Liquids critical point

With these simplifications, and with various values of the as and bs, van Laar (1906-1910) calculated a wide variety of phase diagrams, detennining critical lines, some of which passed continuously from liquid-liquid critical points to liquid-gas critical points. Unfortunately, he could only solve the difficult coupled equations by hand and he restricted his calculations to the geometric mean assumption for a to equation (A2.5.10)). For a variety of reasons, partly due to the eclipse of the van der Waals equation, this extensive work was largely ignored for decades. 

Detailed measurements of the solubility between the lower and upper critical end points have been made only for the solutions in ethylene of naphthalene,14 hexachlorethane,30 and />-iodochloro-benzene.21 Atack and Schneider2 have used dilute solutions of the last-named substance to study the formation of clusters near the gas-liquid critical point of ethane. 

Figure 7.4 Phase diagram of 4He, showing the solid, gas, and two liquid phases (He-I, superfluid He-II), the A-line (dashed) of liquid-liquid transitions (upper terminus 1.76K, 29.8 atm lower terminus 2.17K, 0.0497 atm), and the gas-liquid critical point (circle-x 5.20K, 2.264 atm).

As shown, CP varies in a continuous manner, but appears to diverge at Tx (= 2.17K at latm), resembling in this respect the behavior at the gas-liquid critical point (discussed... 

About 30 years after Buback and Franck s pioneering study on the critical point of molten NH4CI and 10 years after Singh and Pitzer s report on a mean-field nature of the liquid-liquid critical point in an electrolyte solution,... 

The Lw-H-V line has no upper pressure or temperature limit because the pure methane (or nitrogen) vapor-liquid critical points (at 191 and 126 K respectively) are far below the quadruple point Qi. Such low critical temperatures prevent intersection of the vapor pressure line with the Lw-H-V line above 273 K to produce an upper quadruple point. 

Bansal, V., Christiansen, R.L., Sloan, E.D., Influence of Guest Vapor-Liquid Critical Point on Hydrate Formation Conditions, AIChE/., 39(10), 1735 (1993). 

The Clapeyron equation does not apply to a continuous transition, since both the entropy (or enthalpy) change and the volume change are zero. For such a transition, in the region of the critical point, the change in the thermodynamic variable given by the second derivative of G can be represented by an exponential equation. For example, in the region of the (vapor + liquid) critical point, AFyap and T are related byp... 

At low temperatures, rj will be unity because all of the Cu atoms will be localized on A sites. 1 But the degree of disorder increases as the temperature increases until the Cu and Zn atoms are mixed randomly on the two sublattices and 77 = 0. This process, called a positional (order + disorder) transition, is often described as a cooperative phenomenon because it becomes easier to produce additional disorder once some disorder is generated. In the vicinity of a critical temperature, the order parameter rj behaves like the density difference (pi — pg) near the gas-liquid critical point. Thus,... 

Referring again to Figure 14.14, the isotherms at temperatures T3 and T4 are of the typical (gas + liquid) type, but at T2, a temperature below u, two critical points occur, one at f and the other at h. The one at f is a typical (liquid + liquid) critical point while the one at h is better characterized as a (gas + liquid) critical point. In most systems with type III behavior, the critical locus bh occurs over a narrow temperature range, and the double critical points occur only over this small range of temperature. 

Fig. 7.17. Ortho-positronium annihilation rates at various values of ethane gas density D, at a temperature of 305.45 K. The solid line is a weighted average of the annihilation rates between 120 and 180 amagat. The broken line is the prediction for free ortho-positronium. The data are due to Sharma, Kafle and Hart (1984). Reprinted from Physical Review Letters 52, Sharma, Kafle and Hart, New features in the behaviour of ortho-positronium annihilation rates near the vapour-liquid critical point of ethane, 2233-2236, copyright 1984 by the American Physical Society.

Sharma, S.C., Kafle, S.R. and Hart, J.S. (1984). New features in the behaviour of orthopositronium annihilation rates near the vapour-liquid critical point of ethane. Phys. Rev. Lett. 52 2233-2236. 

Values of nc at the gas-liquid critical point aft derived from radii corresponding to the principal maxima in the radial distribution functions using relativistic wave functions. 

