Triple point derived

The four coefficients A, B, C, and D have been derived, for example, with selected hydrocarbons [25, 26], Equation 7.4.3 accurately represents the vapor pressure function over the entire temperature range between the triple point and the critical point. If the coefficients are not available for a given compound, they can be calculated. D is calculated from the pressure van der Waals constant, a, which can be estimated from group contributions. B is calculated directly from group contributions. Then the coefficients A and C can be estimated from two pv/T points (e.g., normal boiling point and critical point). This approach has been evaluated for various classes of hydrocarbons commonly encountered in petroleum technology [25, 26]. 

Note that the only approximation made in the derivation of Eq. (177) is the use of the Kirkwood superposition approximation for the triplet distribution function of the liquid [21]. In a dense liquid at low temperature (near its triple point), this is not a bad approximation [21],... 

We must emphasize that this derivation requires that the variables be extensive. We illustrate the necessity of this by considering a one-component system at its triple point. In this case there are no degrees of freedom, and... 

In summary, we refer to Figure 5.5, which may be considered as the projection of the entire equilibrium surface on the entropy-volume plane. All of the equilibrium states of the system when it exists in the single-phase fluid state lie in the area above the curves alevd. All of the equilibrium states of the system when it exists in the single-phase solid state lie in the area bounded by the lines bs and sc. These areas are the projections of the primary surfaces. The two-phase systems are represented by the shaded areas alsb, lev, and csvd. These areas are the projections of the derived surfaces for these states. Finally, the triangular area slv represents the projection of the tangent plane at the triple point, and represents all possible states of the system at the triple point. This area also is a projection of a derived surface. 

Figure 3. (a) The derivative of f with respect to <5i. The lamellar phase cannot exist in the regions where the derivative is positive, (h) The derivative of f with respect to 62. The lamellar phase cannot exist in the regions where the derivative is positive, (c) The contour plot of the derivatives off with respect to hi and 62, respectively. At line 1, df/ddi = 0 and at line 2, 3/7 3 2 = 0 (inside each of the marked domains delimited by lines 1 and 2, an excess phase separates). The intersections of the line 1 and 2 are triple points, where the lamellar phase is in equilibrium with both excess phases. 

At low temperatures the nematic gel coexists with excess solvent, i.e., the i-N biphasic coexistence. Above the triple point TJ, excess solvent coexists with an isotropic gel, i.e., the i-I coexistence. Also, above this temperature, the isotropic and the nematic phases of a gel can coexist, i.e., I-N phases. Beyond the phase gap, a single nematic phase exists. The I-N region terminates at the = 1 axis. This is at T the N-I transition temperature of the undiluted network which, in turn, is close to Tni, the transition temperature of the uncrosslinked polymer melt from which the network derives. The nematic order is not expected for any 4> for T above. This is the limit of stability for even the undiluted case. 

The above derivations are limited to the unique case of Fig. 4.16b. The solutions in terms of the limiting (asterisked) quantities are valid whenever the equilibrium curve intersects the operating lines only at x =x. If the equilibrium line is sufficiently curved, as illustrated in Fig. 4.16c, one of the operating lines may first intersect its equilibrium line at some point other than Xp, thereby making invalid the above limiting-condition equations for the triple-point intersection. 

Vapor Pressures. Approaching the triple point, propane vapor pressures become immeasurably small. From Ref. 5 we derived new data from the triple point to the boiling point by thermal loops in a procedure... 

Derivative data given in order m p, crystal color, solvent from which crystallized T = triple point, S = at saturation pressure... 

Values of AHu, in g-cal/g-mole, were derived from orthobaric densities and the vapor pressure equations via the Clapeyron equation P ]. An approximate representation of the temperature dependence from the triple point to 32.7°K has recently been evaluated by least-squares ... 

Sonic velocities have been computed from P-F-T data and the derived specific heats. They are tabulated on isobars from 1 to 340 atm at temperatures from the triple point to 100°K, including the saturated fluid phases. Uncertainty is estimated at 0.5%, increasing to 2 % at the high-pressure boundary, for the saturated liquid, and for temperatures below 18°K. It is unknown close to the critical point p ]. Figure 6 depicts these results. 

The transition of polyethylene from the orthorhombic to the hexagonal phase was observed at very high pressures (3000-5000 bar) by proton NMR and nuclear magnetic relaxation time measurements the phase diagram derived from this spectroscopy is in substantial agreement with DSC data. According to these results (see Fig. 1), the hexagonal phase can be observed above the triple point at 490 K and 3000 bar [13,14]. 

Theoretically Based Equations.— Although semi-theoretical derivations can be produced for some vapour-pressure equations, for example equation (12), these are valid at fairly low pressures only, and there are no theoretically based equations which relate vapour pressure to temperature over the whole range from the triple point to the critical point. Equations can be... 

CP, (374C, 221 bar), and triple point, TP, are indicated on the gas-liquid coexistence curve. The points on the broken line extending from TP to the right denote the transitions between different high pressure modifications of ice. At pressures above 25 kbar, water densities have been derived from shock wave measurements ( 7) At 500 C and 1000 C, pressures of about 8 and 20 kbar are needed to produce the triple point density of liquid water. 

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