Intermolecular separation

In a similar manner, the diffusion of hexane into dichloromethane solutions containing mixtures of the alkylammonium salts of bromide and the olefinic acceptors o-CA and TCNE result in the formation of brown-red crystals [23]. X-ray analysis reveals the (1 1) complex of bromide with o-CA, in which the anion is located over the center of the C - C bond of the acceptor moiety (Fig. 15b) and Br - C contacts are shortened by as much as 0.6 A relative to the sum of van der Waals radii (Table 3). In bromide complexes with TCNE, the location of the anion relative to the acceptor is variable. In fact, a 2 1 complex [(Br )2,TCNE] is isolated in which both anions reside over the olefinic bond when the tetraethylammonium salt of bromide is used. In comparison, if the tetrapropyl- or tetrabutylammonium salts of the same anion are employed, the (1 1) complexes [Br ,TCNE] are formed in which the bromide donors are shifted toward the cyano substituents (Fig. 15a). In both cases however, the short intermolecular separations that are characteris-... 

As we have already pointed out, the theoretical basis of free energy calculations were laid a long time ago [1,4,5], but, quite understandably, had to wait for sufficient computational capabilities to be applied to molecular systems of interest to the chemist, the physicist, and the biologist. In the meantime, these calculations were the domain of analytical theories. The most useful in practice were perturbation theories of dense liquids. In the Barker-Henderson theory [13], the reference state was chosen to be a hard-sphere fluid. The subsequent Weeks-Chandler-Andersen theory [14] differed from the Barker-Henderson approach by dividing the intermolecular potential such that its unperturbed and perturbed parts were associated with repulsive and attractive forces, respectively. This division yields slower variation of the perturbation term with intermolecular separation and, consequently, faster convergence of the perturbation series than the division employed by Barker and Henderson. 

Indicated by a thermal de Broglie wavelength which is a substantial fraction of the average intermolecular separation. 

Developed into a power series in R 1, where R is the intermolecular separation, H exhibits the dipole-dipole, dipole-quadrupole terms in increasing order. When nonvanishing, the dipole-dipole term is the most important, leading to the Forster process. When the dipole transition is forbidden, higher-order transitions come into play (Dexter, 1953). For the Forster process, H is well known, but 0. and 0, are still not known accurately enough to make an a priori calculation with Eq. (4.2). Instead, Forster (1947) makes a simplification based on the relative slowness of the transfer process. Under this condition, energy is transferred between molecules that are thermally equilibriated. The transfer rate then contains the same combination of Franck-Condon factors and vibrational distribution as are involved in the vibrionic transitions for the emission of the donor and the adsorptions of the acceptor. Forster (1947) thus obtains... 

Intermolecular vibration involving nearest-neighbor molecules are easily excited in liquid hydrocarbons, since the de Broglie wavelength of the electron at a few tenths of electron-volt energy is comparable to the intermolecular separation. The quantum for this vibration lies in the far IR and can be observed indirectly by Raman spectra. Raman shifts in many hydrocarbon liquids have... 

This expression can be generalized in the presence of an external field, and the ratio of the escape probability as a negative ion to that as an electron in the absence of a scavenger computed as a function of the external field. From such an analysis and taking L = 4 A, a typical intermolecular separation, Mozumder and Tachiya obtained electron attachment cross sections in NP as 4 x 10-16,5 X 10 17, and 1 X 1CF18 cm2, respectively, for SF6, CC14, and CS2 with -15% uncertainty. 

The foregoing treatment can be extended to cases where the electron-ion recombination is only partially diffusion-controlled and where the electron scattering mean free path is greater than the intermolecular separation. Both modifications are necessary when the electron mobility is - 100 cm2v is-1 or greater (Mozumder, 1990). It has been shown that the complicated random trajectory of a diffusing particle with a finite mean free path can have a simple representation in fractal diffusivity (Takayasu, 1982). In practice, this means the diffusion coefficient becomes distance-dependent of the form... 

According to Eyring (Moore and Eyring, 1938) and Joly (1956), Newtonian flow in a monolayer is the result of a cohesive attraction between surfactant molecules. For every molecule that flows from higher to lower surface pressure in a motion parallel to the canal walls, there is another molecule ready to fill the hole vacated by the first. The mechanism for this cohesive flow is presumably attractive van der Waals interactions between hydrocarbon chains. This model assumes that the average intermolecular separation in a surface-continuous monolayer does not exceed the cross-sectional area of the molecule as defined by the average molecular area A of the film at the surface pressure n in the pressurized compartment of the viscometer. 

London introduced a second approximation by expanding Vinl as a multipole power series in 1/7 (inverse intermolecular separation)... 

Figure 5.50 The optimized structure (a) and leading 7i

As a direct consequence of the large intermolecular separations, we can safely say no interactions form between the molecules in ammonia gas. The molecules are simply too far apart. We saw in the previous chapter how the property known as pressure is a macroscopic manifestation of the microscopic collisions occurring between gas particles and, say, a solid object such as a container s walls. But the gas particles can also strike each other on the same microscopic scale we say the resultant interactions between molecules are intermolecular. 

Structure analysis has shifted completely from intramolecular to intermolecular structure. Distributions of intermolecular distances can be statistically analyzed over hundreds of thousands of reliable data these distributions should be properly normalized to be statistically significant. The chemical interpretation must, however, take into account the unavoidable fact that intermolecular separations are a combination of steric and electronic effects, and that near to does not always mean bound to . 

The "force of interaction , F, between two spherical non-polar molecules is a function of the intermolecular separation , r. For most purposes, however, it is more convenient to use the "potential energy of interaction , 0(r), rather than the force of interaction F(r). These two functions are simply related ... 

These relations are valid only for force laws and potential functions which are functions of die intermolecular separation alone. For an angular dependent potential, the force on molecule is different and, in addition, there is a torque tending to rotate the molecule (Ref 8e, p 22)... 

In this section we have examined the three major contributions to what is generally called the van der Waals attraction between molecules. All three originate in dipole-dipole interactions of one sort or another. There are two consequences of this (a) all show the same functional dependence on the intermolecular separation, and (b) all depend on the same family of molecular parameters, especially dipole moment and polarizability, which are fairly readily available for many simple substances. Many of the materials we encounter in colloid science are not simple, however. Hence we must be on the lookout for other measurable quantities that depend on van der Waals interactions. Example 10.2 introduces one such possibility. We see in Section 10.7 that some other difficulties arise with condensed systems that do not apply to gases. 

The average intermolecular separation depends on the concentration of the solution. The critical concentration of the acceptor A0, is related to R0 by the expression... 

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