Critical point measurements

Literature data [108] review of critical point measurements using pulse-heating. Speed of sound... 

Figure 2 shows the thermal conductivities of the saturated liquid between the triple point and the critical point (measured values given by smoothed solid lines and extrapolated... 

The bond critical point properties of an electron density distribution are evaluated at the bond critical point, rc, of a bonded interaction. Collectively, they consist of the curvatures and the Laplacian of the distribution, the value of /9(rc) and the bonded radii of the bonded atoms. The curvatures of p(rc) determine the local concentration or local depletion of the electron density distribution in the vicinity of the bond critical point measured in three mutually perpendicular directions. As observed by Bader and Essen (1984), the curvatures in these directions are found by evaluating the eigenvalues and eigenvectors of the Hessian matrix of p(rc), Hy = p(r l dXjdxj, (ij = 1,3). The three... 

Koningsveld and Kleintjens report on spinodal and critical point measurements for polymer mixtures. They point out that complexity of shape is the most striking feature shared by the majority of experimental cloud point curves in polymer mixtures. Curves with a shoulder and even bimodal curves have been reported. More recently spinodal curves obtained by PICS (p. 315) have also shown complicated shapes including evidence of bimodality. ... 

As the critical point is approached vanishes proportionally to IT —Tl, where P-T is the distance from that critical point measured as the difference between the temperature T and the critical temperature P. and where t (not to be confused with chemical potential) is the critical point surface-tension exponent with the universal value fi —1-26 (Chapter 9). In Fig. 8.10, also due to Cahn. the ordinate of one of the curves is a, shown vanishing at the origin with a power i>l of JT - T). The ordinate of the other curve is the difference where... 

The distance Zt Zf between the F) =0 and F -0 dividing surfaces, given by (2.49), also shows interesting critical behaviour describable by the exponents already introduced. This distance, by (2.49), is F (j,/Aft. The relative adsorption ru/), by (2.47), is the rate at which the interfadal tension varies with the thermodynamic field P(. The temperature is representative of such a field, so, with distance from the critical point measured by T -T, say. F, is seen to vanish proportionally to (T -T) as the critical point is approached. At the same time the density difference Ap, vanishes proportionally to (T "-T) with the result, then, that ... 

It should be pointed out here that the asymptotic description of the thermal conductivity is valid only extremely close to the critical point. Measurements on He + He (Cohen et al. 1982), methane + ethane (Friend Roder 1985 Roder Friend 1985) and CO2 + ethane (Mostert et al. 1992) seem to indicate that the thermal conductivity exhibits a critical enhancement similar to that observed for pure fluids. In Figure 6.7, as an example the experimental thermal-conductivity results for CO2 + ethane for a mole fraction of 25% CO2 in the one-phase region close to the critical isochore are presented, which were obtained by Mostert (1991). To reconcile the experimental data with the asymptotic result of equation (6.54), again a crossover theory is needed. Thermophysical quantities in fluid mixtures near a plait point undergo two types of crossover as the... 

To check the viability of Eq. (16) the system di-phenylether/branched polyethylene has been used as a test case [78]. Experimental liquid-liquid critical points, measured with the phase volume ratio method [44] for samples varying in degree of branching, may serve the purpose together with some experimental spi-nodal temperatures, measured with the Pulse-Induced Critical Scattering technique of Gordon et al. [41 3]. The data were used to calculate values for the parameters in Eqs. (17) and (18) (details of the fitting procedure can be found in [54]). 

It is possible to identify particular spectral features in the modulated reflectivity spectra to band structure features. For example, in a direct band gap the joint density of states must resemble that of critical point. One of the first applications of the empirical pseudopotential method was to calculate reflectivity spectra for a given energy band. Differences between the calculated and measured reflectivity spectra could be assigned to errors in the energy band... 

Many of the earlier uncertainties arose from apparent disagreements between the theoretical values and experimental detemiinations of the critical exponents. These were resolved in part by better calculations, but mainly by measurements closer and closer to the critical point. The analysis of earlier measurements assumed incorrectly that the measurements were close enough. (Van der Waals and van Laar were right that one needed to get closer to the critical point, but were wrong in expectmg that the classical exponents would then appear.) As was shown in section A2.5.6.7. there are additional contributions from extended scaling. 

Whereas this two-parameter equation states the same conclusion as the van der Waals equation, this derivation extends the theory beyond just PVT behavior. Because the partition function, can also be used to derive aH the thermodynamic functions, the functional form, E, can be changed to describe this data as weH. Corresponding states equations are typicaHy written with respect to temperature and pressure because of the ambiguities of measuring volume at the critical point. 

Reduced Equations of State. A simple modification to the cubic van der Waals equation, developed in 1946 (72), uses a term called the ideal or pseudocritical volume, to avoid the uncertainty in the measurement of volume at the critical point. 

The Kellogg and DePriester charts and their subsequent extensions and generahzations use the molar average boiling points of the liquid and vapor phases to represent the composition effect. An alternative measure of composition is the convergence pressure of the system, which is defined as that pressure at which the Kvalues for aU the components in an isothermal mixture converge to unity. It is analogous to the critical point for a pure component in the sense that the two... 

Application of this method or Eq. (3-25 ) in the presence of stray currents is conceivable but would be very prone to error. It is particularly valid for good coating. Potential measurement is then only significant if stray currents are absent for a period, e.g., when the source of the stray current is not operating. In other cases only local direct measurements with the help of probes or test measurements at critical points can be considered. The potential test probes described in Section 3.3.3.2 have proved true in this respect. 

A further critical point are the intensities correlated to spectra of the pure elements. Calculated and experimentally determined values can diverge considerably, and the best data sets for 7 measured on pure reference samples still show a scatter of up to 10%. The use of an internal standard or a simultaneously measured external standard seems to be the most successful way to reducing the inaccuracy below 10%. (Eor a more detailed discussion of background subtraction and quantification see, e.g., Seah [2.9].)... 

Dokoupil, van Soest, and Swenker18 have examined this system in detail from 25° to 70°K and at pressures up to 50 atm. The critical points of the pure substances are at 33° and 126°K and the lower and upper critical end points are at about 37° and 61°K. The majority of their measurements lie between the critical end points. 

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. 

As standard liquid may be taken fluorbenzene, the specific volumes of which have been measured by Young up to the critical point (3 = 560° Abs.). 

In both equations, k and k are proportionality constants and 0 is a constant known as the critical exponent. Experimental measurements have shown that 0 has the same value for both equations and for all gases. Analytic8 equations of state, such as the Van der Waals equation, predict that 0 should have a value of i. Careful experimental measurement, however, gives a value of 0 = 0.32 0.01.h Thus, near the critical point, p or Vm varies more nearly as the cube root of temperature than as the square root predicted from classical equations of state. 

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