Locus of critical points

SFC (see Figure 7.6) occurs when both the critical temperature and critical pressure of the mobile phase are exceeded. (The locus of critical points is indicated in Figure 7.2 by the dashed line over the top of the two-phase region. It is also visible or partly visible in Figures 7.3-7.8). Compressibility, pressure tunability, and diffusion rates are higher in SFC than in SubFC and EFLC, and are much higher than in LC. 

Figure 2-15 shows phase data for eight mixtures of methane and ethane, along with the vapor-pressure lines for pure methane and pure ethane.3 Again, observe that the saturation envelope of each of the mixtures lies between the vapor pressure lines of the two pure substances and that the critical pressures of the mixtures lie well above the critical pressures of the pure components. The dashed line is the locus of critical points of mixtures of methane and ethane. 

V - Op/A. The turning point of each diagram is special there are no steady states to the left of this point two steady states exist on the right. We say that the turning point is a critical point. If we consider the family of V - aP/A diagrams traced for different values of the additional parameter T, a surface is obtained. This is a two-dimensional manifold in the three-dimensional V - oP/A - T space. As each diagram has its own turning point defined by one additional equation, the locus of critical points is a line, a one-dimensional manifold in the V - cP/A - T space. It is easy to see that the dimension of the critical manifold is na - 1, where na is the number of parameters included in the analysis. 

If we now vary the temperature, the phase diagram is of the form indicated in fig. 16.21. Each section at constant temperature gives a curve of the form shown in fig. 16.20, and the locus of critical points is the line k- k k k. ... 

The simplest class of binary phase diagram is class I as shown in Figure 1.2-3. The component with the lower critical temperature is designated as component 1. The solid lines in Figure 1.2-3(b) represent the pure component liquid-vapor coexistence curves which terminate at the pure component critical points (Cj and C2). The feature of importance in this phase diagram is that the mixture critical line (dashed line in Figure 1.2-3(b)) is continuous between the two critical points. The mixture critical line represents the locus of critical points for all mixtures of the two components. The area bounded by the solid and dashed lines represents the two-phase, liquid-vapor (LV) region. The mixture-critical... 

Fig. 50. MiscibUity gap in solutions of a binary polymer (components Pi and Pg) in a single solvent S. AAgCjB quasi-binary section (cloud-point curve) A precipitation threshold CC5C locus of critical points DCjE quasi-binary section of the spinodal surface (—. —)...

The phase behaviour of blends of homopolymers containing block copolymers is governed by a competition between macrophase separation of the homopolymer and microphase separation of the block copolymers. The former occurs at a wavenumber q = 0, whereas the latter is characterized by q + 0. The locus of critical transitions at q, the so-called X line, is divided into q = 0 and q + 0 branches by the (isotropic) Lifshitz point. The Lifshitz point can be described using a simple Landau-Ginzburg free-energy functional for a scalar order parameter rp(r), which for ternary blends containing block copolymers is the total volume fraction of, say, A monomers. The free energy density can be written (Selke 1992)... 

On the other hand, the electronic specific heat reveals an important field dependence [62] (Fig. 10). The linear specific heat eoefficient 7 increases from 10 mJ mol K at low field (although larger than the critical field = 1 kOe along c ), and passes through a maximum of 25 mJ mol K at H = 20 kOe for r == 1 K. The locus of the points corresponding to the maximum of 7 in the H -T plane is located in the paramagnetic domain of (TMTSF)2C104, i.e. not to... 

The points Ci and are the critical points of pure methane and ethane, respectively. The line connecting these two points, which is the intersection of the bubble point and dew point surfaces, is the critical locus. This is the set of critical points for the various mixtures of methane and ethane. The black curve connecting points A and Ci is the vapor pressure curve of pure methane, and the violet curve connecting points B and C2 is the vapor pressure curve of pure ethane. 

Bubble points and dew points may be generated as described above for a given mixture over ranges of temperature and pressure. The locus of bubble points is the bubble point curve and the locus of dew points is the dew point curve. The two curves together define the phase envelope. In addition to the bubble point curve (total liquid saturated) and the dew point curve (total vapor saturated), other curves may be drawn representing constant vapor mole fraction. All these curves meet at one point, the critical point, where the vapor and liquid phases lose their distinctive characteristics and merge into a single, dense phase. 

