Critical curve

Glassification of Phase Boundaries for Binary Systems. Six classes of binary diagrams have been identified. These are shown schematically in Figure 6. Classifications are typically based on pressure—temperature (P T) projections of mixture critical curves and three-phase equiHbria lines (1,5,22,23). Experimental data are usually obtained by a simple synthetic method in which the pressure and temperature of a homogeneous solution of known concentration are manipulated to precipitate a visually observed phase. 

To evaluate the fibrillation behavior of dispersed TLCP domains according to the - 5 relation discussed previously, different - 5 graphs were calculated by eliminating the thickness variable x. The result is reported in Fig. 18. It is obvious that all the points obtained are found to be relatively close to the critical curve by Taylor. The Taylor-limit is also shown in the figure with a solid curve. One finds that all the values calculated on sample 1 are completely above the limit, while all those determined on sample 4 are completely below the limit. The other two samples, 2 and 3, have the We - 5 relation just over the limit. 

According to the criteria, the dispersed phase embedded in the matrix of sample 1 must have been deformed to a maximum aspect ratio and just began or have begun to break up. By observing the relative position of the experimental data to the critical curve, the deformational behavior of the other samples can be easily evaluated. Concerning the fibrillation behavior of the PC-TLCP composite studied, the Taylor-Cox criteria seems to be valid. 

The only parts of Fig. 5 which can meaningfully be described as solubility in a compressed gas are WX and XV. However, a very different situation arises if the saturated vapor pressure curve cuts the critical curve (M—N of Fig. 3). Figure 4 shows that this does not happen for the three sodium halides. The complete course of the critical curve is not known, but enough is known in the case of the sodium chloride system51 75 for it to be clear that it rises well above the maximum of the saturated vapor pressure curve. However, it is cut by the vapor pressure curves of less soluble salts such as sodium carbonate and sodium sulphate.40 87 The (p, T) projection of a system of the type water + sodium chloride is... 

Systems in which the saturated vapor pressure curve does not cut the critical curve (as in Figs. 7 and 11). Example, ethane + />-dichlorobenzene.74... 

Liquid-Fluid Equilibria Nearly all binary liquid-fluid phase diagrams can be conveniently placed in one of six classes (Prausnitz, Licntenthaler, and de Azevedo, Molecular Thermodynamics of Fluid Phase Blquilibria, 3d ed., Prentice-Hall, Upper Saddle River, N.J., 1998). Two-phase regions are represented by an area and three-phase regions by a line. In class I, the two components are completely miscible, and a single critical mixture curve connects their criticsu points. Other classes may include intersections between three phase lines and critical curves. For a ternary wstem, the slopes of the tie lines (distribution coefficients) and the size of the two-phase region can vary significantly with pressure as well as temperature due to the compressibility of the solvent. 

We notice that the positive blackbody contributions for E and P dominate in the high-temperature limit, while the energy and the pressure are negative for low T. From Eq. (32), we can determine the critical curve (/3C = XoL) for the transition from negative to positive values of P, by searching for the value of the ratio x = j3/L for which the pressure vanishes this value, xo, is the solution of the transcendental equation... 

As before, for any given value of L, the pressure changes from negative to positive when the temperature is raised. The critical curve for this transition is f3c = xoL, where Xo is the value of ratio % = f3/L for which the pressure vanishes, that is the solution of the transcendental equation... 

Fig. 5.12 Two different 3-D representations of the phase diagram of 3-methylpyridine plus wa-ter(H/D). (a) T-P-x(3-MP) for three different H2O/D2O concentration ratios. The inner ellipse (light gray) and corresponding critical curves hold for (0 < W(D20)/wt% < 17). Intermediate ellipses stand for (17(D20)/wt% < 21), and the outer ellipses hold for (21(D20)/wt% < 100. There are four types of critical lines, and all extrema on these lines correspond to double critical points, (b) Phase diagram at approximately constant critical concentration 3-MP (x 0.08) showing the evolution of the diagram as the deuterium content of the solvent varies. The white line is the locus of temperature double critical points whose extrema (+) corresponds to the quadruple critical point. Note both diagrams include portions at negative pressure (Visak, Z. P., Rebelo, L. P. N. and Szydlowski, J. J. Phys. Chem. B. 107, 9837 (2003))...

