Isotherm critical

Because a phase change is usually accompanied by a change in volume the two-phase systems of a pure substaiice appear on a P- V (or a T- V) diagram as regions with distinct boundaries. On a P- V plot, the triple point appears as a horizontal line, and the critical point becomes a point of inflection of the critical isotherm, T = T (see Figure 2-78 and Figure 2-80). 

On uptake of hydrogen, at lower temperatures distant enough from that of the critical isotherm, the physical properties of palladium or nickel re-... 

The critical temperature is the highest temperature at which a gas may be liquefied by pressure, and, since the pressure increases with the temperature, there will correspond to the critical temperature a critical pressure (pK), which is the greatest pressure which will produce liquefaction. This pressure is given by the ordinate of the critical point K, or point of inflexion, on the critical isotherm. 

The most important case is the critical isotherm on the p, r diagram. This has a point of inflexion at the critical point, there becoming parallel to the volume axis, and everywhere else slopes constantly from right to left upwards (Rule of Sarrau, 1882). 

Figure 8.7 Experimental p- V isotherms for CCK Point a is the critical point where the critical isotherm (7=304.19 K) has zero slope. The dashed line encloses the (liquid + vapor) two-phase region. At 7=293.15 K, points b and c give the molar volumes of the liquid and vapor in equilibrium, while points d and e give the same information at 273.15 K.

At the temperature of the critical isotherm (71 = 304.19 K for C02), the coexistence region has collapsed to a single point and represents a point of inflection in the isotherm. From calculus we know that at an inflection point, the first and second derivatives are equal to zero so that... 

For sub-critical isotherms (T < Tc), the parts of the isotherm where (dp/dV)T < 0 become unphysical, since this implies that the thermodynamic system has negative compressibility. At the particular reduced volumes where (dp/dV)T =0, (spinodal points that correspond to those discussed for solutions in the previous section. This breakdown of the van der Waals equation of state can be bypassed by allowing the system to become heterogeneous at equilibrium. The two phases formed at T[Pg.141]

Figure 5.5. Isotherms for CO2 taken from the work of Andrews [3]. The shaded area indicates the region of stability of a one-phase liquid system. Liquid and vapor exist together at equilibrium under the dashed curve. Above the critical isotherm, 31.1°C, no distinction exists between liquid and gas. Andrews suggested that the term vapor be used only to represent the region to the right of the dashed curve below the critical temperature. The dashed curve ABCD represents the van der Waals equation. (By permission, from J. R. Partington, An Advanced Treatise on Physical Chemistry, Vol. 1, Longman Group, Ltd., London, 1949,...

If we now repeat compression along the critical isotherm 7, we reach (at the critical point) a flexus condition, analytically defined by the partial derivatives... 

The supercritical vapor region, below the critical isobar and above the critical isotherm. 

The vapor region, limited by the vaporization boundary and the critical isotherm. 

Such a diagram is presented in Fig 4.1-1, p 238 of Ref 3. The adjective "critical is also applied to temperature, pressure, volume and density existing at that point (Ref 1, p 269). Methods for determining critical point on the "critical isotherm are given in Ref 3, pp 357-63 (See also under "Corresponding States and under "Critical Phenomena )... 

Once (a) is evaluated, three points can be selected on the critical isotherm and b calcd for each point. The three values substituted into eq (3) give three eqs that can be solved simultaneously for A, B and C Rush Gamson (Addnl Ref Ag) showed chat the constants of the Macleod equation are functions of the "critical constants ... 

For temperatures below the vapor—liquid critical temperature, T, isotherms to the left of the liquid saturation curve (see Fig. 3) represent states of subcooled liquid isotherms to the right of the vapor saturation curve are for superheated vapor. For sufficiently large molar volumes, V, all isotherms are approximated by the ideal gas equation, P = RTjV. Isotherms in the two-phase liquid—vapor region are horizontal. The critical isotherm at temperature T exhibits a horizontal inflection at the critical state, for which... 

Cv Pc 8 a an P Y c f specific heat reduced density critical exponent for the critical isotherm critical exponent for the specific heat critical exponent for the specific heat along isot r critical exponent for the order parameter critical exponent for the susceptibility reduced temperature friction coefficient... 

Figure 7. Conductance-viscosity products Ai of tetra-n-butylammonium picrate dissolved in 1-tridecanol [72] (O) and 1-chloroheptane ( ) [137] as a function of the salt concentration along near-critical isotherms normalized to the values at the critical points.

In the literature there have been repeated reports on an apparent mean-field-like critical behavior of such ternary systems. To our knowledge, this has first been noted by Bulavin and Oleinikova in work performed in the former Soviet Union [162], which only more recently became accessible to a greater community [163], The authors measured and analyzed refractive index data along a near-critical isotherm of the system 3-methylpyridine (3-MP) + water -I- NaCl. The shape of the refractive index isotherm is determined by the exponent <5. Bulavin and Oleinikova found the mean-field value <5 = 3 (cf. Table I). Viscosity data for the same system indicate an Ising-like exponent, but a shrinking of the asymptotic range by added NaCl [164],... 

Figure 10. Degree of dissociation a of the RPM as a function of the total ion density along the critical isotherms of Debye-Httckel-Ebeling (or Bjerrum) theory (DHEb), Fisher-Levin theory (FL), and Weiss-Schroer theory (WS) [138]. The asterisks show the critical points of the three models.

