Hard-sphere radius

FIGURE 1.22. Solvent reorganization energies derived from the standard rate constants of the electrochemical reduction of aromatic hydrocarbons in DMF (with n-Bu4N+ as the cation of the supporting electrolyte) uncorrected from double-layer effects. Variation with the equivalent hard-sphere radii. Dotted line, Hush s prediction. Adapted from Figure 4 in reference 13, with permission from the American Chemical Society. 

In the classical formalism it is assumed that bimolecular electron transfer occurs in a precursor complex in which the inner-coordination shells of the reactants are in contact, that is, r - , where a2 and a3 are the hard-sphere radii of the reactants (16). Under these conditions the... 

In the homogeneous case, Aq is given by (7), where is the electron charge, Dop and Dg the optical and static dielectric constants of the solvent respectively, and a I and 2 e equivalent hard sphere radii of the two reactants (and products). For the electrochemical case, there are two versions for the expression of A., . In Marcus s treatment (Marcus, 1965) the reaction site is assumed to be located at a distance from the electrode equal to its radius, a, and the effect of image forces in the electrode is taken into account (8). In Hush s treatment (Hush, 1961) the reaction site is assumed to be located farther from the electrode surface and the effect of image forces is neglected (9). 

It is interesting to note that the effective hard-sphere radii of the adsorbed alkalis are only 5% less than the bcc metallic values. By contrast, as listed in Table 4, the corresponding hard-sphere radii of alkali atoms adsorbed in on-top sites in the ( 3 x y3)R30° phases formed by adsorption at low temperature are 20% less than the bcc metallic values. It can also be seen from Table 5 that the adsorption leads to a small, 2-3% contraction of the first interlayer spacing in the substrate. Somewhat surprisingly, the enhanced vibrations of first layer Al atoms in the clean Al(l 11) surface are not reduced by the adsorption of the alkalis, except for K. 

The surface geometries of the substitutional Al(lll)—( 3 x y 3)R30°—Li, Na, K, and Rb phases. The second column of the table lists the effective hard-sphere radii r (A) of the adsorbed alkali atoms, and the third column lists the corresponding values r cc for the bcc metallic phases. For these systems, relaxations of A1 atoms parallel to the surface are found to be less than the estimated uncertainties. Thus the geometry can be specified in terms of the interlayer spacings dij (A) between the j th and y th layers, and the RMS vibrational amplitudes m,- (A), doi is the distance between the alkali layer and the first A1 layer, and o is the vibrational amplitude of an adsorbed alkali atom. The final column lists the R factor values from the LEED analyses. 

In summary, collision theory provides a good physical picture of bimolecular reactions, even though the structure of the molecules is not taken into account. Also, it is assumed that reaction takes place instantaneously in practice, the reaction itself requires a certain amount of time. The structure of the reaction complex must evolve, and this must be accounted for in a reaction rate theory. For some reactions, the rate coefficient actually decreases with increasing temperature, a phenomenon that collision theory does not describe. Finally, real molecules interact with each other over distances greater than the sum of their hard-sphere radii, and in many cases these interactions can be very important. For example, ions can react via long-range Coulomb forces at a rate that exceeds the collision limit. The next level of complexity is transition state theory. 

According to Equation 6.3, this factor is equivalent to the Arrhenius A-factor. In the collision model it is a measure of the standard rate at which reactant species collide that is it is a measure of the number of collisions per second when the concentrations of the reactant species are both 1 mol dm"-. It is necessary to specify standard conditions since, in general, the collision rate depends on the concentrations of the species present (cf. Section 4.1). The value of Atheory a given bimolecular reaction depends on the hard-sphere radii and masses of the reactant species. Calculations show that it does not vary significantly from reaction to reaction with values usually of the order of 10 dm mol s . Table 7.1 compares the calculated values of Atheory for gas-phase bimolecular reactions with those derived from experiment. 

NAin the test configuration. Cx are the standard deviations for parameter type X Ij and Lj are the measured and target jth bond lengths, respectively. Similar definitions are given for bond angle roj and Qj, dihedrals 9j and roj, coordinates xj and Xj, and the distance djk between non-bonded atoms j and k with the atomic hard-sphere radii ri and ij, respectively (Hellinga Richards, 199 1). 

The NAST [16, 34] model represents each nucleotide by one pseudoatom at the C3 atom of the ribose group. NAST utilizes MD simulations and a force field parameterized from solved rRNA structures. NAST relies upon information from an accurate secondary structure and can also include experimental constraints. These constraints are modeled by a harmonic energy term. The bonded energy terms of distance, angle, and dihedral are further modeled by a harmonic potential, parameterized according to a Boltzmann inversion. Non-bonded interactions are modeled by a Lennard-Jones potential with a hard sphere radii of 5 A. Due to the low-resolution representation of one pseudoatom per nt, the conversion from the CG model to the all-atom model is complex and may produce steric overlaps. In order to overcome this difficulty, Jonikas et al. developed a program C2A [35] which is able to insert and minimize the all atom structure. 

Ci2E5-alkane-H20 systems. The apparent hydrodynamic radii are shown in 8(a) and the hard sphere radii in 8(b). The figures are adapted and data taken from ref. [118]. 

Fig. 12 Absolute truncation error of the reduced potential energy from the Ewald summation as a function of a at Rent = 13.0i M and cut = 14 using an equilibrium (filled spheres) and random (open spheres) configiuation of System IV. For a > 0.38J j, the symbols for the equilibrium and random configurations are indistinguishable. Nm = 80 and L = 32.224J m- The hard-sphere radii were retained when generating the random configuration. The corresponding estimated truncation errors of the real-space and reciprocal-space terms of the reduced potential energy according to Eqs. 21 and 22, respectively, are also shown (solid curves)...

One type of measure of an atomic size are van der Waals radii, van der Waals radii are strictly hard sphere radii measured using atomic distances in closest packed crystals. 

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