Fig. 6. Qualitative pressure—temperature diagrams depicting critical curves for the six types of phase behaviors for binary systems, where Ca or C corresponds to pure component critical point G, vapor L-, liquid U, upper critical end point and U, lower critical end point. Dashed curves are critical lines or phase boundaries (5). (a) Class I, the Ar—Kr system (b) Class II, the C02—C8H18 system (c) Class III, where the dashed lines A, B, C, and D correspond to the H2-CO, CH4-H2S, He-H2, and He-CH4 system, respectively (d) Class IV, the CH4 C6H16 system (e) Class V, the C2H6 C2H5OH...

Often the essentials of phase diagrams in P,7,x-space are represented in a P,7-projection. In this type of diagrams only non-variant (F=0) and monovariant (F=l) equilibria can be represented. Since pressure and temperature of phases in equilibrium are equal, a four-phase equilibrium is now represented by one point and a three-phase equilibrium with one curve. Also the critical curve and the azeotropic curve are projected as a curve on the P, 7-plane. A four-phase point is the point of intersection of four three-phase curves. The point of intersection of a three-phase curve and a critical curve is a so-called critical endpoint. In this intersection point both the three-phase curve and the critical curve terminate. 

In Figure 2.2-3 the curves Ig are the vapour pressure curves of the pure components which end in a critical point l=g. The curves l=g, h=g and h=g are vapour-liquid critical curves and the curves h=h are curves on which two liquid phases become critical. The points of intersection of a critical curve with a three-phase curve hhg is a critical endpoint. Distinction can be made between upper critical endpoints (UCEP) and lower critical endpoints (LCEP). The UCEP is highest temperature of a three-phase curve, the LCEP is the lowest temperature of a three-phase curve. The point of intersection of the hhg curve with a l/=g curve is a critical endpoint in which the li liquid phase and the vapour phase are critical in the presence of a non-critical l2 phase (h+(h=g)) and the point of intersection of the hhg curve with a h=h curve is a critical endpoint in which the two liquid phases h and // are critical in the presence of a non-critical vapour phase (h=h)+g)-... 

In the case of type I phase behaviour there is only one critical curve, the l=g critical curve, which runs continously from the critical point of component A to the critical point of component B. In Figure 2.2-4 some Ppc-sections are shown and in Figure 2.2-5 a P.T-section. 

In the P,7-section the two-phase envelope is tangent to the binary critical curve in the critical point. 

Type V fluid phase behaviour shows at temperatures close to 7C-A a three-phase curve hhg which ends at low temperature in a LCEP (h=h)+g and at high temperature in a UCEP (h+h) g The critical curve shows two branches. The branch h=g runs from the critical point of pure component A to the UCEP. The second branch starts in the LCEP and ends in the critical point of pure component B. This branch of the critical curve is at low temperature h=h in nature and at high temperature its character is changed into h=g- The h=h curve is a critical curve which represents lower critical solution temperatures. In Figure 2.2-7 four isothermal P c-sections are shown. 

Type IV fluid phase behaviour is a combination of type II and type V behaviour. These systems show two branches of the l2lig curve, three branches of the critical curve and three critical endpoints. At low temperature the P c-sections for this type of systems are similar to Figure 2.2-6, at temperatures close to the critical temperature of the pure component A PResections similar to Figure 2.2-7 are found. 

In type HI phase behaviour the two branches of the l2lig equilibrium are combined and also the two branches of the l2=li critical curve are united. Only one critical endpoint is left. The h=h/h=g branch of the critical curve can have the shape as is shown in Figure 2.2-3, but it is also possible that this curve goes from the critical point of component B to high pressure via a temperature minimum or that dP/dT is always positive [10]. 

In type VI phase behaviour a three-phase curve l2hg is found with an LCEP and an UCEP. Both critical endpoints are of the type (l2=li)+g and are connected by a l2=h critical curve which shows a pressure maximum. For this type of phase behaviour at constant pressure closed loop isobaric regions of l2+li equilibria are found with a lower critical solution temperature and an upper critical solution temperature. 

As discussed for CO2 + n-alkane systems at carbon numbers n<24 the three-phase curve hhg ends a low temperature in a quadruple point s2l2lig. This is shown schematically in Figure 2.2-9a and b. In the quadrupel point three other three-phase curves terminate. The s2hh curve runs steeply to high pressure and ends in a critical endpoint where this curve intersects the critical curve. The s2l2g curve runs to the triple point of pure component B and the s/l/g curve runs to lower temperature and ends at low temperature in a second quadruple point s2silig (not shown). 