These fluctuations, which are referred to as order-parsmeter fluctuations in studies of critical phenomena (3). comprise the driving forces for transport in the system. For liquid mixtures near a critical mixing point, the order parameter is concentration, and for pure gases near the vapor-liquid critical point, the order parameter is density. For gas mixtures such as supercritical solutions near the critical line, the order parameter is again density, which is a function of composition and temperature compared to a pure gas where density is a function of only temperature at constant pressure. 

The dec8y rate of the order-parameter fluctuations is proportional to the thermal diffusivity in case of pure gases near the vapor-liquid critical point and is proportional to the binary diffusion coefficient in case of liquid mixtures near the critical mixing point (6). Recently, we reported (7) single-exponential decay rate of the order-parameter fluctuations in dilute sugercritical solutions of liquid hydrocarbons in CO for T - T 10 C. This implied that the time scales associated with thermal diffusion and mass diffusion are similar in these systems. 

This LCEP is a gas-liquid critical point in the presence of the solid phase. A LCEP was not observed for toluene-TPP mixtures at temperatures below 350°C. Measurements were not made at higher temperatures because of thermal degradation of the porphyrin. The results in Tables I-III are also shown on PT projections in Figures 1 and 2. The mixture critical points in Figure 2 are obtained from Figure 4. 

The question of whether there is a tme glassy nature of amorphous ices is of interest when speculating about possible liquid-liquid transitions in (deeply) supercooled water. For true glasses, the amorphous-amorphous transitions described here can be viewed as the low-temperature extension of liquid-liquid transitions among LDL, HDL, and possibly VHDL. That is, the first-order like LDA <-> HDA transition may map into a first-order LDL HDL transition, and the continuous HDA <-> VHDA transition may map into a smeared HDL VHDL transition. Many possible scenarios are used how to explain water s anomalies [40], which share the feature of a liquid-liquid transition [202, 207-212]. They differ, however, in the details of the nature of the liquid-liquid transition Is it continuous or discontinuous Does it end in a liquid-liquid critical point or at the reentrant gas-liquid spinodal ... 

Figure 13.15 is drawn for a single constant pressure equilibrium phase compositions, and hence the locations of the lines, change with pressure, but the general nature of the diagram is the same over a range of pressures. For the majority of systems the species become more soluble in one another as the temperature increases, as indicated by lines CG and DH of Fig. 13.15. If this diagram is drawn for successively higher pressures, the corresponding three-phase equilibrium temperatures increase, and lines CG and DH extend further and further until they meet at the liquid/liquid critical point Af, as shown by Fig. 13.16. The temperature at which this occurs is known as the upper critical solution temperature, and at this temperature the two liquid phases become identical and merge into a single phase.

Some indirect experimental evidence exists for the liquid-liquid critical point hypothesis from the changing slope of the melting curves, which was observed for different ice polymorphs (30, 31). A more direct route to the deeply supercooled region, by confining water in nanopores to avoid crystallization, has been used more recently by experimentalists. These researchers applied neutron-scattering, dielectric, and NMR-relaxation measurements (32-35). These studies focus on the dynamic properties and will be discussed later. They indicate a continuous transition from the high to the low-density liquid at ambient pressure. The absence of a discontinuity in this case could be explained by a shift of the second critical point to positive pressures in the confinement. This finding correlated with simulations, which yield such a shift when water is confined in a hydrophilic nanopore (36). 

Mishima O. Liquid-liquid critical point in heavy water. Phys. 

Figure 4 Cartoon for hidden liquid-liquid spinodal in a polymer melt, calculated for a Flory Rotational Isomeric State chain with a simple coupling between density and conformational order A is the trans-gauche energy gap and a dimensionless density. Shown is the path of an isochoric quence into the unstable regime. r, is the spinodal temperature, T the melting temperature, T the liquid-liquid critical point, and Tp the temperature at which the harrier between dense liquid and crystal is of order ksT ...

Thus close to the critical point of phase separation the behavior is first nonmonotonic and then the scaling behavior of Ig e) becomes a two-index and then a one-index. Similar expressions can be derived for gas-liquid critical point behavior. 

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