When these values of a and b are used, the van der Waals equation fits the critical point and the slope and curvature of the critical isotherm. By continuity, the equation should also be a good fit to experimental data at temperatures slightly above the critical point. Other values of the constants may provide a better fit to data at conditions far from the critical point. Because it is an analytical function, the van der Waals equation cannot reproduce the discontinuities characteristic of vaporization shown at the two-phase regions in Figs. 7-9. Equation (21) (as well as other two-parameter equations, such as those of Berthelot, and Redlich and... 

Fluids are highly compressible along near-critical isotherms (L01-1.2 Tc) and display properties ranging from gas-like to Liquid-Like with relatively small pressure variations around the critical pressure. The liquid-like densities and better-than-liquid transport properties of supercritical fluids (SCFs) have been exploited for the in situ extraction of coke-forming compounds from porous catalysts [1-6], For i-hexene reaction on a low activity, macroporous a catalyst, Tiltschcr el al. [1] demonstrated that reactor operation at supercritical... 

The horizontal segments of the isotherms in the two-phase region become progressively shorter at higher temperatures, being ultimately reduced to a point at C. Thus, the critical isotherm, labeled T exhibits a horizontal inflection at the critical point C at the top of the dome. Here the liquid and vapor phases cannot be distinguished from one another, because their properties are the same. 

The constants in an equation of state may of course be evaluated by a fit to available PVT data. For simple cubic equations of state, however, suitable estimates come from the critical constants Tc and Pc. Since the critical isotherm exhibits a horizontal inflection at the critical point, we may impose the mathematical conditions ... 

In spite of the presence of the type 1 part of the surface, one can frequently estimate the two-dimensional critical temperature of the phase transition, provided that the type 2 part of the surface is uniform enough to make the transition manifest. Thus we can use this procedure, for example, with the supra- and sub-critical isotherms of methane, ethane, and xenon adsorbed by the jlOOl face of sodium chloride, published by Ross and Clark (12). Using estimates of aTc from these data in Equation 4 yields a value for the surface field of F = 1.5 0.5 X 10r> e.s.u. per sq. cm. the poor precision of the result is due to the difficulty of interpolating the temperature of the critical isotherm by eye. The surface field of the jlOOj face of sodium bromide can be estimated in a similar way, using the data of Fisher and McMillan (6) for the adsorption of methane and krypton. The result for the surface field is again between 1 and 2 X 105 e.s.u. per sq. cm. 

Very few experiments have been performed on vibrational dynamics in supercritical fluids (47). A few spectral line experiments, both Raman and infrared, have been conducted (48-58). While some studies show nothing unique occurring near the critical point (48,51,53), other work finds anomalous behavior, such as significant line broadening in the vicinity of the critical point (52,54-60). Troe and coworkers examined the excited electronic state vibrational relaxation of azulene in supercritical ethane and propane (61-64). Relaxation rates of azulene in propane along a near-critical isotherm show the three-region dependence on density, as does the shift in the electronic absorption frequency. Their relaxation experiments in supercritical carbon dioxide, xenon, and ethane were done farther from the critical point, and the three-region behavior was not observed. The measured density dependence of vibrational relaxation in these fluids was... 

C, for ethane, fluoroform, and C02, respectively. The near-critical isotherm temperatures for the three solvents were chosen so that the reduced temperatures (T/Tc) are essentially the same. For the near-critical isotherm, the lifetime decreases rapidly as the density is increased from low density. As the critical density, pc, is approached, the rate of change in the lifetime with density decreases (pc = 6.88, 7.56, and 10.6 mol/L, for ethane, fluoroform, and C02, respectively). The change in slope is more pronounced for C02. This is even more evident when the gas phase (zero density) contribution to the lifetime is removed from the data (see below). The difference between the C02 data and the other data will be discussed in Section V. The higher temperature isotherm data show a somewhat more uniform density dependence. Although the slope decreases with increasing density, there is a somewhat smaller change in slope than in the near-critical isotherm data. 

Figure 12 shows Ti(p, T) measured in fluoroform. Figure 12a shows data on the near critical isotherm of 28°C, i.e., 2 K above Tc. The calculated curve is scaled to match the data at the critical density, 7.5 M. (In... 

Figure 12 (a) Ti (p, T) data measured in fluoroform on the near critical isotherm (28°C), 2 K above Tc, and the calculated curve, which is scaled to match the data at the critical density, 7.56 mol/L. The fluoroform hard sphere diameter was adjusted since a good value at experimental temperatures is not available. A diameter of 3.28 A yielded the optimal fit. co is not adjustable. It is set equal to 150 cm-1, the value obtained in the fit of the ethane data. The theory does a very good job of reproducing the shape of the data with only the adjustment in the solvent size as a fitting parameter that affects the shape of the calculated curve, (b) Ti(p, T) data taken at 44°C, which is the equivalent increase in temperature above Tc as the higher temperature data taken in ethane (Fig. 10b). The theory curve is calculated using the same scaling factor, frequency, and solvent hard sphere diameter as at the lower temperature. Considering that there are no free parameters, the theory does an excellent job of reproducing the higher temperature data.

The isothermal data in the three supercritical solvents all show similar trends. For the near-critical isotherm, the lifetime initially decreases rapidly with density. At an intermediate density less than pc, the slope changes and develops a much weaker density dependence. Along the higher temperature isotherm, the density dependence is somewhat smoother and does not exhibit as substantial a change of slope. The isopycnic (constant density) data, however, display an interesting solvent dependence. 

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