Critical reviews of existing methods to model phase behaviour at high pressure using equations of state have been made in recent years, and we refer to these papers for further details [24, 25]. The general conclusion is that modelling is still case-specific. When the critical point is approached, predictions and even correlations of critical curves and solubilities are extremely difficult because of the nonclassical behaviour in this region. 

The phase behaviour of many polymer-solvent systems is similar to type IV and type HI phase behaviour in the classification of van Konynenburg and Scott [5]. In the first case, the most important feature is the presence of an Upper Critical Solution Temperature (UCST) and a Lower Critical Solution Temperature (LCST). The UCST is the temperature at which two liquid phases become identical (critical) if the temperature is isobarically increased. The LCST is the temperature at which two liquid phases critically merge if the system temperature is isobarically reduced. At temperatures between the UCST and the LCST a single-phase region is found, while at temperatures lower than the UCST and higher than the LCST a liquid-liquid equilibrium occurs. Both the UCST and the LCST loci end in a critical endpoint, the point of intersection of the critical curve and the liquid liquid vapour (hhg) equilibrium line. In the two intersection points the two liquid phases become critical in the presence of a... 

The S-L-V curve intersects the gas-liquid critical curve in two points the lower critical end point (LCEP) and the upper critical end point (UCEP). At these two points, the liquid and gas phases merge into a single fluid-phase in the presence of excess solid. At temperatures between Tlcep and Tucep a S-V equilibrium is observed. The solubility of the heavy component in the gas phase increases very rapidly with pressure near the LCEP and the UCEP. Near the LCEP the solubility of heavy component in the light one is limited by the low temperatures. In contrast, near the UCEP the solubility of heavy component in the light one is high, owing to the much higher temperatures [6],... 

For electrolyte solutions such as NaCl + water the critical temperatures of the pure components differ by about a factor of five. From the perspective of nonelectrolyte thermodynamics, the absence of a liquid-liquid immiscibility then comes as a great surprise. It is a major challenge for theory to explain why this salt, as well as similar salts such as KC1 or CaCl2, seems to show a continuous critical line. Perhaps there is a slight indication for a transition toward an interrupted critical curve in Marshall s study [151] of the critical line of NaCl + H20. Marshall observed a dip in the TC(XS) curve some K away from the critical point of pure water, which at first glance seems obscure. It was suggested [152] that the vicinity to an upper critical end point leaves its mark by this dip. 

The simplest salt-solvent model comprises charged spheres mixed with neutral spheres. Intuitively, one expects wide liquid-liquid miscibility gaps, owing to the very different nature of the constituents. In fact, MSA calculations indicate type III phase behavior with a wide liquid-liquid miscibility gap in the solvent-rich regime that breaks the L-G critical curve [149, 230-232]. 

Mixtures of equisized charged spheres were also treated by the MSA. Such a system is then uniquely characterized by the ratio of the critical temperatures of the pure components. Harvey [235] found that a continuous critical curve from the dipolar solvent to the molten salt is maintained until the critical temperature of the ionic component exceeds that of the dipolar component by a factor of about 3.6. This ratio is much higher than theoretically predicted for nonionic model fluids. We recall that for NaCl the critical line is still continuous at a critical temperature ratio of about 5. Thus, the MSA of the charged-hard-sphere-dipolar-hard-sphere system captures, at least in part, some unusual features of real salt-water systems with regard to their critical curves. 

Figure 3. Behaviour of the exponents PP and Pt along the critical curve for water+methyl-di(n-amyl)phosphynoxide.

A PVT-data-code was generated to calculate phase equilibria, critical curves and thermodynamic excess quantities of quaternary systems. The program uses the Christoforakis-Franck-equation of state [11]... 

The sets of concentrations (xi, x2) of inhibitors that are in agree with equation (21) are critical and represent multitude reflected on plane (x, y) as a line which may be called as critical curve of cytoside action of toxicants combination (Xi X2). 

Solid-Fluid Equilibria The phase diagrams of binary mixtures in which the heavier component (the solute) is normally a solid at the critical temperature of the light component (the solvent) include solid-liquid-vapor (SLV) curves which may or may not intersect the LV critical curve. The solubility of the solid is very sensitive to pressure and temperature in compressible regions where the solvent s density and solubility parameter are highly variable. In contrast, plots of the log of the solubility versus density at constant temperature exhibit fairly simple linear behavior. 

Figure 1 Isotherms of the NaCl-H20 system illustrating the composition of coexisting vapor (low XNaci) and brine (high XNaci) in P-X space. Note that the critical curve (c.c.) is everywhere above the three-phase equilibrium...

---