Critical point behavior near

Bilinear, and higher, nonlinear variables containing Sn are also negligi-ble near a critical point. In estimating critical point ( ) behavior of various contributions to we have been approximating various correlation... 

Critical phenomena, behavior near or beyond the critical point, was first reported in 1869. Above the critical point temperature, a substance exists at all pressures as one pure phase, a critical fluid phase. Above the critical point temperature, the system is homogeneous with no appearance of a discontinuity, and the density is uniform even throughout a process that takes the substance from very low to very high pressure no pressure increase can bring about a phase change in this region. 

The form of equations (8.11) and (8.12) turns out to be general for properties near a critical point. In the vicinity of this point, the value of many thermodynamic properties at T becomes proportional to some power of (Tc - T). The exponents which appear in equations such as (8.11) and (8.12) are referred to as critical exponents. The exponent 6 = 0.32 0.01 describes the temperature behavior of molar volume and density as well as other properties, while other properties such as heat capacity and isothermal compressibility are described by other critical exponents. A significant scientific achievement of the 20th century was the observation of the nonanalytic behavior of thermodynamic properties near the critical point and the recognition that the various critical exponents are related to one another ... 

Sengers, J. V. Levelt-Sengers, J. N. H., Thermodynamic behavior of fluids near the critical point, Ann, Rev. Phys. Chem. 1986, 37, 187-222... 

Other quantities associated with second derivatives of the thermodynamic potential are also enhanced near the critical point demonstrating typical 1 / / l behavior, cf. [21], However numerical coefficients depend strongly on what quantity is studied. E.g. fluctuation contributions above Tc to the color diamagnetic susceptibilities... 

One of the consequences of the suppression of the phase transition is the presence of a special critical point, Tc = 0 K. This point, called the quantum displacive limit, is characterized by special critical exponents. Its presence gives rise to classical quantum crossover phenomena. Quantum suppression and the response at and near this limit, Tc = 0 K, have been extensively studied on the basis of lattice dynamic models solved within the framework of both classical and quantum statistical mechanics. Figure 8 is a log-log plot of the 6 T) results for ST018 [15]. The expectation from theory is that in the quantum regime, y = 2 at 0.7 kbar, after which y should decrease. The results in Fig. 8 quantitatively show the expected behavior however, y is < 2 at 0.70 kbar. Despite the difference in the methods to suppress Tc in ST018, the results in Fig. 4a and Fig. 8 are quite similar. As shown in the results in Fig. 3b, uniaxial pressure also can be a critical parameter S for the evolution of ferroelectricity in STO. 

Chemical reactions at supercritical conditions are good examples of solvation effects on rate constants. While the most compelling reason to carry out reactions at (near) supercritical conditions is the abihty to tune the solvation conditions of the medium (chemical potentials) and attenuate transport limitations by adjustment of the system pressure and/or temperature, there has been considerable speculation on explanations for the unusual behavior (occasionally referred to as anomalies) in reaction kinetics at near and supercritical conditions. True near-critical anomalies in reaction equilibrium, if any, will only appear within an extremely small neighborhood of the system s critical point, which is unattainable for all practical purposes. This is because the near-critical anomaly in the equilibrium extent of the reaction has the same near-critical behavior as the internal energy. However, it is not as clear that the kinetics of reactions should be free of anomalies in the near-critical region. Therefore, a more accurate description of solvent effect on the kinetic rate constant of reactions conducted in or near supercritical media is desirable (Chialvo et al., 1998). 

In a supercritical extraction process a solvent is contacted with a solute at conditions near a critical point of the solvent plus solute mixture. The mixture may exhibit multiphase behavior invoving vapor, liquid, and solid phases, depending on the mixture composition and temperature and pressure conditions. 

Figure 2.10 Representative Van der Waals PV isotherms for CO2 near the critical point (x), showing (a) contrasting monotonic behavior above Tc (at T = 31 OK) compared with oscillatory loops below Tc (at T = 280 and 290K) (b) Maxwell s equal-area construction for finding pressure P0 (horizontal dashed line) that cuts off equal areas in the upper loop (between VM and VG) and the lower loop (between VL and VM) P0 = 52.2 atm for T = 280K, 60.4 atm for 290K.

An older water model deserves mention, if not for its likely authenticity, at least for its excellent numerical agreement between calculated and observed values. Eucken (39) treated water as a mixture of distinct, associated species, such as dimers, tetramers, and octamers. It now seems unlikely that this theory can be correct although Wicke (156) recently argued that at least near the critical point, dimers of the type implied by Eucken s theory may exist and be responsible for the anomalous behavior at that temperature (see Ref. 101). Wicke (156) has also dis-... 

