Critical point divergences

It would be convenient if the critical temperature of a mixture were the mole weighted average of the critical temperatures of its pure components, and the critical pressure of a mixture were simply a mole weighted average of the critical pressures of the pure components (the concept used in Kay s rule), but these maxims simply are not true, as shown in Fig. 3.22. The pseudocritical temperature falls on the dashed line between the critical temperatures of CO2 and SO2, whereas the actual critical point for the mixture lies somewhere else. The solid line in Fig. 3.22 illustrates how the locus of the actual critical points diverges from the locus of the pseudo critical points. 

A third exponent y, usually called the susceptibility exponent from its application to the magnetic susceptibility x in magnetic systems, governs what m pure-fluid systems is the isothennal compressibility k, and what in mixtures is the osmotic compressibility, and detennines how fast these quantities diverge as the critical point is approached (i.e. as > 1). 

However, the discovery in 1962 by Voronel and coworkers [H] that the constant-volume heat capacity of argon showed a weak divergence at the critical point, had a major impact on uniting fluid criticality widi that of other systems. They thought the divergence was logaritlnnic, but it is not quite that weak, satisfying equation (A2.5.21) with an exponent a now known to be about 0.11. The equation applies both above and... 

That analyticity was the source of the problem should have been obvious from the work of Onsager (1944) [16] who obtained an exact solution for the two-dimensional Ising model in zero field and found that the heat capacity goes to infinity at the transition, a logarithmic singularity tiiat yields a = 0, but not the a = 0 of the analytic theory, which corresponds to a finite discontinuity. (Wliile diverging at the critical point, the heat capacity is synnnetrical without an actual discontinuity, so perhaps should be called third-order.)... 

If the finite size of the system is ignored (after all, A is probably 10 or greater), the compressibility is essentially infinite at the critical point, and then so are the fluctuations. In reality, however, the compressibility diverges more sharply than classical theory allows (the exponent y is significantly greater dian 1), and thus so do the fluctuations. 

The field-density concept is especially usefiil in recognizing the parallelism of path in different physical situations. The criterion is the number of densities held constant the number of fields is irrelevant. A path to the critical point that holds only fields constant produces a strong divergence a path with one density held constant yields a weak divergence a path with two or more densities held constant is nondivergent. Thus the compressibility Kj,oi a one-component fluid shows a strong divergence, while Cj in the one-component fluid is comparable to (constant pressure and composition) in the two-component fluid and shows a weak... 

A feature of a critical point, line, or surface is that it is located where divergences of various properties, in particular correlation lengths, occur. Moreover it is reasonable to assume that at such a point there is always an order parameter that is zero on one side of the transition and tliat becomes nonzero on the other side. Nothing of this sort occurs at a first-order transition, even the gradual liquid-gas transition shown in figure A2.5.3 and figure A2.5.4. 

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].)... 

Now, assume that we are getting closer to the critical point of our transition, i.e., to the point of the second-order transition. In the case of a uniform system the critical region can be described by the divergent correlation length of statistical fluctuations [138]... 

Ising model harbors a critical point. It can be shown (see [bax82]) that the correlation length = [ln(Ai/A2)] h If H = 0, however, then it can also be shown that limy g+(Ai/Aj) = 1 and, thus, that oo at // = 7 = 0. Since one commonly associates a divergent correlation length with criticality, it is in this sen.se that 7/ = T = 0 may be thought of as a critical point. 

Raveau now calculated the values of p, v from van der Waals equation, plotted the logarithms, and compared the diagram with a similar one drawn from the experimental results. The results showed that the diagrams could not be made to fit in the ease of carbon-dioxide and acetylene, the divergencies being very marked near the critical point. 

Again within the Matsubara technique one still should do the replacement lu -> tjn - 2mnT, -i f - T"=-oo/We dropped an infinite constant term in (14). However expression (14) still contains a divergent contribution. To remove the regular term that does not depend on the closeness to the critical point we find the temperature derivative of (14) (entropy per unit volume) ... 

Very large isotope effects like those shown in Fig. 5.12 seem to be limited to the hypercritical regions of phase diagrams, i.e. not too far from thermodynamic divergences of the type (dP/dT)c = 00 or (dT/dP)c = 00 (i.e. pressure-double critical points (p-DCP) or temperature-double critical points (T-DCP), respectively). 

Equations 8.16 and 8.17 or 8.16 and 8.18 show that, at the critical point, the specific first and second derivative properties of any representative equation of state will be divergent (Johnson and Norton, 1991). This inherent divergency has profound consequences on the thermodynamic and transport properties of H2O in the vicinity of the critical point. Figure 8.7 shows, for example, the behav-... 

Many physical properties diverge near (i.e. show large values as is approached from either side). Interestingly, divergences of similar quantities in different phase transitions are strikingly similar as shown by the typical Cp-temperature curves in Fig. 4.8. These divergences can be quantified in terms of the so-called critical exponents. A critical point exponent is given by... 

Fig. 24a-c. a. Equilibrium radius of a NIPA gel sphere as a function of temperature. At lower temperatures the gel is swollen and at higher temperatures it is shrunken. At about 34 °C the swelling curve becomes infinitely sharp, which corresponds to the critical point, b. Relaxation time of gel volume change in response to a temperature jump, as a function of temperature, c. Thermal expansion coefficient, the relative radius change per temperature increment, also diverges at the critical point... 

Tanaka et al. found a critical point at the zero-osmotic pressure condition in ionic gels by varying the degree of ionization and diminishing the volume-discontinuity at first-order changes. At such a critical point, the first three derivatives of F with respect to V should vanish from Eqs. (2.6) and (2.26). On the other hand, the so-called spinodal point is given by K = 0, at which the volume fluctuations diverges as shown by Eq. (2.10). 

The mysteries of the helium phase diagram further deepen at the strange A-line that divides the two liquid phases. In certain respects, this coexistence curve (dashed line) exhibits characteristics of a line of critical points, with divergences of heat capacity and other properties that are normally associated with critical-point limits (so-called second-order transitions, in Ehrenfest s classification). Sidebar 7.5 explains some aspects of the Ehrenfest classification of phase transitions and the distinctive features of A-transitions (such as the characteristic lambda-shaped heat-capacity curve that gives the transition its name) that defy classification as either first-order or second-order. Such anomalies suggest that microscopic understanding of phase behavior remains woefully incomplete, even for the simplest imaginable atomic components. 

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... 

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