Van der Waals properties

Some workers take e for a non-identical pair of atoms to be an independent parameter, not related to the e-values for the identical pairs. This difference in choosing e seems to be the main cause of the difference in van der Waals properties of atoms used by different workers. To fit available structural information, hydrogen needs to be a good deal harder if e for the carbon-hydrogen interaction is taken to be the product of the square roots of the e-values of the separate atoms than it does if one can use a smaller e for the carbon-hydrogen interaction. In that case, the hydrogens can... 

VLE data, the results of two thermodynamic consistency tests, and the parameters of different -models, such as the Wilson, NRTL, and UNIQUAC equation. Additionally, the parameters of the Margules [28] and van Laar [29] equation are listed. Furthermore, the calculated results for the different models are given. For the model which shows the lowest mean deviation in vapor phase mole fraction the results are additionally shown in graphical form together with the experimental data and the calculated activity coefficients at infinite dilution. In the appendix of the data compilation the reader will find the additionally required pure component data, such as the molar volumes for the Wilson equation, the relative van der Waals properties for the UNIQUAC equation, and the parameters of the dimerization constants for carboxylic acids. Usually, the Antoine parameter A is adjusted to A to start from the vapor pressure data given by the authors, and to use the -model parameters only to describe the deviation from Raoult s law. Since in this data compilation only VLE data up to 5000 mm Hg are presented, ideal vapor phase behavior is assumed when fitting the parameters. For systems with carboxylic acids the association model is used to describe the deviation from ideal vapor phase behavior. 

For the UN I FAC group contribution method the relative van der Waals properties r, and q[ can be obtained using the relative van der Waals group volumes Rk and relative van der Waals group surface areas Qc, which can be derived from x-ray data. Tabulated values for Rk and Qjc can be found by Hansen et al. [53]. They can also be derived from the tabulated van der Waals properties published by Bondi [54. For selected groups the Rk and Qjt values are given in the Appendix H ... 

I 5 Phase Equilibria in Fluid Systems van der Waals properties ... 

First of all the van der Waals properties of the two compounds can be calculated with the help of the van der Waals properties of the groups ... 

Using these van der Waals properties for xi = 0.5 the following values are obtained for Vi and Fr. 

Table 7.6 Relative van der Waals volumes (rsoiv) and van der Waals surface areas (c soiv) for selected solvents (the relative van der Waals properties of the ions r.on and q,on were set to a value of 1).

A part of the published LIQUAC parameters [21] and van der Waals properties are given in Tables 7.5 and 7.6. 

In the last step, the SR term is calculated using the UNIQUAC equation. The required van der Waals properties and interaction parameters are summarized in the table below. 

For most of the structural groups the group interaction parameters of the group contribution equation of state are identical with the UNIFAC group interaction parameters. Therefore only van der Waals properties of a few selected gases are given below. 

Runeberg N, Pyykkd P (1998) Relativistic pseudopotential calculations on Xe2, RnXe, and Rn2 the van der Waals properties of radon. Int J Quantum Chem 66 131-140... 

Smirnov BM (1993) Mechanisms of melting of rare gas solids. Physica Scripta 48 483-486 Runeberg N, Pyykko P (1998) Relativistic pseudopotential calculations on Xe2, RnXe, and Ru2 the van der Waals properties of radon. Int J Quantum Chem 66 131-140 Batsanov SS (1998) Some characteristics of van der Waals interaction of atoms. Russ J Phys Chem 72 894-897... 

Then we have some key pieces of information, two of which are the van der Waals characteristics of helium and neon. (Larger rare gas atoms also become important when one considers atoms further down in the periodic table.) How closely can we approximate the van der Waals characteristics of hydrogen with those of helium And how closely can we approximate the van der Waals properties of carbon with those of neon We would expect that carbon and neon would be rather similar in their properties. We know that as we go across the periodic table from left to right, the van der Waals radii of the atoms become smaller, and we know by approximately how much. The value of e for carbon would be expected to be similar to that for neon. The polarizabilities of atoms decrease somewhat as we go to the right in the periodic table. The number of electrons may increase, but they are more tightly held. But we also know that there is a big difference in polarizability between the first-row elements and second-row elements. To a first approximation, the first-row elements are similar. On the other hand, one expects a sizable difference between hydrogen and helium. 

There is a real problem here in the following way. We can measure the van der Waals properties of a molecule such as methane, for example. But we cannot individually measure the van der Waals properties for the hydrogen atoms and for the carbon atom in methane. And the numbers that we would like to use will vary somewhat, depending on the exact function chosen for the van der Waals interaction, and on the other approximations that have been previously outlined. And from this point, different workers managed to deduce rather different values for van der Waals functions, as discussed earlier. 

