Coefficient critical lines

CO -benzene, and CO -n-decane. The critical densities and the corresponding compositions are plotted in Figure 1. The three hydrocarbons in order of higher to lower solubility in C0 were heptane, benzene, and decane. The measured binary diffusion coefficients or the decay rates of the order-parameter fluctuations at various temperatures and pressures are listed in Tables I, II, and III for CO -heptane, CO -benzene, and CO -decane systems respectively. In Figure 2, the critical lines of the three binary systems in the dilute hydrocarbon range are shown in the pressure-temperature space. dP/dT along the critical lines of CO.-heptane and CO -benzene systems are similar and lower than dP/dT along the critical line of CO -decane system, which indicates that C02 and decane form more asymmetric mixtures relative to CO with heptane or benzene. 

Fig. 23. Schematic phase diagram of a system where by a variation of a parameter p the coefficient K (p) of the gradient energy (1/2)Jf (p)(v0)2 vanishes at a Lifshitz point Kl = 0, r, (pL) = Tl. For p < pl one has a ferromagnetic structure, while for p > pi where fC (p) < 0 one has a modulated structure, with a characteristic wavenumber q describing the modulation. For p -> p from above one has — 0 along the critical line...

The cosolvency phenomenon was discovered in 1920 s experimentally for cellulose nitrate solution systems. Thereafter cosolvency has been observed for numerous polymer/mixed solvent systems. Polystyrene (PS) and polymethylmethacrylate (PMMA) are undoubtedly the most studied polymeric solutes in mixed solvents. Horta et al. have developed a theoretical expression to calculate a coefficient expressing quantitatively flie cosolvent power of a mixture (dTydx)o, where T,. is the critical temperature of the system and x is the mole fraction of hquid 2 in the solvent mixture, and subscript zero means x—>0. This derivative expresses the initial slope of the critical line as a function of solvent composition (Figure 5.4.1). Large negative values of (dT/dx) are the characteristic feature of the powerful cosolvent systems reported. The theoretical expression developed for (dT dx)o has been written in terms of the interaction parameters for the binary systems ... 

The coefficient of sell-diffusion does not appear to have an anomaly near the critical point. For the engineer, however, the mutual dift usion coefficient is the more important property. The binary dilfusion coefficient approaches zero at the mixture critical point ("critical slowing-down"). In dilute mixtures, however, the decrease of the binary dilfusion coefficient is not seen until the critical line is approached very closely. For many practical purposes, such as supercritical extraction and chromatography, the mixture is dilute, and it can be assumed that the coefficient of binary diffusion is intermediate between that in the vapor and that in the liquid. Since the diffusion coefficient decreases roughly inversely proportional to the density, dilfusion in supercritical solvents is much more rapid than in liquid solvents, thus increasing the speed of diffusion-controlled chemical processes. 

Iiifomiation about the behaviour of the 3D Ising ferromagnet near the critical point was first obtained from high- and low-temperatnre expansions. The expansion parameter in the high-temperatnre series is tanli K, and the corresponding parameter in the low-temperatnre expansion is exp(-2A ). A 2D square lattice is self-dual in the sense that the bisectors of the line joining the lattice points also fomi a square lattice and the coefficients of the two expansions, for the 2D square lattice system, are identical to within a factor of two. The singularity occurs when... 

Figure A2.5.17. The coefficient Aias a fimction of temperature T. The line IRT (shown as dashed line) defines the critical point and separates the two-phase region from the one-phase region, (a) A constant K as assumed in the simplest example (b) a slowly decreasing K, found frequently in experimental systems, and (c) a sharply curved K T) that produces two critical-solution temperatures with a two-phase region in between.

Coefficient A and exponent a can be evaluated readily from data on Re and T. The dimensionless groups are presented on a single plot in Figure 15. The plot of the function = f (Re) is constructed from three separate sections. These sections of the curve correspond to the three regimes of flow. The laminar regime is expressed by a section of straight line having a slope P = 135 with respect to the x-axis. This section corresponds to the critical Reynolds number, Re < 0.2. This means that the exponent a in equation 53 is equal to 1. At this a value, the continuous-phase density term, p, in equation 46 vanishes. 