This chapter deals with critical phenomena in simple ionic fluids. Prototypical ionic fluids, in the sense considered here, are molten salts and electrolyte solutions. Ionic states occur, however, in many other systems as well we quote, for example, metallic fluids or solutions of complex particles such as charged macromolecules, colloids, or micelles. Although for simple atomic and molecular fluids thermodynamic anomalies near critical points have been extensively studied for a century now [1], for a long time the work on ionic fluids remained scarce [2, 3]. Reviewing the rudimentary information available in 1990, Pitzer [4] noted fundamental differences in critical behavior between ionic and nonionic fluids. 

There is a large body of experimental work on ternary systems of the type salt + water + organic cosolvent. In many cases the binary water + organic solvent subsystems show reentrant phase transitions, which means that there is more than one critical point. Well-known examples are closed miscibility loops that possess both a LCST and a UCST. Addition of salts may lead to an expansion or shrinking of these loops, or may even generate a loop in a completely miscible binary mixture. By judicious choice of the salt concentration, one can then achieve very special critical states, where two or even more critical points coincide [90, 160,161]. This leads to very peculiar critical behavior—for example, a doubling of the critical exponent y. We shall not discuss these aspects here in detail, but refer to a comprehensive review of reentrant phase transitions [90], We note, however, that for reentrant phase transitions one has to redefine the reduced temperature T, because near a given critical point the system s behavior is also affected by the existence of the second critical point. An improper treatment of these issues will obscure results on criticality. 

From a global assessment of these results, it seems inescapable to conclude that mean-field behavior does not remain valid asymptotically close to the critical point. Rather, ionic systems seem to show Ising-to-mean-field crossover. Such a crossover has been a recurring result observed near liquid-liquid consolute points in Coulombic electrolyte solutions, in ternary aqueous electrolyte solutions containing an organic cosolvent, and in binary aqueous solutions of NaCl near the liquid-vapor critical line. 

The remnants of the critical behavior is seen even at conditions significantly away from the critical point. For example, the specific heat and compressibilities are still quite large at the reduced temperature T = 1.41 and density p = 0.3, whereas the critical values are T = 1.32 and p = 0.32. This makes study of diffusion near the critical point a challenging theoretical problem. 

Tn the critical region of mixtures of two or more components some physical properties such as light scattering, ultrasonic absorption, heat capacity, and viscosity show anomalous behavior. At the critical concentration of a binary system the sound absorption (13, 26), dissymmetry ratio of scattered light (2, 4-7, II, 12, 23), temperature coefficient of the viscosity (8,14,15,18), and the heat capacity (15) show a maximum at the critical temperature, whereas the diffusion coefficient (27, 28) tends to a minimum. Starting from the fluctuation theory and the basic considerations of Omstein and Zemike (25), Debye (3) made the assumption that near the critical point, the work which is necessary to establish a composition fluctuation depends not only on the average square of the amplitude but also on the average square of the local... 

The first example of a free-radical chain reaction successfully conducted in sc C02, which demonstrated the potential of this solvent for preparative scale chemistry, was a report from the McHugh group (Suppes et al., 1989) dealing with the oxidation of cumene (eq. 4.4). The propagation steps for this reaction are depicted in Scheme 4.11. Pressure (and thus viscosity) had little effect on the initiation, propagation, or termination rate constants. No unusual kinetic behavior was observed near the critical point. 

It also runs into difficulties of a different kind at the other end of the coexistence curve, as one approaches the critical point. GEMC effectively supposes that criticality may be identified by the coalescence of the two peaks in the separate branches of the density distribution captured by the two simulation boxes. The limiting critical behavior of the full density distribution in a system of finite size (Fig. 10, to be discussed below) shows that this is not so the critical point cannot be reliably located this way. These difficulties are reflected in the strong finite-size dependence of the shape of the coexistence curve evident in GEMC studies [74]. To make sense of the GEMC behavior near the... 

The relaxation of a local mode is characterized by the time-dependent anomalous correlations the rate of the relaxation is expressed through the non-stationary displacement correlation function. The non-linear integral equations for this function has been derived and solved numerically. In the physical meaning, the equation is the self-consistency condition of the time-dependent phonon subsystem. We found that the relaxation rate exhibits a critical behavior it is sharply increased near a specific (critical) value(s) of the interaction the corresponding dependence is characterized by the critical index k — 1, where k is the number of the created phonons. In the close vicinity of the critical point(s) the rate attains a very high value comparable to the frequency of phonons. 

Several different measurements were made as a result of the different phase behavior observed for the two binary mixtures. Gas-solid equilibrium was observed for mixtures of TPP and pentane at conditions near the critical point of pentane. Hence, solid solubilities for TPP in supercritical pentane were measured. 

---