Small metal clusters are also of interest because of their importance in catalysis. Despite the fact that small clusters should consist of mostly surface atoms, measurement of the photon ionization threshold for Hg clusters suggest that a transition from van der Waals to metallic properties occurs in the range of 20-70 atoms per cluster [88] and near-bulk magnetic properties are expected for Ni, Pd, and Pt clusters of only 13 atoms [89] Theoretical calculations on Sin and other semiconductors predict that the stmcture reflects the bulk lattice for 1000 atoms but the bulk electronic wave functions are not obtained [90]. Bartell and co-workers [91] study beams of molecular clusters with electron dirfraction and molecular dynamics simulations and find new phases not observed in the bulk. Bulk models appear to be valid for their clusters of several thousand atoms (see Section IX-3). 

This simple model is adequate for some properties of rare gas fluids. When it is combined with an accurate description of the electrostatic interactions, it can rationalize the structures of a large variety of van der Waals... 

Hutson J M and Howard B J 1980 Spectroscopic properties and potential surfaces for atom-diatom van der Waals molecules Mol. Phys. 41 1123... 

Buckingham A D, Fowler P W and Stone A J 1986 Electrostatic predictions of shapes and properties of van der Waals molecules Int. Rev. Phys. Chem. 5 107... 

This is the well known equal areas mle derived by Maxwell [3], who enthusiastically publicized van der Waal s equation (see figure A2.3.3. The critical exponents for van der Waals equation are typical mean-field exponents a 0, p = 1/2, y = 1 and 8 = 3. This follows from the assumption, connnon to van der Waals equation and other mean-field theories, that the critical point is an analytic point about which the free energy and other themiodynamic properties can be expanded in a Taylor series. 

Van der Waals complexes can be observed spectroscopically by a variety of different teclmiques, including microwave, infrared and ultraviolet/visible spectroscopy. Their existence is perhaps the simplest and most direct demonstration that there are attractive forces between stable molecules. Indeed the spectroscopic properties of Van der Waals complexes provide one of the most detailed sources of infonnation available on intennolecular forces, especially in the region around the potential minimum. The measured rotational constants of Van der Waals complexes provide infonnation on intennolecular distances and orientations, and the frequencies of bending and stretching vibrations provide infonnation on how easily the complex can be distorted from its equilibrium confonnation. In favourable cases, the whole of the potential well can be mapped out from spectroscopic data. 

Atomistically detailed models account for all atoms. The force field contains additive contributions specified in tenns of bond lengtlis, bond angles, torsional angles and possible crosstenns. It also includes non-bonded contributions as tire sum of van der Waals interactions, often described by Lennard-Jones potentials, and Coulomb interactions. Atomistic simulations are successfully used to predict tire transport properties of small molecules in glassy polymers, to calculate elastic moduli and to study plastic defonnation and local motion in quasi-static simulations [fy7, ( ]. The atomistic models are also useful to interiDret scattering data [fyl] and NMR measurements [70] in tenns of local order. 

The Hamaker constant can be evaluated accurately using tire continuum tlieory, developed by Lifshitz and coworkers [40]. A key property in tliis tlieory is tire frequency dependence of tire dielectric pennittivity, (cij). If tills spectmm were tlie same for particles and solvent, then A = 0. Since tlie refractive index n is also related to f (to), tlie van der Waals forces tend to be very weak when tlie particles and solvent have similar refractive indices. A few examples of values for A for interactions across vacuum and across water, obtained using tlie continuum tlieory, are given in table C2.6.3. 

Molecular volumes are usually computed by a nonquantum mechanical method, which integrates the area inside a van der Waals or Connolly surface of some sort. Alternatively, molecular volume can be determined by choosing an isosurface of the electron density and determining the volume inside of that surface. Thus, one could find the isosurface that contains a certain percentage of the electron density. These properties are important due to their relationship to certain applications, such as determining whether a molecule will fit in the active site of an enzyme, predicting liquid densities, and determining the cavity size for solvation calculations. 

Once the molecules are aligned, a molecular field is computed on a grid of points in space around the molecule. This field must provide a description of how each molecule will tend to bind in the active site. Field descriptors typically consist of a sum of one or more spatial properties, such as steric factors, van der Waals parameters, or the electrostatic potential. The choice of grid points will also affect the quality of the final results. 

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