From the standpoint of a stress-strain curve, the constitutive behavior discussed in the previous section corresponds to a limited range of loads and strains. In particular, as seen in fig. 2.10, for stresses beyond a certain level, the solid suffers permanent deformation and if the load is increased too far, the solid eventually fails via fracture. From a constitutive viewpoint, more challenges are posed by the existence of permanent deformation in the form of plasticity and fracture. Phenomenologically, the onset of plastic deformation is often treated in terms of a yield surface. The yield surface is the locus of all points in stress space at which the material begins to undergo plastic deformation. The fundamental idea is that until a certain critical load is reached, the deformation is entirely elastic. Upon reaching a critical state of stress (the yield stress), the material then undergoes plastic deformation. Because the state of stress at a point can be parameterized in terms of six numbers, the tensorial character of the stress state must be reflected in the determination of the yield surface. 

At the critical temperature and pressure the liquid and vapor phases are indistinguishable. One such point exists for each envelope. The curve connecting the points for all envelopes is called the locus of criticals. ... 

Figure 7.4. Phase diagrams for type I (top) and type II (bottom) binary mixtures with carbon dioxide as one component (L = liquid and v = vapor). The UCST line indicates the temperature at which the two immiscible liquids merge to form a single liquid phase. The critical mixture curve is the locus of critical mixture points spanning the entire composition range. (From ref. [44] American Chemical Society).

In Fig. 11.5.C-2 the locus of the partial pressure and temperature in the maximum of the temperature profile and the locus of the inflection points before the hot spot are shown as p and (pj), respectively. Two criteria were derived from this. The first criterion is based on the observation that extreme sensitivity is found for trajectories—the p-T relations in the reactor—intersecting the maxima curve p beyond its maximum. Therefore, the trajectory going through the maximum of the p -curve is considered as critical. This is a criterion for runaway based on an intrinsic property of the system, not on an arbitrarily limited temperature increase. The second criterion states that runaway will occur when a trajectory intersects (Pi)i, which is the locus of inflection points arising before the maximum. Therefore, the critical trajectory is tangent to the (pi)i-curve. A more convenient version of this criterion is based on an approximation for this locus represented by p in... 

Fig. 10.5. Hypothetical low-temperature phase diagram for La (Balster and Wittig). Following the conventional notation, the d-hcp phase is called a. The high-temperature fee -phase can metastably exist at low temperatures in the d-hcp range O). Solid circles show the T-P dependence of the points of inflection from fig. 10.4. The locus of the points of inflection was identified with the 0l0 phase boundary. Another subtle phase transformation between two fee phases (fi and fi") has been postulated at about 50kbar. The phase boundary may terminate in a critical point at higher temperatures (asterisk).

All phases with chemical reaction equilibria simultaneously. Prediction of critical points and the critical locus of a mixture. 

The conditions that apply for the saturated liquid-vapor states can be illustrated with a typical p-v, or (1 /p), diagram for the liquid-vapor phase of a pure substance, as shown in Figure 6.5. The saturated liquid states and vapor states are given by the locus of the f and g curves respectively, with the critical point at the peak. A line of constant temperature T is sketched, and shows that the saturation temperature is a function of pressure only, Tsm (p) or psat(T). In the vapor regime, at near normal atmospheric pressures the perfect gas laws can be used as an acceptable approximation, pv = (R/M)T, where R/M is the specific gas constant for the gas of molecular weight M. Furthermore, for a mixture of perfect gases in equilibrium with the liquid fuel, the following holds for the partial pressure of the fuel vapor in the mixture ... 

J. As with the alkane - water systems, the interaction parameters for the aqueous liquid phase were found to be temperature - dependent. However, the compositions for the benzene - rich phases could not be accurately represented using any single value for the constant interaction parameter. The calculated water mole fractions in the hydrocarbon - rich phases were always greater than the experimental values as reported by Rebert and Kay (35). The final value for the constant interaction parameter was chosen to fit the three phase locus of this system. Nevertheless, the calculated three-phase critical point was about 9°C lower than the experimental value. 

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))...

Figure 2-10 shows a more nearly complete pressure-volume diagram.2 The dashed line shows the locus of all bubble points and dew points. The area within the dashed line indicates conditions for which liquid and gas coexist. Often this area is called the saturation envelope. The bubble-point line and dew-point line coincide at the critical point. Notice that the isotherm at the critical temperature shows a point of horizontal inflection as it passes through the critical pressure. 

Notice that the locus of the critical points connects the critical pressure of ethane, 708 psia, to the critical pressure of methane, 668 psia. When the temperature exceeds the critical temperature of both components, it is not possible for any mixture of the two components to have two phases. 

Third, plot weight-averaged critical point on Figure 14-3, interpolate a locus of convergence pressures, and read pk at 160°F. 

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