethane is classed aa a two-component system when Gibbs phase rule is applied since it could be formed from water and ethane. What is the variance of this system when a solid, a liquid, and a vapor phase coexist in equilibrium If the temperature of this three-phase system is specified, would it be possible to alter the pressure without the disappaaranoe of a phase ... 

Solid lines represent the liquid-vapor phase equilibria of the two pure components that end in critical points marked by arrows. When a small amount of solvent is added to the pure pol3uner, the liquid-vapor coexistence shifts and so does the critical point. The loci of critical points for the binary system form a critical line that is shown by the dashed line with squares for = 1 and triangles for = 0.886. In the former case - phase behavior of t3q>e I -the critical line connects the critical points of the two pure components and the two coexisting phases gradually change from vapor and solvent-rich liquid... 

Figure 4.2 Variation of the vapor pressure, Pv, of a substance with the temperature, 7, showing the phase transition between solid, liquid and vapor phases. Two phases can coexist in equilibrium only at pressures and temperatures defined by the phase boundary lines in the phase diagram, such as liquid-vapor, solid-liquid and solid-vapor lines. The liquid-vapor phase boundary terminates at the critical point, 7C. All three phases can coexist in equilibrium only at the triple point, 73, which is the intersection of the three two-phase boundaries.

An equation of state, applicable to all fluid phases, is paitiodariy useful for phase-equilibrium calculations where a liquid phase and a vapor phase coexist at high pressures. At such conditions, conventional activity coefficients are not useful because, with rare exceptions, at least one of the mixture s components is supercritical that is, (he system temperature is above (hat component s critical temperature. In that event, one must employ special standard states for the activity coefficients of the supercritical components (see Section 1.5-2). That complication is avoided when ail fugacities are calculated front en equation of state. 

One experimental observation in phase equilibrium is that the two coexisting equilibrium phases must have the same temperature and pressure. Clearly, the arguments given in Secs. 7.1 and 7.2 establish this. Another experimental observation is that as the pressure is lowered along an isotherm on which a liquid-vapor phase transition occurs, the actual volume-pressure behavior is as shown in Fig. 7.3-3, and not as in Fig. 7.3-2. 

Figure 4. The equilibrium fraction of monomers in the coexisting liquid-vapor phase of an associating fluid with one square-well bonding site. The liquid-phase fractions of monomers are on the left-hand side of the figure. The circles are data from RCMC-Gibbs ensemble simulations, and the lines are calculations from three different implementations of a theory for associating fluids. The solid line uses exact values of the reference fluid radial distribution function the dashed and long dashed-short dashed lines use the WCA and modified WCA approximations to the radial distribution function, respectively. (Reprinted with permission from Muller et al. [43]. Copyright 1995 American Institute of Physics.)...

VP indicates vapor pressure point CVGT indicates constant volume gas thermometer point TP indicates triple point (equilibrium temperature at which the solid, liquid, and vapor phases coexist) FP indicates freezing point, and MP indicates melting point (the equihbrium temperatures at which the solid and liquid phases coexist under a pressure of 101 325 Pa, one standard atmosphere). The isotopic composition is that naturally occurring. 

The plots of the total vapor pressure as functions of the mole fraction of A in both the liquid and vapor phases are shown in Figure 9.12(a) and (b), respectively. The combined plot shown in Eigure 9.12(c) is a liquid-vapor phase diagram for an ideal binary solution at a fixed temperature T—often called a pressure-composition diagram. At any pressure and composition above the upper curve (the liquid line) the mixtnre is a liquid. Below the lower curve (the vapor line), the mixture is entirely vapor. The region between the two curves is a region of phase coexistence, that is, both liquid and vapor phases are present in the system. 

The curve AD that divides the soUd region from the gaseous region gives the vapor pressures of the sohd at various temp atures. This curve intersects the other curves at point A, the triple point, which is the point on a phase diagram representing the temperature and pressure at which three phases of a substance coexist in equilibrium. For water, the triple point occurs at 0.01°C, 0.00603 atm (4.58 mmHg), and the solid, liquid, and vapor phases coexist. ... 

---