The turbulent regime for Cq is characterized by the section of line almost parallel to the x-axis (at the Re" > 500). In this case, the exponent a is equal to zero. Consequently, viscosity vanishes from equation 46. This indicates that the friction forces are negligible in comparison to inertia forces. Recall that the resistance coefficient is nearly constant at a value of 0.44. Substituting for the critical Reynolds number, Re > 500, into equations 65 and 68, the second critical values of the sedimentation numbers are obtained ... 

In the earlier discussion of critical depth (6.5), it was pointed out that this depth is determined by the density, and by the mass absorption coefficients of the sample for the incident beam and for the analytical line. With light elements, the coefficient for the analytical line can be so large that the coefficient of the incident beam may be neglected in... 

As mentioned earlier, the physical properties of a liquid mixture near a UCST have many similarities to those of a (liquid + gas) mixture at the critical point. For example, the coefficient of expansion and the compressibility of the mixture become infinite at the UCST. If one has a solution with a composition near that of the UCEP, at a temperature above the UCST, and cools it, critical opalescence occurs. This is followed, upon further cooling, by a cloudy mixture that does not settle into two phases because the densities of the two liquids are the same at the UCEP. Further cooling results in a density difference and separation into two phases occurs. Examples are known of systems in which the densities of the two phases change in such a way that at a temperature well below the UCST. the solutions connected by the tie-line again have the same density.bb When this occurs, one of the phases separates into a shapeless mass or blob that remains suspended in the second phase. The tie-lines connecting these phases have been called isopycnics (constant density). Isopycnics usually occur only at a specific temperature. Either heating or cooling the mixture results in density differences between the two equilibrium phases, and separation into layers occurs. 

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. 

From the physicochemical point of view, the prevailing one in the present review, the question is how these coefficients relate to the characteristics of the constituant molecules and how this information can be extracted from (2,3). The answer relies on the fact that the essential contributions to these integrals come from only few nonoverlapping critical regions in the joint density of states (18,19) these are points, lines and surfaces depending on the spatial extension of the conjugated electron distribution. They are defined by the condition... 

Figure 9 Chain center of mass self-diffusion coefficient for the bead-spring model as a function of temperature (open circles). The full line is a fit with the Vogel-Fulcher law in Eq. [3]. The dashed and dotted lines are two fits with a power-law divergence at the mode-coupling critical temperature. 

Stimulated Rayleigh scattering from localised thermal fluctuations in gases 258) and liquids 259) has been reported with measurements of the line shifts, thresholds and critical absorption coefficients. 

The approach to the critical point, from above or below, is accompanied by spectacular changes in optical, thermal, and mechanical properties. These include critical opalescence (a bright milky shimmering flash, as incident light refracts through intense density fluctuations) and infinite values of heat capacity, thermal expansion coefficient aP, isothermal compressibility /3r, and other properties. Truly, such a confused state of matter finds itself at a critical juncture as it transforms spontaneously from a uniform and isotropic form to a symmetry-broken (nonuniform and anisotropically separated) pair of distinct phases as (Tc, Pc) is approached from above. Similarly, as (Tc, Pc) is approached from below along the L + G coexistence line, the densities and other phase properties are forced to become identical, erasing what appears to be a fundamental physical distinction between liquid and gas at all lower temperatures and pressures. 

Equation 15-14 is a cubic equation with real coefficients. Thus, three values of z-factor cause the equation to equal zero. These three roots are all real when pressure and temperature are on the vapor pressure line—that is, when liquid and gas are present. One real root and two complex roots exist when the temperature is above the critical temperature. 

For many such systems the maximum yield of the desired product corresponds to a middle steady state that may be unstable as shown earlier. For such systems, an efficient adiabatic operation is not possible and nonadiabatic operation is mandatory. Flowever, the choice of the heat-transfer coefficient U, the area of heat transfer Ah and the cooling jacket temperature are critical for the stable operation of the system. The value of the dimensionless heat-transfer coefficient Kc should exceed a critical value KCtCrit in order to stabilize an unstable middle steady state. The value of iFc,crit corresponds to the line... 

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