Extrapolated from neighboring values

Where published values of F2, F4, Fq, and were not available, initial assignments were based on levels calculated from the parameters F and f which were obtained by extrapolation from neighboring members of the series (4). Both F2 and can be assumed to be approximately linear functions of Z (atomic number) within the series, and by using the ratios F4/F2 and Fq/F2 calculated for a 4/ or 5f hydrogenic eigenfunction (15), all three electrostatic parameters can be evaluated from F2 alone. 

Covalent radii estimated from homonuclear bond lengths where available and from selected heteronuclear bonds otherwise. Bond lengths from Tables of Interactomic Distances and Configuration in Molecules and Jons) Sutton. L., Ed. Spec. Publ. Nos. 11 and 18 The Chemical Society London, 1958, 1%5, except where noted. Values in parentheses are for noble gases not known to form compounds and are extrapolated from the values of neighboring nonmetals Allen, L. C. Huheey, J. E. J. Jnorg. NucJ. Chem. 1980, 42, 1523. 

In the two-dimensional physisorbed environment the nearest-neighbor distance increases from 3.99 A in the bulk a-phase to 4.26 A in the commensurate monolayer on graphite. This change reduces the quadrupole-quadmpole and dipole-dipole interaction energies to about 72% and 82% of the bulk values, respectively. Additional features complicating the extrapolation from bulk to expected monolayer behavior are the different number of nearest neighbors and the geometry of the coordination shells, as well as the surface interactions with the substrate. How the similarities and differences between bulk N2 and CO carry over to the two-dimensional physisorbed situation is the focus of Sections III and IV and Sections V and VI, respectively. 

The wealth of experimental thermodynamic data has been used to develop a powerful predictive capacity. As early as 1986, Breslauer [9] illustrated a useful empirical method to predict DNA duplex stability based on nearest-neighbor interactions in the base sequence. AH and AS of all 10 possible nearest-neighbor interactions were extrapolated from a small library of 28 DNA duplexes. The overall AW and AS " of DNA duplex formation could be predicted from the sum of the appropriate empirical values. This rough but robust method has been verified and further refined by more and more available databases of DNA thermodynamics [10,11]. The predictive power of this empirical methodology is a powerful tool for those interested in using DNA in supramolecular (or other) applications for which specific thermodynamics are an important criterion. 

The extrapolated line of log S-log C crossed each other at a critical concentration Cq at which S stays constant and independent of temperature. These results suggest that the temperature dependence of the cholesteric pitch would inflect at the concentration higher than Cq This is analogous to the behavior of thermotropic liquid crystals composed of cholesteric solute and nematic solvent, where the sign of dS/dT reverses at a critical concentration. It is understood that the behavior of both thermotropic and lyotropic liquid crystals is comparable provided that the nematic substances of the former are substituted with the solvents of the latter. The critical concentration Cq is about 0.41 vol/vol and this value is very close to the concentration at which the side chains on neighboring molecules of the polymer come to contact each other ( refer to fig.5 ). From these results, it is expected that the origin or mechanism of twist would change at this concentration Cq. The... 

FIGURE 5.5 Electrostatic interaction energy, E -g, of a layer of polarizable point-dipoles above the metal surface as a function of the nearest neighbor dipole-dipole distance for two limiting examples, that is, a layer of dipoles oriented parallel ( ) and perpendicular (X) to the surface. The plotted range of / nn and the parameters of the polarizable point dipole model are chosen characteristically for BTAH (m = 4.1 D, a = 91 bohr, ril im = 7 bohrs, and ri im = 9 bohrs). The two thin horizontal lines designate the zero coverage (R =N) extrapolated values [70]. With permission from Peljhan and Kokalj [70]. 

The parameters of the second critical endpoint Q (Li = L2-S) and neighboring portions of ctuwes QTb (L1-L2-S) and QKb (Li = L2-S) are characterized by pressure values that are considerably higher than fliose of the supercritical transition of water from a gas-like to a hquid-like state (critical isochore or supercritical extrapolation of liquid-gas curve for pure water or those which correspond to die critical curves L = G in water-salt systems of type la (two-dots-dashed line in Figure 1.32) at the same temperature). 

Each rod in the isolated fractal can point in any direction if excluded volume is ignored, but as a fractal is interpenetrated by its neighbor the freedom of rotation of each rod is reduced. Thus the angular freedom of a rod is reduced from 4it to a value which will be of the order of the ratio of the density of the intruding fractal polymer to the rod length. A crude extrapolation formula for the entropy could then be... 

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